Fact.cs

        RecalculateTransform();
    }

    /// \copydoc Fact.parseFact(Scroll.ScrollFact)
    public new static OrthogonalCircleLineFact parseFact(Scroll.ScrollFact fact)
    {
        OMA tp = (OMA)((Scroll.ScrollSymbolFact)fact).tp;
        if (tp == null)
            return null;

        string uri = fact.@ref.uri;

        string CircleUri = ((OMS)((OMA)((OMA)((Scroll.ScrollSymbolFact)fact).tp).arguments[0]).arguments[0]).uri;
        string LineUri = ((OMS)((OMA)((OMA)((Scroll.ScrollSymbolFact)fact).tp).arguments[0]).arguments[1]).uri;

        if (StageStatic.stage.factState.ContainsKey(CircleUri)
         && StageStatic.stage.factState.ContainsKey(LineUri))

            return new OrthogonalCircleLineFact(CircleUri, LineUri, uri, StageStatic.stage.factState);

        else    //If dependent facts do not exist return null
            return null;
    }

    /// \copydoc Fact.generateLabel
    protected override string generateLabel()
        => Circle.Label + "⊥" + Ray.Label;

    /// <summary>
    /// Constructs struct for not-right-angled MMT %Fact <see cref="AddFactResponse"/>
    /// </summary>
    /// <param name="c1URI"> Uri for <see cref="Cid1"/></param>
    /// <param name="l1URI"> Uri for <see cref="Lid1"/></param>
    /// <returns>struct for <see cref="AddFactResponse"/></returns>
    private MMTDeclaration generateMMTDeclaration(string c1URI, string l1URI)
    {
        SOMDoc tp = new OMA(
            new OMS(MMT_OMS_URI.Ded),
            new List<SOMDoc>{
                new OMA(
                    new OMS(MMT_OMS_URI.OrthoCircleLine),
                    new List<SOMDoc>{
                        new OMS(c1URI),
                        new OMS(l1URI),
                    }
                ),
            }
        );

        SOMDoc df = null;

        return new MMTSymbolDeclaration(this.Label, tp, df);
    }

    /// \copydoc Fact.hasDependentFacts
    public override bool hasDependentFacts()
        => true;

    /// \copydoc Fact.getDependentFactIds
    public override string[] getDependentFactIds()
        => new string[] { Cid1, Lid1 };

    /// \copydoc Fact.instantiateDisplay(GameObject, Transform)
    public override GameObject instantiateDisplay(GameObject prefab, Transform transform)
    {
        var obj = GameObject.Instantiate(prefab, Vector3.zero, Quaternion.identity, transform);
        obj.transform.GetChild(0).gameObject.GetComponent<TextMeshProUGUI>().text = Circle.Label;
        obj.transform.GetChild(1).gameObject.GetComponent<TextMeshProUGUI>().text = Ray.Label;

        obj.GetComponent<FactWrapper>().fact = this;
        return obj;
    }

    /// \copydoc Fact.GetHashCode
    /// uhhh is this a problem?
    public override int GetHashCode()
        => this.Cid1.GetHashCode() ^ this.Lid1.GetHashCode();

    /// \copydoc Fact.Equivalent(Fact, Fact)
    protected override bool EquivalentWrapped(OrthogonalCircleLineFact f1, OrthogonalCircleLineFact f2)
        => DependentFactsEquivalent(f1, f2);

    protected override Fact _ReInitializeMe(Dictionary<string, string> old_to_new, FactOrganizer organizer)
        => new OrthogonalCircleLineFact(old_to_new[this.Cid1], old_to_new[this.Lid1], organizer);
}

/// <summary>
/// The volume of a cone A  defined by a base area  <see cref="CircleFact">CircleFact</see>, a top area <see cref="CircleFact">CircleFact</see> and the volume as float
/// </summary>
public class TruncatedConeVolumeFact : FactWrappedCRTP<TruncatedConeVolumeFact>
{
    ///  <summary> a <see cref="CircleFact">CircleFact</see> describing the base area </summary>
    public string Cid1;
    [JsonIgnore]
    public CircleFact Circle1 { get => (CircleFact)FactOrganizer.AllFacts[Cid1]; }

    ///  <summary> a <see cref="CircleFact">CircleFact</see> describing the top area  </summary>
    public string Cid2;
    [JsonIgnore]
    public CircleFact Circle2 { get => (CircleFact)FactOrganizer.AllFacts[Cid2]; }

    ///  <summary> the volume of Truncated the cone as a float </summary>
    public float vol;
    /// <summary> a proof that both circles have not the same size </summary>
    public string unequalCirclesProof;
    ///  <summary> OMA proof that the two circles are parallel  </summary>
    public OMA proof;


