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 :: Jordan's Curve Theorem
  for C being Simple_closed_curve ex A1, A2 being Subset of TOP-REAL 2 st
  C` = A1 \/ A2 & A1 misses A2 & (Cl A1) \ A1 = (Cl A2) \ A2 &
  for C1, C2 being Subset of (TOP-REAL 2)|C` st C1 = A1 & C2 = A2 holds
  C1 is a_component & C2 is a_component
proof
  let C;
  C is Jordan by Lm92;
  hence thesis;
end;
