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 Satz6p3:
  p out a,b iff (a <> p & b <> p &
  (ex c st c <> p & between a,p,c & between b,p,c))
  proof
    thus p out a,b implies (a <> p & b <> p &
    (ex c st (c <> p & between a,p,c & between b,p,c)))
    proof
      assume
A1:   p out a,b;
      consider c be POINT of S such that
A2:   between a,p,c & p,c equiv p,a by GTARSKI1:def 8;
A3:   p <> c by A1,A2,Satz2p2,GTARSKI1:def 7;
A4:   between p,a,b implies (ex c st (c <> p & between a,p,c & between b,p,c))
      proof
        assume between p,a,b;
        between b,p,c by A1,A2,A3,Satz6p2;
        hence thesis by A2,A3;
      end;
      between p,b,a implies (ex c st (c <> p & between a,p,c & between b,p,c))
      proof
        assume between p,b,a;
        between b,p,c by A1,A2,A3,Satz6p2;
        hence thesis by A2,A3;
      end;
      hence thesis by A1,A4;
    end;
    thus a <> p & b <> p & (ex c st (c <> p & between a,p,c & between b,p,c))
      implies p out a,b by Satz6p2;
  end;
