reserve            S for satisfying_CongruenceSymmetry
                         satisfying_CongruenceEquivalenceRelation
                         TarskiGeometryStruct,
         a,b,c,d,e,f for POINT of S;
reserve S for satisfying_CongruenceSymmetry
              satisfying_CongruenceEquivalenceRelation
              satisfying_CongruenceIdentity
              satisfying_SegmentConstruction
              satisfying_SAS
              TarskiGeometryStruct,
        q,a,b,c,a9,b9,c9,x1,x2 for POINT of S;
reserve S for satisfying_CongruenceIdentity
              satisfying_SegmentConstruction
              satisfying_BetweennessIdentity
              satisfying_Pasch
              TarskiGeometryStruct,
        a,b,c,d for POINT of S;
reserve       S for satisfying_Tarski-model TarskiGeometryStruct,
        a,b,c,d for POINT of S;
reserve         S for satisfying_CongruenceIdentity
                      satisfying_SegmentConstruction
                      satisfying_BetweennessIdentity
                      satisfying_Pasch
                      TarskiGeometryStruct,
        a,b,c,d,e for POINT of S;
reserve       S for satisfying_Tarski-model
                    TarskiGeometryStruct,
      a,b,c,d,p for POINT of S;
reserve                   S for satisfying_Tarski-model TarskiGeometryStruct,
        a,b,c,d,a9,b9,c9,d9 for POINT of S;
reserve S for satisfying_Tarski-model
              TarskiGeometryStruct,
        a,b,c,d,a9,b9,c9,d9,p,q for POINT of S;
reserve                       S for satisfying_Tarski-model
                                    TarskiGeometryStruct,
        a,b,c,d,e,f,a9,b9,c9,d9 for POINT of S;
reserve p for POINT of S;
reserve r for POINT of S;
reserve x,y for POINT of S;
reserve S for non empty satisfying_Tarski-model TarskiGeometryStruct;
reserve p,q,r,s for POINT of S;
reserve S for non empty satisfying_CongruenceIdentity
              satisfying_SegmentConstruction
              satisfying_BetweennessIdentity
              satisfying_Pasch
              TarskiGeometryStruct,
  a,b,p,q for POINT of S;
reserve S for non empty satisfying_Tarski-model TarskiGeometryStruct,
                  A,B for Subset of S,
        a,b,c,p,q,r,s for POINT of S;
reserve S for satisfying_CongruenceIdentity
              satisfying_CongruenceSymmetry
              satisfying_CongruenceEquivalenceRelation
              satisfying_SegmentConstruction
              satisfying_BetweennessIdentity
              satisfying_Pasch
              TarskiGeometryStruct,
        a,b,m for POINT of S;
reserve S for satisfying_CongruenceIdentity
              satisfying_CongruenceSymmetry
              satisfying_CongruenceEquivalenceRelation
              satisfying_SegmentConstruction
              satisfying_BetweennessIdentity
              TarskiGeometryStruct,
        a,b,m for POINT of S;
reserve S for satisfying_CongruenceIdentity
              satisfying_CongruenceSymmetry
              satisfying_CongruenceEquivalenceRelation
              satisfying_SegmentConstruction
              satisfying_SAS
              TarskiGeometryStruct,
        a for POINT of S;
reserve S for satisfying_CongruenceIdentity
              satisfying_CongruenceSymmetry
              satisfying_CongruenceEquivalenceRelation
              satisfying_SegmentConstruction
              satisfying_BetweennessIdentity
              satisfying_SAS
              TarskiGeometryStruct,
  a,p,p9 for POINT of S;
reserve S for satisfying_CongruenceIdentity
              satisfying_CongruenceSymmetry
              satisfying_CongruenceEquivalenceRelation
              satisfying_SegmentConstruction
              satisfying_BetweennessIdentity
              satisfying_SAS
              satisfying_Pasch
              TarskiGeometryStruct,
  a,p,p9 for POINT of S;
reserve S for satisfying_CongruenceIdentity
                satisfying_CongruenceSymmetry
                satisfying_CongruenceEquivalenceRelation
                satisfying_SegmentConstruction
                satisfying_BetweennessIdentity
                satisfying_SAS
                TarskiGeometryStruct,
        a,p for POINT of S;
reserve              S for satisfying_Tarski-model TarskiGeometryStruct,
