theorem Satz5p6:
  a,b <= c,d & a,b equiv a9,b9 & c,d equiv c9,d9 implies a9,b9 <= c9,d9
  proof
    assume
A1: a,b <= c,d & a,b equiv a9,b9 & c,d equiv c9,d9;
    then
    consider x be POINT of S such that
A2: between a,b,x & a,x equiv c,d by Satz5p5;
    Collinear a,b,x by A2;
    then consider y be POINT of S such that
A3: a,b,x cong a9,b9,y by A1,Satz4p14;
    a9,y equiv c9,d9
    proof
      a9,y equiv a,x by A3,Satz2p2;
      then a9,y equiv c,d by A2,Satz2p3;
      hence thesis by A1,Satz2p3;
    end;
    hence a9,b9 <= c9,d9 by A2,A3,Satz4p6,Satz5p5;
  end;
