theorem Satz7p4uniqueness:
  for p,p1,p2 being POINT of S st Middle p,a,p1 & Middle p,a,p2 holds p1 = p2
  proof
    let p,p1,p2 be POINT of S;
    assume
A1: Middle p,a,p1 & Middle p,a,p2;
    per cases;
    suppose
A2:   p <> a;
      a,p1 equiv a,p & a,p2 equiv a,p by A1,Satz2p2;
      hence thesis by A1,A2,Satz2p12;
    end;
    suppose p = a;
      then a = p1 & a = p2 by A1,Satz2p2,GTARSKI1:def 7;
      hence thesis;
    end;
end;