    /// <summary> \copydoc Fact.Fact </summary>
    public TruncatedConeVolumeFact() : base()
    {
        this.Cid1 = null;
        this.Cid2 = null;
        this.vol = 0.0f;
        this.unequalCirclesProof = null;
        this.proof = null;
    }

    /// <summary>
    /// Standard Constructor
    /// </summary>
    /// <param name="cid1">sets <see cref="Cid1"/></param>
    /// <param name="cid2">sets <see cref="Cid2"/></param>
    /// <param name="vol">sets <see cref="vol"/></param>
    /// <param name="proof">sets <see cref="proof"/></param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public TruncatedConeVolumeFact(string cid1, string cid2, float vol, string unequalproof, OMA proof, FactOrganizer organizer) : base(organizer)
        => init(cid1, cid2, vol, unequalproof, proof);

    /// <summary>
    /// sets variables and generates MMT Declaration
    /// </summary>
    /// <param name="cid1">sets <see cref="Cid1"/></param>
    /// <param name="cid2">sets <see cref="Cid2"/></param>
    /// <param name="vol">sets <see cref="vol"/></param>
    /// <param name="proof">sets <see cref="proof"/></param>
    private void init(string cid1, string cid2, float vol, string unequalproof, OMA proof)
    {
        this.Cid1 = cid1;
        this.Cid2 = cid2;
        this.proof = proof;
        this.unequalCirclesProof = unequalproof;
        this.vol = vol;

        RecalculateTransform();

        MMTDeclaration mmtDecl = generateMMTDeclaration(cid1, cid2, vol, unequalproof, proof);
        AddFactResponse.sendAdd(mmtDecl, out this._URI);
    }

    protected override void RecalculateTransform()
    {
        Position = Circle2.Position;
        Rotation = Circle2.Rotation;
    }

    /// <summary>
    /// Bypasses initialization of new MMT %Fact by using existend URI, _which is not checked for existence_.
    /// </summary>
    /// <param name="Cid1">sets <see cref="Cid1"/></param>
    /// <param name="Cid2">sets <see cref="Cid2"/></param>
    /// <param name="volume">sets <see cref="vol"/></param>
    /// <param name="proof">sets <see cref="proof"/></param>
    /// <param name="backendURI">MMT URI</param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public TruncatedConeVolumeFact(string Cid1, string Cid2, float volume, string unequalproof, OMA proof, string backendURI, FactOrganizer organizer) : base(organizer)
    {
        this.Cid1 = Cid1;
        this.Cid2 = Cid2;
        this.vol = volume;
        this.proof = proof;
        this.unequalCirclesProof = unequalproof;

        this._URI = backendURI;
        _ = this.Label;

        RecalculateTransform();
    }

    /// \copydoc Fact.parseFact(Scroll.ScrollFact)
    public new static TruncatedConeVolumeFact parseFact(Scroll.ScrollFact fact)
    {
        string uri = fact.@ref.uri;

        string Circle1Uri = ((OMS)((OMA)((OMA)((Scroll.ScrollValueFact)fact).lhs).arguments[0]).arguments[0]).uri;
        string Circle2Uri = ((OMS)((OMA)((OMA)((Scroll.ScrollValueFact)fact).lhs).arguments[0]).arguments[1]).uri;
        float volume = ((OMF)((Scroll.ScrollValueFact)fact).value).f;

        string UnEqualCirclesProof = ((OMS)(((OMA)((OMA)((Scroll.ScrollValueFact)fact).lhs).arguments[0]).arguments[2])).uri;
        OMA proof = (OMA)(((OMA)((OMA)((Scroll.ScrollValueFact)fact).lhs).arguments[0]).arguments[3]);


        if (StageStatic.stage.factState.ContainsKey(Circle1Uri) && StageStatic.stage.factState.ContainsKey(Circle2Uri))

            return new TruncatedConeVolumeFact(Circle1Uri, Circle2Uri, volume, UnEqualCirclesProof, proof, uri, StageStatic.stage.factState);

        else    //If dependent facts do not exist return null
            return null;
    }

    /// \copydoc Fact.generateLabel
    protected override string generateLabel()
        => "V(" + Circle1.Label + "," + Circle2.Label + ")";

    /// <summary>
    /// Constructs struct for not-right-angled MMT %Fact <see cref="AddFactResponse"/>
    /// </summary>
    /// <param name="c1URI"> Uri for <see cref="Cid1"/></param>
    /// <param name="c2URI"> Uri for <see cref="Cid2"/></param>
    /// <param name="val"> <see cref="vol"/></param>
    /// <returns>struct for <see cref="AddFactResponse"/></returns>
    private MMTDeclaration generateMMTDeclaration(string c1URI, string c2URI, float val, string unequalproof, OMA proof)
    {
        SOMDoc lhs =
            new OMA(
                new OMS(MMT_OMS_URI.TruncatedVolumeCone),

                new List<SOMDoc> {
                    new OMS(c1URI),
                    new OMS(c2URI),
                    new OMS(unequalproof),
                    proof,
                }
            );

