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 Th50:
  a < b & c < d implies
  Int closed_inside_of_rectangle(a,b,c,d) = inside_of_rectangle(a,b,c,d)
proof
  assume that
A1: a < b and
A2: c < d;
  set P = rectangle(a,b,c,d);
  set R = closed_inside_of_rectangle(a,b,c,d);
  set P1 = inside_of_rectangle(a,b,c,d);
  set P2 = outside_of_rectangle(a,b,c,d);
A3: P = {p where p is Point of T2: 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;
A4: R misses P2 by Th48;
  thus Int R = (Cl P2``)` by Th47
    .= (P2 \/ P)` by A1,A2,A3,JORDAN1:44
    .= P2` /\ P` by XBOOLE_1:53
    .= R /\ P` by Th47
    .= R /\ (P1 \/ P2) by A1,A2,A3,JORDAN1:36
    .= R /\ P1 \/ R /\ P2 by XBOOLE_1:23
    .= R /\ P1 \/ {} by A4
    .= P1 by Th49;
end;
