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 Satz6p2:
  a <> p & b <> p & c <> p & between a,p,c implies
  (between b,p,c iff p out a,b)
  proof
    assume
A1: a <> p & b <> p & c <> p & between a,p,c;
    thus between b,p,c implies p out a,b
    proof
      assume between b,p,c;
      then between c,p,b & between c,p,a by A1,Satz3p2;
      hence thesis by A1,Satz5p2;
    end;
    thus p out a,b implies between b,p,c
    proof
      assume
A2:   p out a,b;
A3:   between p,a,b implies between b,p,c
      proof
        assume between p,a,b;
        then between b,a,p by Satz3p2;
        hence thesis by A1,Satz3p7p1;
      end;
      between p,b,a implies between b,p,c
      proof
        assume between p,b,a;
        then between a,b,p by Satz3p2;
        hence thesis by A1,Satz3p6p1;
      end;
      hence thesis by A2,A3;
    end;
  end;
