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 Th12:
  for D being with_the_max_arc Subset of TOP-REAL 2 holds proj2.:(
  D /\ Vertical_Line((W-bound D + E-bound D) / 2)) is not empty
proof
  let D be with_the_max_arc Subset of TOP-REAL 2;
  set w = (W-bound D + E-bound D) / 2;
  D meets Vertical_Line w by Def1;
  then D /\ Vertical_Line w is non empty;
  hence thesis by Lm1,RELAT_1:119;
end;
