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 Th22:
  Lower_Arc C /\ Vertical_Line((W-bound C + E-bound C) / 2) is not
  empty & proj2.:(Lower_Arc C /\ Vertical_Line((W-bound C + E-bound C) / 2)) is
  not empty
proof
  set w = (W-bound C + E-bound C) / 2;
A1: W-bound C < E-bound C by SPRECT_1:31;
  (E-max C)`1 = E-bound C by EUCLID:52;
  then
A2: w <= (E-max C)`1 by A1,XREAL_1:226;
  Lower_Arc C is_an_arc_of E-max C,W-min C by JORDAN6:def 9;
  then
A3: Lower_Arc C is_an_arc_of W-min C,E-max C by JORDAN5B:14;
  (W-min C)`1 = W-bound C by EUCLID:52;
  then (W-min C)`1 <= w by A1,XREAL_1:226;
  then Lower_Arc C meets Vertical_Line w by A3,A2,JORDAN6:49;
  then Lower_Arc C /\ Vertical_Line w is non empty;
  hence thesis by Lm1,RELAT_1:119;
end;
