theorem Satz5p3:
  between a,b,d & between a,c,d implies between a,b,c or between a,c,b
  proof
    assume
A1:  between a,b,d & between a,c,d;
    per cases;
    suppose a = b;
      hence thesis by Satz3p1,Satz3p2;
    end;
    suppose
A2:   a <> b;
      consider p be POINT of S such that
A3:   between d,a,p & a,p equiv a,b by GTARSKI1:def 8;
A4:   a <> p by A3,Satz2p2,A2,GTARSKI1:def 7;
      between p,a,d by A3,Satz3p2;
      then between p,a,b & between p,a,c by A1,Satz3p5p1;
      hence thesis by A4,Satz5p2;
    end;
  end;
