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;

theorem Satz6p11pb:
  r <> a & b <> c & (a out x,r & a,x equiv b,c) & (a out y,r & a,y equiv b,c)
  implies x = y
  proof
    assume
A1:  r <> a & b <> c & (a out x,r & a,x equiv b,c) & (a out y,r &
       a,y equiv b,c);
    consider s be POINT of S such that
A2: s <> a & between x,a,s & between r,a,s by A1,Satz6p3;
    consider t be POINT of S such that
A3: t <> a & between y,a,t & between r,a,t by A1,Satz6p3;
A4: between a,s,t implies x = y
    proof
      assume between a,s,t;
      then between x,s,t & between x,a,t by A2,Satz3p7p1,Satz3p7p2;
      then between t,a,x & between t,a,y by A3,Satz3p2;
      hence thesis by A1,A3,Satz2p12;
    end;
    between a,t,s implies x = y
    proof
      assume between a,t,s;
      then between y,t,s & between y,a,s by A3,Satz3p7p1,Satz3p7p2;
      then between s,a,y & between s,a,x by A2,Satz3p2;
      hence thesis by A1,A2,Satz2p12;
    end;
    hence thesis by A4,A1,A2,A3,Satz5p2;
  end;
