theorem Th15:
  (Lower_Middle_Point C)`2 <> (Upper_Middle_Point C)`2
proof
  set l = Lower_Middle_Point C, u = Upper_Middle_Point C, w = (W-bound C+
  E-bound C)/2;
A1: l`1 = w by JORDAN6:64;
A2: u`1 = w by JORDAN6:65;
  assume l`2 = u`2;
  then l = u by A1,A2,TOPREAL3:6;
  then l in Lower_Arc C & l in Upper_Arc C by JORDAN6:66,67;
  then l in Lower_Arc C /\ Upper_Arc C by XBOOLE_0:def 4;
  then l in {W-min(C),E-max(C)} by JORDAN6:def 9;
  then l = W-min(C) or l = E-max(C) by TARSKI:def 2;
  then W-bound C = w or E-bound C = w by A1,EUCLID:52;
  hence thesis by SPRECT_1:31;
end;