        SOMDoc valueTp = new OMS(MMT_OMS_URI.RealLit);
        SOMDoc value = new OMF(val);

        return new MMTValueDeclaration(this.Label, lhs, valueTp, value);
    }

    /// \copydoc Fact.hasDependentFacts
    public override Boolean hasDependentFacts()
        => true;

    /// \copydoc Fact.getDependentFactIds
    public override string[] getDependentFactIds()
        => new string[] { Cid1, Cid2 };

    /// \copydoc Fact.instantiateDisplay(GameObject, Transform)
    public override GameObject instantiateDisplay(GameObject prefab, Transform transform)
    {
        var obj = GameObject.Instantiate(prefab, Vector3.zero, Quaternion.identity, transform);
        obj.transform.GetChild(0).gameObject.GetComponent<TextMeshProUGUI>().text = Circle1.Label + Circle2.Label;
        obj.GetComponent<FactWrapper>().fact = this;

        return obj;
    }

    /// \copydoc Fact.Equivalent(Fact, Fact)
    protected override bool EquivalentWrapped(TruncatedConeVolumeFact f1, TruncatedConeVolumeFact f2)
        => DependentFactsEquivalent(f1, f2);

    protected override Fact _ReInitializeMe(Dictionary<string, string> old_to_new, FactOrganizer organizer)
        => new TruncatedConeVolumeFact(old_to_new[this.Cid1], old_to_new[this.Cid2], this.vol, old_to_new[this.unequalCirclesProof], this.proof, organizer);
}

/// <summary>
/// The volume of a cylinder defined by a base area  <see cref="CircleFact">CircleFact</see>, a top area <see cref="CircleFact">CircleFact</see> and the volume as float
/// </summary>
public class CylinderVolumeFact : FactWrappedCRTP<CylinderVolumeFact>
{
    ///  <summary> a <see cref="CircleFact">CircleFact</see> describing the base area </summary>
    public string Cid1;
    [JsonIgnore]
    public CircleFact Circle1 { get => (CircleFact)FactOrganizer.AllFacts[Cid1]; }

    ///  <summary> a <see cref="CircleFact">CircleFact</see> describing the top area  </summary>
    public string Cid2;
    [JsonIgnore]
    public CircleFact Circle2 { get => (CircleFact)FactOrganizer.AllFacts[Cid2]; }

    ///  <summary> the volume of the cylinder as a float </summary>
    public float vol;
    /// <summary> a proof that both circles have the same size </summary>
    public string equalCirclesProof;
    ///  <summary> OMA proof that the two circles are parallel  </summary>
    public OMA proof;

    /// <summary> \copydoc Fact.Fact </summary>
    public CylinderVolumeFact() : base()
    {
        this.Cid1 = null;
        this.Cid2 = null;
        this.vol = 0.0f;
        this.proof = null;
        this.equalCirclesProof = null;
    }

    /// <summary>
    /// Standard Constructor
    /// </summary>
    /// <param name="cid1">sets <see cref="Cid1"/></param>
    /// <param name="cid2">sets <see cref="Cid2"/></param>
    /// <param name="vol">sets <see cref="vol"/></param>
    /// <param name="proof">sets <see cref="proof"/></param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public CylinderVolumeFact(string cid1, string cid2, float vol, string eqProof, OMA proof, FactOrganizer organizer) : base(organizer)
        => init(cid1, cid2, vol, eqProof, proof);

    /// <summary>
    /// sets variables and generates MMT Declaration
    /// </summary>
    /// <param name="cid1">sets <see cref="Cid1"/></param>
    /// <param name="cid2">sets <see cref="Cid2"/></param>
    /// <param name="vol">sets <see cref="vol"/></param>
    /// <param name="proof">sets <see cref="proof"/></param>
    private void init(string cid1, string cid2, float vol, string eqProof, OMA proof)
    {
        this.Cid1 = cid1;
        this.Cid2 = cid2;
        this.proof = proof;
        this.equalCirclesProof = eqProof;
        this.vol = vol;

        RecalculateTransform();

        MMTDeclaration mmtDecl = generateMMTDeclaration(cid1, cid2, vol, eqProof, proof);
        AddFactResponse.sendAdd(mmtDecl, out this._URI);
    }

    protected override void RecalculateTransform()
    {
        Position = Circle2.Position;
        Rotation = Circle2.Rotation;
    }

