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 Th45:
  a <= b & c <= d implies
  rectangle(a,b,c,d) c= closed_inside_of_rectangle(a,b,c,d)
proof
  assume that
A1: a <= b and
A2: c <= d;
  let x be object;
  assume x in rectangle(a,b,c,d);
  then x in {p: p`1 = a & p`2 <= d & p`2 >= c or
  p`1 <= b & p`1 >= a & p`2 = d or p`1 <= b & p`1 >= a & p`2 = c or
  p`1 = b & p`2 <= d & p`2 >= c} by A1,A2,SPPOL_2:54;
  then ex p st x = p & (p`1 = a & p`2 <= d & p`2 >= c or
  p`1 <= b & p`1 >= a & p`2 = d or p`1 <= b & p`1 >= a & p`2 = c or
  p`1 = b & p`2 <= d & p`2 >= c);
  hence thesis by A1,A2;
end;
