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;

theorem Lemmapsegcon2:
  ex x being POINT of S st (between p,a,x or between p,x,a) & p,x equiv b,c
  proof
    set q = p;
    consider r be POINT of S such that
A1: between a,q,r & q,r equiv a,q by GTARSKI1:def 8;
    consider x be POINT of S such that
A2: between r,q,x & q,x equiv b,c by GTARSKI1:def 8;
A3: r = q implies ((between q,a,x or between q,x,a) & q,x equiv b,c)
      by A1,A2,Satz2p2,GTARSKI1:def 7;
    r <> q implies ((between q,a,x or between q,x,a) & q,x equiv b,c)
    proof
      assume
A4:   r <> q;
      between r,q,a by A1,Satz3p2;
      hence thesis by A2,A4,Satz5p2;
    end;
    hence thesis by A3;
  end;
