reserve a, b, c, d, r, s for Real,
  n for Element of NAT,
  p, p1, p2 for Point of TOP-REAL 2,
  x, y for Point of TOP-REAL n,
  C for Simple_closed_curve,
  A, B, P for Subset of TOP-REAL 2,
  U, V for Subset of (TOP-REAL 2)|C`,
  D for compact with_the_max_arc Subset of TOP-REAL 2;

theorem Th47:
  closed_inside_of_rectangle(a,b,c,d) = outside_of_rectangle(a,b,c,d)`
proof
  set R = closed_inside_of_rectangle(a,b,c,d);
  set O = outside_of_rectangle(a,b,c,d);
  thus R c= O`
  proof
    let x be object;
    assume x in R;
    then consider p such that
A1: x = p and
A2: a <= p`1 and
A3: p`1 <= b and
A4: c <= p`2 and
A5: p`2 <= d;
    now
      assume p in O;
      then ex p1 st p1 = p &
      not(a <= p1`1 & p1`1 <= b & c <= p1`2 & p1`2 <= d);
      hence contradiction by A2,A3,A4,A5;
    end;
    hence thesis by A1,SUBSET_1:29;
  end;
  let x be object;
  assume
A6: x in O`;
  then
A7: not x in O by XBOOLE_0:def 5;
  reconsider x as Point of T2 by A6;
A8: a <= x`1 by A7;
A9: x`1 <= b by A7;
A10: c <= x`2 by A7;
  x`2 <= d by A7;
  hence thesis by A8,A9,A10;
end;
