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;

theorem Satz5p12:
  Collinear a,b,c implies (between a,b,c iff (a,b <= a,c & b,c <= a,c))
  proof
    assume
A1: Collinear a,b,c;
    thus between a,b,c implies a,b <= a,c & b,c <= a,c
    proof
      assume
A2:   between a,b,c;
      hence a,b <= a,c by Satz2p1;
      thus b,c <= a,c
      proof
        between c,b,a by A2,Satz3p2;
        then c,b <= c,a by Satz2p1;
        then b,c <= c,a by Satz2p4;
        hence thesis by Lemma5p12p2;
      end;
    end;
    thus a,b <= a,c & b,c <= a,c implies between a,b,c
    proof
      assume
A3:   a,b <= a,c & b,c <= a,c;
A4:   between c,a,b implies between a,b,c
      proof
        assume
A5:     between c,a,b;
        c,b <= a,c by A3,Satz2p4;
        then a = b by A5,Lemma5p12p2,Lemma5p12p4;
        hence thesis by Satz3p1,Satz3p2;
      end;
      between b,c,a implies between a,b,c
      proof
        assume between b,c,a;
        then b = c by A3,Satz3p2,Lemma5p12p4;
        hence thesis by Satz3p1;
      end;
      hence thesis by A1,A4;
    end;
  end;
