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;

theorem Satz3p13:
  for S being satisfying_CongruenceIdentity
              satisfying_SegmentConstruction
              satisfying_Lower_Dimension_Axiom
              TarskiGeometryStruct holds
  ex a,b,c being POINT of S st not between a,b,c &
  not between b,c,a & not between c,a,b & a <> b & b <> c & c <> a
  proof
    let S be satisfying_CongruenceIdentity
             satisfying_SegmentConstruction
             satisfying_Lower_Dimension_Axiom
             TarskiGeometryStruct;
    consider a,b,c be POINT of S such that
A1: not between a,b,c & not between b,c,a & not between c,a,b
      by GTARSKI2:def 7;
    take a,b,c;
    thus thesis by A1,Satz3p1;
  end;