    /// <summary>
    /// Bypasses initialization of new MMT %Fact by using existend URI, _which is not checked for existence_.
    /// </summary>
    /// <param name="Cid1">sets <see cref="Cid1"/></param>
    /// <param name="Cid2">sets <see cref="Cid2"/></param>
    /// <param name="volume">sets <see cref="vol"/></param>
    /// <param name="proof">sets <see cref="proof"/></param>
    /// <param name="backendURI">MMT URI</param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public CylinderVolumeFact(string Cid1, string Cid2, float volume, string eqProof, OMA proof, string backendURI, FactOrganizer organizer) : base(organizer)
    {
        this.Cid1 = Cid1;
        this.Cid2 = Cid2;
        this.vol = volume;
        this.proof = proof;
        this.equalCirclesProof = eqProof;

        this._URI = backendURI;
        _ = this.Label;

        RecalculateTransform();
    }

    /// \copydoc Fact.parseFact(Scroll.ScrollFact)
    public new static CylinderVolumeFact parseFact(Scroll.ScrollFact fact)
    {
        string uri = fact.@ref.uri;

        string Circle1Uri = ((OMS)((OMA)((OMA)((Scroll.ScrollValueFact)fact).lhs).arguments[0]).arguments[0]).uri;
        string Circle2Uri = ((OMS)((OMA)((OMA)((Scroll.ScrollValueFact)fact).lhs).arguments[0]).arguments[1]).uri;
        float volume = ((OMF)((Scroll.ScrollValueFact)fact).value).f;
        string EqualCirclesProof = ((OMS)(((OMA)((OMA)((Scroll.ScrollValueFact)fact).lhs).arguments[0]).arguments[2])).uri;

        OMA proof = (OMA)(((OMA)((OMA)((Scroll.ScrollValueFact)fact).lhs).arguments[0]).arguments[3]);

        if (StageStatic.stage.factState.ContainsKey(Circle1Uri) && StageStatic.stage.factState.ContainsKey(Circle2Uri))

            return new CylinderVolumeFact(Circle1Uri, Circle2Uri, volume, EqualCirclesProof, proof, uri, StageStatic.stage.factState);

        else    //If dependent facts do not exist return null
            return null;
    }

    /// \copydoc Fact.generateLabel
    protected override string generateLabel()
        => "V(" + Circle1.Label + "," + Circle2.Label + ")";

    /// <summary>
    /// Constructs struct for not-right-angled MMT %Fact <see cref="AddFactResponse"/>
    /// </summary>
    /// <param name="c1URI"> Uri for <see cref="Cid1"/></param>
    /// <param name="c2URI"> Uri for <see cref="Cid2"/></param>
    /// <param name="val"> <see cref="vol"/></param>
    /// <returns>struct for <see cref="AddFactResponse"/></returns>
    private MMTDeclaration generateMMTDeclaration(string c1URI, string c2URI, float val, string p1URI, OMA proof)
    {
        SOMDoc lhs =
            new OMA(
                new OMS(MMT_OMS_URI.CylinderVolume),

                new List<SOMDoc> {
                    new OMS(c1URI),
                    new OMS(c2URI),
                    new OMS(p1URI),
                    proof,
                }
            );

        SOMDoc valueTp = new OMS(MMT_OMS_URI.RealLit);
        SOMDoc value = new OMF(val);

        return new MMTValueDeclaration(this.Label, lhs, valueTp, value);
    }

    /// \copydoc Fact.hasDependentFacts
    public override bool hasDependentFacts()
        => true;

    /// \copydoc Fact.getDependentFactIds
    public override string[] getDependentFactIds()
        => new string[] { Cid1, Cid2, equalCirclesProof };

    /// \copydoc Fact.instantiateDisplay(GameObject, Transform)
    public override GameObject instantiateDisplay(GameObject prefab, Transform transform)
    {
        var obj = GameObject.Instantiate(prefab, Vector3.zero, Quaternion.identity, transform);
        obj.transform.GetChild(0).gameObject.GetComponent<TextMeshProUGUI>().text = Circle1.Label + Circle2.Label;
        obj.GetComponent<FactWrapper>().fact = this;

        return obj;
    }

    /// \copydoc Fact.Equivalent(Fact, Fact)
    protected override bool EquivalentWrapped(CylinderVolumeFact f1, CylinderVolumeFact f2)
        => DependentFactsEquivalent(f1, f2);

    protected override Fact _ReInitializeMe(Dictionary<string, string> old_to_new, FactOrganizer organizer)
        => new CylinderVolumeFact(old_to_new[this.Cid1], old_to_new[this.Cid2], this.vol, old_to_new[this.equalCirclesProof], this.proof, organizer);
}

