reserve C for Simple_closed_curve,
  P for Subset of TOP-REAL 2,
  p for Point of TOP-REAL 2,
  n for Element of NAT;
reserve D for compact with_the_max_arc Subset of TOP-REAL 2;

theorem Th36:
  (LMP C)`2 < (UMP C)`2
proof
  set w = (E-bound C + W-bound C) / 2;
  set X = C /\ Vertical_Line (w);
A1: (UMP C)`2 = upper_bound (proj2.:X) & (LMP C)`2 = lower_bound (proj2.:X)
 by EUCLID:52;
  set u = Upper_Middle_Point C, l = Lower_Middle_Point C;
A2: l`2 = proj2.l by PSCOMP_1:def 6;
  l`1 = w by JORDAN6:64;
  then
A3: l in Vertical_Line (w) by JORDAN6:31;
  l in C by Th14;
  then l in X by A3,XBOOLE_0:def 4;
  then
A4: l`2 in proj2.:X by A2,FUNCT_2:35;
  X is bounded by TOPREAL6:89;
  then
A5: proj2.:X is real-bounded by JCT_MISC:14;
  u`1 = w by JORDAN6:65;
  then u in C & u in Vertical_Line (w) by JORDAN6:31,68;
  then
A6: u in X by XBOOLE_0:def 4;
  u`2 = proj2.u by PSCOMP_1:def 6;
  then
A7: u`2 in proj2.:X by A6,FUNCT_2:35;
  u`2 <> l`2 by Th15;
  hence thesis by A1,A5,A7,A4,SEQ_4:12;
end;
