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
  a,b <= c,d & c,d <= e,f implies a,b <= e,f
  proof
    assume
A1: a,b <= c,d & c,d <= e,f;
    then consider x be POINT of S such that
A2: between a,b,x & a,x equiv c,d by Satz5p5;
    consider y be POINT of S such that
A3: between c,d,y & c,y equiv e,f by A1,Satz5p5;
    Collinear c,d,y by A3;
    then consider q be POINT of S such that
A4: c,d,y cong a,x,q by A2,Satz2p2,Satz4p14;
A5: between a,b,q
    proof
      between a,x,q by A3,A4,Satz4p6;
      hence thesis by A2,Satz3p6p2;
    end;
    a,q equiv e,f
    proof
      a,q equiv c,y by A4,Satz2p2;
      hence thesis by A3,Satz2p3;
    end;
    hence thesis by A5,Satz5p5;
  end;
