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;

theorem
  r <> a & b <> c implies ex x being POINT of S st (a out x,r & a,x equiv b,c)
  proof
    assume
A1: r <> a & b <> c;
    consider x be POINT of S such that
A2: (between a,r,x or between a,x,r) & a,x equiv b,c by Lemmapsegcon2;
    a out x,r by A1,A2,Satz2p2,GTARSKI1:def 7;
    hence thesis by A2;
  end;
