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 Th31:
  LMP D in D
proof
  set w = (W-bound D + E-bound D) / 2;
  set X = D /\ Vertical_Line w;
A1: proj2.:X is bounded_below by Th13;
  proj2.:X is non empty & proj2.:X is closed by Th12,Th13;
  then consider x being Point of TOP-REAL 2 such that
A2: x in X and
A3: lower_bound (proj2.:X) = proj2.x by A1,Lm2,RCOMP_1:13;
  x in Vertical_Line w by A2,XBOOLE_0:def 4;
  then
A4: x`1 = w by JORDAN6:31
    .= (LMP D)`1 by EUCLID:52;
  x`2 = lower_bound (proj2.:X) by A3,PSCOMP_1:def 6
    .= (LMP D)`2 by EUCLID:52;
  then x = LMP D by A4,TOPREAL3:6;
  hence thesis by A2,XBOOLE_0:def 4;
end;
