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 Th68:
  U = P & U is a_component &
  V is a_component & U <> V implies Cl P misses V
proof
  assume that
A1: U = P and
A2: U is a_component and
A3: V is a_component and
A4: U <> V;
  assume Cl P meets V;
  then
A5: ex x being object st x in Cl P & x in V by XBOOLE_0:3;
  the carrier of T2|C` = C` by PRE_TOPC:8;
  then reconsider V1 = V as Subset of T2 by XBOOLE_1:1;
  reconsider T2C = T2|C` as non empty SubSpace of T2;
  T2C is locally_connected by JORDAN2C:81;
  then V is open by A3,CONNSP_2:15;
  then V1 is open by TSEP_1:17;
  then P meets V1 by A5,PRE_TOPC:def 7;
  hence thesis by A1,A2,A3,A4,CONNSP_1:35;
end;