        a,b,c,d,m,p,p9,q,r,s for POINT of S;

theorem Satz7p13:
  p,q equiv reflection(a,p),reflection(a,q)
  proof
    reconsider p9 = reflection(a,p),
               q9 = reflection(a,q) as POINT of S;
    per cases;
    suppose
A1:   p = a;
      Middle q,a,reflection(a,q) by DEFR;
      hence thesis by A1,Satz7p10;
    end;
    suppose
A2:   p <> a;
A3:   Middle p,a,p9 by DEFR;
A4:   Middle q,a,q9 by DEFR;
      consider x be POINT of S such that
A5:   between p9,p,x and
A6:   p,x equiv q,a by GTARSKI1:def 8;
      consider y be POINT of S such that
A7:   between q9,q,y and
A8:   q,y equiv p,a by GTARSKI1:def 8;
      consider x9 be POINT of S such that
A9:   between x,p9,x9 and
A10:  p9,x9 equiv q,a by GTARSKI1:def 8;
      consider y9 be POINT of S such that
A11:  between y,q9,y9 and
A12:  q9,y9 equiv p,a by GTARSKI1:def 8;
A13:  between5 x,p,a,p9,x9 & between5 y,q,a,q9,y9
      proof
A14:    between x9,p9,x by A9,Satz3p2;
        between p9,a,p by A3,Satz3p2;
        then between5 x9,p9,a,p,x by A5,A14,Satz3p11p3pb,Satz3p11p4pb;
        hence between5 x,p,a,p9,x9 by Satz3p2;
A15:    between y9,q9,y by A11,Satz3p2;
        between q9,a,q by A4,Satz3p2;
        then between5 y9,q9,a,q,y by A7,A15,Satz3p11p3pb,Satz3p11p4pb;
        hence between5 y,q,a,q9,y9 by Satz3p2;
      end;
A16:  a,x equiv a,y
      proof
A17:    p,a equiv q,y by A8,Satz2p2;
        x,p equiv q,a by A6,Satz2p4; then
A18:    x,p equiv a,q by Satz2p5;
        between x,p,a & between a,q,y by A13,Satz3p2;
        then x,a equiv a,y by A17,A18,Satz2p11;
        hence thesis by Satz2p4;
      end;
A19:  a,y equiv a,x9
      proof
        q,a equiv p9,x9 by A10,Satz2p2;
        then a,q equiv p9,x9 by Satz2p4; then
A20:    a,q equiv x9,p9 by Satz2p5;
        q,y equiv a,p by A8,Satz2p5;
        then q,y equiv a,p9 by A3,Satz2p3; then
A21:    q,y equiv p9,a by Satz2p5;
        between a,q,y & between x9,p9,a by A13,Satz3p2;
        then a,y equiv x9,a by A20,A21,Satz2p11;
        hence thesis by Satz2p5;
      end;
A22:  a,x9 equiv a,y9
      proof
        q,a equiv a,q9 by A4,Satz2p4;
        then p9,x9 equiv a,q9 by A10,Satz2p3; then
A23:    x9,p9 equiv a,q9 by Satz2p4;
        p,a equiv a,p9 by A3,Satz2p4;
        then q9,y9 equiv a,p9 by A12,Satz2p3;
        then a,p9 equiv q9,y9 by Satz2p2; then
A24:    p9,a equiv q9,y9 by Satz2p4;
        between x9,p9,a & between a,q9,y9 by A13,Satz3p2;
        then x9,a equiv a,y9 by A23,A24,Satz2p11;
        hence thesis by Satz2p4;
      end;
A25:  x,a,x9,y9 AFS y9,a,y,x
      proof
        now
          thus between x,a,x9 by A13;
          thus between y9,a,y by A13,Satz3p2;
          a,x equiv a,x9 by A16,A19,Satz2p3; then
A26:      a,x equiv a,y9 by A22,Satz2p3;
          hence a,y9 equiv a,x by Satz2p2;
          a,x equiv y9,a by A26,Satz2p5;
          hence x,a equiv y9,a by Satz2p4;
          thus a,x9 equiv a,y by A19,Satz2p2;
          thus x,y9 equiv y9,x by Satz2p1,Satz2p5;
        end;
        hence thesis;
      end;
A27:  S is satisfying_SST_A5;
      x <> a by A13,A2,GTARSKI1:def 10; then
A28:  x9,y9 equiv y,x by A27,A25;
A29:  y,q,a,x IFS y9,q9,a,x9
      proof
        now
          thus between y,q,a & between y9,q9,a by A13,Satz3p2;
          a,y equiv a,y9 by A19,A22,Satz2p3;
          then a,y equiv y9,a by Satz2p5;
          hence y,a equiv y9,a by Satz2p4;
          a,q equiv q9,a by A4,Satz2p5;
          hence q,a equiv q9,a by Satz2p4;
          y,x equiv x9,y9 by A28,Satz2p2;
          hence y,x equiv y9,x9 by Satz2p5;
          thus a,x equiv a,x9 by A16,A19,Satz2p3;
        end;
        hence thesis;
      end;
      x,p,a,q IFS x9,p9,a,q9
      proof
        now
          thus between x,p,a & between x9,p9,a by A13,Satz3p2;
          a,x equiv a,x9 by A16,A19,Satz2p3;
          then a,x equiv x9,a by Satz2p5;
          hence x,a equiv x9,a by Satz2p4;
          a,p equiv p9,a by A3,Satz2p5;
          hence p,a equiv p9,a by Satz2p4;
          q,x equiv x9,q9 by A29,Satz4p2,Satz2p5;
          hence x,q equiv x9,q9 by Satz2p4;
          thus a,q equiv a,q9 by A4;
        end;
        hence thesis;
      end;
      hence thesis by Satz4p2;
    end;
  end;