/// <summary>
/// A fact that describes, that two circles have the same size and is comprised of two <see cref="CircleFact">CircleFacts</see> 
/// </summary>
public class EqualCirclesFact : FactWrappedCRTP<EqualCirclesFact>
{
    /// @{ <summary>
    /// two circles that are meant to be equal in area
    /// </summary>
    public string Cid1, Cid2;
    /// @}

    [JsonIgnore]
    public CircleFact Circle1 { get => (CircleFact)FactOrganizer.AllFacts[Cid1]; }
    [JsonIgnore]
    public CircleFact Circle2 { get => (CircleFact)FactOrganizer.AllFacts[Cid2]; }

    /// <summary> \copydoc Fact.Fact </summary>
    public EqualCirclesFact() : base()
    {
        this.Cid1 = null;
        this.Cid2 = null;
    }

    /// <summary>
    /// Standard Constructor
    /// </summary>
    /// <param name="cid1">sets <see cref="Cid1"/></param>
    /// <param name="cid2">sets <see cref="Cid2"/></param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public EqualCirclesFact(string cid1, string cid2, FactOrganizer organizer) : base(organizer)
        => init(cid1, cid2);

    /// <summary>
    /// Initiates <see cref="Cid1"/>, <see cref="Cid2"/>, <see cref="Fact._URI"/> and creates MMT %Fact Server-Side
    /// </summary>
    /// <param name="cid1">sets <see cref="Cid1"/></param>
    /// <param name="cid2">sets <see cref="Cid2"/></param>
    private void init(string cid1, string cid2)
    {
        this.Cid1 = cid1;
        this.Cid2 = cid2;

        RecalculateTransform();

        MMTDeclaration mmtDecl = generateEqualCirclesFactDeclaration(cid1, cid2);
        AddFactResponse.sendAdd(mmtDecl, out this._URI);
    }

    protected override void RecalculateTransform()
    {
        Position = Circle1.Position;
        Rotation = Circle1.Rotation;
    }

    /// <summary>
    /// Bypasses initialization of new MMT %Fact by using existend URI, _which is not checked for existence_.
    /// </summary>
    /// <param name="Cid1">sets <see cref="Cid1"/></param>
    /// <param name="Cid2">sets <see cref="Cid2"/></param>
    /// <param name="backendURI">MMT URI</param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public EqualCirclesFact(string Cid1, string Cid2, string backendURI, FactOrganizer organizer) : base(organizer)
    {
        this.Cid1 = Cid1;
        this.Cid2 = Cid2;

        this._URI = backendURI;
        _ = this.Label;

        RecalculateTransform();
    }

    /// \copydoc Fact.parseFact(Scroll.ScrollFact)
    public new static EqualCirclesFact parseFact(Scroll.ScrollFact fact)
    {
        OMA tp = (OMA)((Scroll.ScrollSymbolFact)fact).tp;
        if (tp == null)
            return null;

        string circleAUri = "";
        string circleBUri = "";

        string uri = fact.@ref.uri;
        OMA proof_OMA = (OMA)((Scroll.ScrollSymbolFact)fact).tp; // proof DED

        OMA parallel_circles_OMA = (OMA)proof_OMA.arguments[0]; // parallel

        if (parallel_circles_OMA.arguments[0] is OMS)
        {
            // Normaler Fall 
            circleAUri = ((OMS)((OMA)((OMA)((Scroll.ScrollSymbolFact)fact).tp).arguments[0]).arguments[0]).uri;
            circleBUri = ((OMS)((OMA)((OMA)((Scroll.ScrollSymbolFact)fact).tp).arguments[0]).arguments[1]).uri;
        }

        if (StageStatic.stage.factState.ContainsKey(circleAUri)
         && StageStatic.stage.factState.ContainsKey(circleBUri))

            return new EqualCirclesFact(circleAUri, circleBUri, uri, StageStatic.stage.factState);

        else    //If dependent facts do not exist return null
            return null;
    }

    /// \copydoc Fact.generateLabel
    protected override string generateLabel()
        => Circle1.Label + " ≠ " + Circle2.Label;

    /// <summary>
    /// Constructs struct for equalCirclesFact <see cref="AddFactResponse"/>
    /// </summary>
    /// <param name="c1URI"><see cref="Cid1"/></param>
    /// <param name="c2URI"><see cref="Cid2"/></param>
    /// <returns>struct for <see cref="AddFactResponse"/></returns>
    private MMTDeclaration generateEqualCirclesFactDeclaration(string c1URI, string c2URI)
    {
        SOMDoc tp = new OMA(
            new OMS(MMT_OMS_URI.Ded),
            new List<SOMDoc> {
                new OMA(
                    new OMS(MMT_OMS_URI.EqualityCircles),
                    new List<SOMDoc> {
                        new OMS(c1URI),
                        new OMS(c2URI),
                    }
                )
            }
        );

