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;
