theorem Satz4p19:
  between a,c,b & a,c equiv a,c9 & b,c equiv b,c9 implies c = c9
  proof
    assume
A1: between a,c,b & a,c equiv a,c9 & b,c equiv b,c9;
A2: a = b implies c = c9
    proof
      assume a = b;
      then a = c by A1,GTARSKI1:def 10;
      hence thesis by A1,Satz2p2,GTARSKI1:def 7;
    end;
    a <> b implies c = c9
    proof
      assume
A3:   a <> b;
      Collinear a,b,c by A1,Satz3p2;
      hence thesis by A1,A3,Satz4p18;
    end;
    hence thesis by A2;
  end;