        SOMDoc df = null;

        MMTSymbolDeclaration mmtDecl = new MMTSymbolDeclaration(this.Label, tp, df);
        AddFactResponse.sendAdd(mmtDecl, out this._URI);

        return mmtDecl;
    }

    /// \copydoc Fact.hasDependentFacts
    public override Boolean hasDependentFacts()
        => true;

    /// \copydoc Fact.getDependentFactIds
    public override string[] getDependentFactIds()
        => new string[] { Cid1, Cid2 };

    /// \copydoc Fact.instantiateDisplay(GameObject, Transform)
    public override GameObject instantiateDisplay(GameObject prefab, Transform transform)
    {
        var obj = GameObject.Instantiate(prefab, Vector3.zero, Quaternion.identity, transform);
        obj.transform.GetChild(0).gameObject.GetComponent<TextMeshProUGUI>().text = Circle1.Label;
        obj.transform.GetChild(1).gameObject.GetComponent<TextMeshProUGUI>().text = Circle2.Label;
        obj.GetComponent<FactWrapper>().fact = this;
        return obj;
    }

    /// \copydoc Fact.Equivalent(Fact, Fact)
    protected override bool EquivalentWrapped(EqualCirclesFact f1, EqualCirclesFact f2)
        => DependentFactsEquivalent(f1, f2);

    protected override Fact _ReInitializeMe(Dictionary<string, string> old_to_new, FactOrganizer organizer)
        => new EqualCirclesFact(old_to_new[this.Cid1], old_to_new[this.Cid2], organizer);
}

/// <summary>
/// A fact that describes, that two circles have not the same size and is comprised of two <see cref="CircleFact">CircleFacts</see> 
/// </summary>
public class UnEqualCirclesFact : FactWrappedCRTP<UnEqualCirclesFact>
{
    /// @{ <summary>
    /// two circles that are meant to be unequal in area
    /// </summary>
    public string Cid1, Cid2;
    /// @}

    [JsonIgnore]
    public CircleFact Circle1 { get => (CircleFact)FactOrganizer.AllFacts[Cid1]; }
    [JsonIgnore]
    public CircleFact Circle2 { get => (CircleFact)FactOrganizer.AllFacts[Cid2]; }

    /// <summary> \copydoc Fact.Fact </summary>
    public UnEqualCirclesFact() : base()
    {
        this.Cid1 = null;
        this.Cid2 = null;
    }

    /// <summary>
    /// Standard Constructor
    /// </summary>
    /// <param name="cid1">sets <see cref="Cid1"/></param>
    /// <param name="cid2">sets <see cref="Cid2"/></param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public UnEqualCirclesFact(string cid1, string cid2, FactOrganizer organizer) : base(organizer)
        => init(cid1, cid2);

    /// <summary>
    /// Initiates <see cref="Cid1"/>, <see cref="Cid2"/>, <see cref="Fact._URI"/> and creates MMT %Fact Server-Side
    /// </summary>
    /// <param name="cid1">sets <see cref="Cid1"/></param>
    /// <param name="cid2">sets <see cref="Cid2"/></param>
    private void init(string cid1, string cid2)
    {
        this.Cid1 = cid1;
        this.Cid2 = cid2;

        RecalculateTransform();

        MMTDeclaration mmtDecl = generateUnEqualCirclesFactDeclaration(cid1, cid2);
        AddFactResponse.sendAdd(mmtDecl, out this._URI);
    }

    protected override void RecalculateTransform()
    {
        Position = Circle2.Position;
        Rotation = Circle2.Rotation;
    }

    /// <summary>
    /// Bypasses initialization of new MMT %Fact by using existend URI, _which is not checked for existence_.
    /// </summary>
    /// <param name="Cid1">sets <see cref="Cid1"/></param>
    /// <param name="Cid2">sets <see cref="Cid2"/></param>
    /// <param name="backendURI">MMT URI</param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public UnEqualCirclesFact(string Cid1, string Cid2, string backendURI, FactOrganizer organizer) : base(organizer)
    {
        this.Cid1 = Cid1;
        this.Cid2 = Cid2;

        this._URI = backendURI;
        _ = this.Label;

        RecalculateTransform();
    }

    /// \copydoc Fact.parseFact(Scroll.ScrollFact)
    public new static UnEqualCirclesFact parseFact(Scroll.ScrollFact fact)
    {
        OMA tp = (OMA)((Scroll.ScrollSymbolFact)fact).tp;
        if (tp == null)
            return null;

        string circleAUri = "";
        string circleBUri = "";

        string uri = fact.@ref.uri;
        OMA proof_OMA = (OMA)((Scroll.ScrollSymbolFact)fact).tp; // proof DED

