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 Satz5p10:
  a,b <= c,d or c,d <= a,b
  proof
    consider x be POINT of S such that
A1: between b,a,x & a,x equiv c,d by GTARSKI1:def 8;
    consider y be POINT of S such that
A2: between x,a,y & a,y equiv c,d by GTARSKI1:def 8;
A3: x = a implies a,b <= c,d or c,d <= a,b
    proof
      assume x = a;
      then
A4:   c = d by A1,Satz2p2,GTARSKI1:def 7;
      ex q be POINT of S st between c,c,q & c,q equiv a,b by GTARSKI1:def 8;
      hence thesis by A4,Satz5p5;
    end;
    x <> a implies a,b <= c,d or c,d <= a,b
    proof
      assume
A5:   x <> a;
A6:   between x,a,b by A1,Satz3p2; then
A7:   between x,y,b or between x,b,y by A2,A5,Satz5p1;
      between x,y,b implies c,d <= a,b
      proof
        assume between x,y,b;
        then between a,y,b by A2,Satz3p6p1;
        hence thesis by A2,Satz2p2;
      end;
      hence thesis by A2,A6,A7,Satz3p6p1,Satz5p5;
    end;
    hence thesis by A3;
  end;
