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 Th42:
  LSeg(UMP C, |[ (W-bound C + E-bound C) / 2, N-bound C]|) misses
  LSeg(LMP C, |[ (W-bound C + E-bound C) / 2, S-bound C]|)
proof
  set w = (W-bound C + E-bound C) / 2;
  assume LSeg(UMP C, |[w, N-bound C]|) meets LSeg(LMP C, |[w, S-bound C]|);
  then consider x being object such that
A1: x in LSeg(UMP C, |[w, N-bound C]|) and
A2: x in LSeg(LMP C, |[w, S-bound C]|) by XBOOLE_0:3;
  reconsider x as Point of TOP-REAL 2 by A1;
  |[w, N-bound C]|`2 = N-bound C by EUCLID:52;
  then (UMP C)`2 <= |[w, N-bound C]|`2 by Th39;
  then
A3: x`2 >= (UMP C)`2 by A1,TOPREAL1:4;
  |[w, S-bound C]|`2 = S-bound C by EUCLID:52;
  then |[w, S-bound C]|`2 <= (LMP C)`2 by Th40;
  then x`2 <= (LMP C)`2 by A2,TOPREAL1:4;
  hence contradiction by A3,Th36,XXREAL_0:2;
end;