        OMA unequal_circles_OMA = (OMA)proof_OMA.arguments[0]; // unequal

        if (unequal_circles_OMA.arguments[0] is OMS)
        {
            // Normaler Fall 
            circleAUri = ((OMS)((OMA)((OMA)((Scroll.ScrollSymbolFact)fact).tp).arguments[0]).arguments[0]).uri;
            circleBUri = ((OMS)((OMA)((OMA)((Scroll.ScrollSymbolFact)fact).tp).arguments[0]).arguments[1]).uri;
        }

        if (StageStatic.stage.factState.ContainsKey(circleAUri)
         && StageStatic.stage.factState.ContainsKey(circleBUri))

            return new UnEqualCirclesFact(circleAUri, circleBUri, uri, StageStatic.stage.factState);

        else    //If dependent facts do not exist return null
            return null;
    }

    /// \copydoc Fact.generateLabel
    protected override string generateLabel()
        => Circle1.Label + " = " + Circle2.Label;

    /// <summary>
    /// Constructs struct for equalCirclesFact <see cref="AddFactResponse"/>
    /// </summary>
    /// <param name="c1URI"><see cref="Cid1"/></param>
    /// <param name="c2URI"><see cref="Cid2"/></param>
    /// <returns>struct for <see cref="AddFactResponse"/></returns>
    private MMTDeclaration generateUnEqualCirclesFactDeclaration(string c1URI, string c2URI)
    {
        SOMDoc tp = new OMA(
            new OMS(MMT_OMS_URI.Ded),
            new List<SOMDoc> {
                new OMA(new OMS(MMT_OMS_URI.UnEqualityCircles),
                new List<SOMDoc> {
                    new OMS(c1URI),
                    new OMS(c2URI),
        }),});

        SOMDoc df = null;

        MMTSymbolDeclaration mmtDecl = new(this.Label, tp, df);
        AddFactResponse.sendAdd(mmtDecl, out this._URI);
        return mmtDecl;
    }

    /// \copydoc Fact.hasDependentFacts
    public override Boolean hasDependentFacts()
        => true;

    /// \copydoc Fact.getDependentFactIds
    public override string[] getDependentFactIds()
        => new string[] { Cid1, Cid2 };

    /// \copydoc Fact.instantiateDisplay(GameObject, Transform)
    public override GameObject instantiateDisplay(GameObject prefab, Transform transform)
    {
        var obj = GameObject.Instantiate(prefab, Vector3.zero, Quaternion.identity, transform);
        obj.transform.GetChild(0).gameObject.GetComponent<TextMeshProUGUI>().text = Circle1.Label;
        obj.transform.GetChild(1).gameObject.GetComponent<TextMeshProUGUI>().text = Circle2.Label;
        obj.GetComponent<FactWrapper>().fact = this;
        return obj;
    }

    /// \copydoc Fact.Equivalent(Fact, Fact)
    protected override bool EquivalentWrapped(UnEqualCirclesFact f1, UnEqualCirclesFact f2)
        => DependentFactsEquivalent(f1, f2);

    protected override Fact _ReInitializeMe(Dictionary<string, string> old_to_new, FactOrganizer organizer)
        => new UnEqualCirclesFact(old_to_new[this.Cid1], old_to_new[this.Cid2], organizer);
}


#pragma warning disable // Testing...

/// TEST FACT
/// use this if you need to test certain implementations of facts.

/// <summary>
/// just for testing purposes  
/// </summary>
public class TestFact : FactWrappedCRTP<TestFact>
{
    /// <summary> \copydoc Fact.Fact </summary>
    public TestFact() : base()
    {

    }

    /// <summary>
    /// Standard Constructor
    /// </summary>
    /// <param name="pid1">sets <see cref="Pid1"/></param>
    /// <param name="pid2">sets <see cref="Pid2"/></param>
    /// <param name="radius">sets <see cref="radius"/></param>
    /// <param name="normal">sets <see cref="normal"/></param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public TestFact(FactOrganizer organizer) : base(organizer)
    {
        init();
    }

    /// <summary>
    /// Initiates <see cref="Pid1"/>, <see cref="Pid2"/>, <see cref="radius"/>,<see cref="dir1"/>,<see cref="dir2"/>, <see cref="Fact._URI"/> and creates MMT %Fact Server-Side
    /// </summary>
    /// <param name="pid1">sets <see cref="Pid1"/></param>
    /// <param name="pid2">sets <see cref="Pid2"/></param>
    /// <param name="radius">sets <see cref="radius"/></param>
    /// <param name="normal">sets <see cref="normal"/></param>
    private void init()
    {


