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;
