theorem Lemma5p12p3:
  between a,b,c & a,c equiv a,b implies c = b
  proof
    assume
A1: between a,b,c & a,c equiv a,b;
    then
A2: between c,b,a by Satz3p2;
A3: c,b,a cong b,c,a
    proof
A4:   c,a equiv a,b by A1,Satz2p4;
      a,b equiv a,c by A1,Satz2p2;
      then b,a equiv a,c by Satz2p4;
      hence thesis by A4,Satz2p5,GTARSKI1:def 5;
    end;
    between b,c,a by A2,A3,Satz4p6;
    hence thesis by A2,Satz3p4;
  end;