        // mmtDecl = generateCircleFactDeclaration(p1URI, p2URI, radius, normal);

        //  AddFactResponse.sendAdd(mmtDecl, out this._URI);
    }

    protected override void RecalculateTransform() { }

    /// <summary>
    /// Bypasses initialization of new MMT %Fact by using existend URI, _which is not checked for existence_.
    /// </summary>
    /// <param name="Pid1">sets <see cref="Pid1"/></param>
    /// <param name="Pid2">sets <see cref="Pid2"/></param>
    /// <param name="radius">sets <see cref="radius"/></param>
    /// <param name="normal">sets <see cref="normal"/></param>
    /// <param name="backendURI">MMT URI</param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public TestFact(string backendURI, FactOrganizer organizer) : base(organizer)
    {

        this._URI = backendURI;
        _ = this.Label;
    }

    /// <summary>
    /// parses the Circlefact response of the MMT-Server
    /// </summary>
    /// \copydoc Fact.parseFact(Scroll.ScrollFact) 
    public new static TestFact parseFact(Scroll.ScrollFact fact)
    {
        string uri = fact.@ref.uri;
        Debug.Log("TestFact Uri:" + uri);
        return new TestFact(uri, StageStatic.stage.factState);

    }

    /// \copydoc Fact.generateLabel
    protected override string generateLabel()
    {

        return "test";
    }

    /// <summary>
    /// Constructs struct for right-angled MMT %Fact <see cref="AddFactResponse"/>
    /// </summary>
    /// <param name="p1URI"> <see cref="Pid1"/></param>
    /// <param name="p2URI"> <see cref="Pid2"/></param>
    /// <param name="radius"> <see cref="radius"/></param>
    /// <returns>struct for <see cref="AddFactResponse"/></returns>
    private MMTDeclaration generateCircleFactDeclaration(string p1URI, string p2URI, float radius, Vector3 normal)
    {
        PointFact p1 = FactOrganizer.AllFacts[p1URI] as PointFact;
        PointFact p2 = FactOrganizer.AllFacts[p2URI] as PointFact;


        List<SOMDoc> normalArgs = new List<SOMDoc>
        {
            new OMF(normal.x),
            new OMF(normal.y),
            new OMF(normal.z)
        };
        OMA NormalVector = new OMA(new OMS(MMT_OMS_URI.Tuple), normalArgs);



        List<SOMDoc> planeArgs = new List<SOMDoc>
        {
            new OMS(p2URI),
            NormalVector //n
        };

        OMA CirclePlane = new OMA(new OMS(MMT_OMS_URI.pointNormalPlane), planeArgs);
        OMS middlePoint = new OMS(p1URI);
        OMF Radius = new OMF(radius);

        List<SOMDoc> outerArguments = new List<SOMDoc>
        {
           CirclePlane,
           middlePoint,
           Radius
        };

        //OMS constructor generates full URI
        // Do i need this here? doubt 
        SOMDoc tp = new OMS(MMT_OMS_URI.CircleType3d);
        SOMDoc df = new OMA(new OMS(MMT_OMS_URI.MkCircle3d), outerArguments);

        MMTSymbolDeclaration mmtDecl = new MMTSymbolDeclaration(this.Label, tp, df);
        AddFactResponse.sendAdd(mmtDecl, out this._URI);


        return mmtDecl;
    }

    /// \copydoc Fact.hasDependentFacts
    public override Boolean hasDependentFacts()
    {
        return false;
    }

    /// \copydoc Fact.getDependentFactIds
    public override string[] getDependentFactIds()
    {
        return new string[] { };
    }

    /// \copydoc Fact.instantiateDisplay(GameObject, Transform)
    public override GameObject instantiateDisplay(GameObject prefab, Transform transform)
    {
        var obj = GameObject.Instantiate(prefab, Vector3.zero, Quaternion.identity, transform);

        // obj.transform.GetChild(1).gameObject.GetComponent<TextMeshProUGUI>().text = _Facts[this.Lid2].Label;

        obj.GetComponent<FactWrapper>().fact = this;
        return obj;
    }

    /// \copydoc Fact.GetHashCode
    public override int GetHashCode()
    {
        return base.GetHashCode();// this.Pid1.GetHashCode() ^ this.Pid2.GetHashCode();
    }

    /// \copydoc Fact.Equivalent(Fact, Fact)
    protected override bool EquivalentWrapped(TestFact f1, TestFact f2)
    {
        return DependentFactsEquivalent(f1, f2);
    }

    protected override Fact _ReInitializeMe(Dictionary<string, string> old_to_new, FactOrganizer organizer)
        => new TestFact(organizer);
}

#pragma warning restore // Testing over