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 Th13:
  for C being compact Subset of TOP-REAL 2 holds proj2.:(C /\
  Vertical_Line((W-bound C + E-bound C) / 2)) is closed bounded_below
  bounded_above
proof
  let C be compact Subset of TOP-REAL 2;
  set w = (W-bound C + E-bound C) / 2;
  set X = C /\ Vertical_Line w;
  Vertical_Line w is closed by JORDAN6:30;
  then
A1: X is closed by TOPS_1:8;
  X is bounded by RLTOPSP1:42,XBOOLE_1:17;
  hence proj2.:X is closed by A1,JCT_MISC:13;
  X c= C by XBOOLE_1:17;
  then proj2.:X is real-bounded by JCT_MISC:14,RLTOPSP1:42;
  hence thesis by XXREAL_2:def 11;
end;
