reserve X for TopSpace;
reserve X for non empty TopSpace;
reserve X1, X2, X3 for non empty SubSpace of X;
reserve X1, X2, X3 for non empty SubSpace of X;
reserve X for TopSpace;
reserve A1, A2 for Subset of X;
reserve A1,A2 for Subset of X;
reserve X for TopSpace,
  A1, A2 for Subset of X;

theorem Th50:
  for C being Subset of X holds A1,A2 are_weakly_separated implies
  A1 \/ C,A2 \/ C are_weakly_separated
proof
  let C be Subset of X;
  (A1 \/ C) \ (A2 \/ C) = (A1 \ (A2 \/ C)) \/ (C \ (A2 \/ C)) by XBOOLE_1:42
    .= (A1 \ (A2 \/ C)) \/ {} by XBOOLE_1:46
    .= (A1 \ A2) /\ (A1 \ C) by XBOOLE_1:53;
  then
A1: (A1 \/ C) \ (A2 \/ C) c= A1 \ A2 by XBOOLE_1:17;
  (A2 \/ C) \ (A1 \/ C) = (A2 \ (A1 \/ C)) \/ (C \ (A1 \/ C)) by XBOOLE_1:42
    .= (A2 \ (A1 \/ C)) \/ {} by XBOOLE_1:46
    .= (A2 \ A1) /\ (A2 \ C) by XBOOLE_1:53;
  then
A2: (A2 \/ C) \ (A1 \/ C) c= A2 \ A1 by XBOOLE_1:17;
  assume A1,A2 are_weakly_separated;
  then A1 \ A2,A2 \ A1 are_separated;
  then (A1 \/ C) \ (A2 \/ C),(A2 \/ C) \ (A1 \/ C) are_separated by A1,A2,
CONNSP_1:7;
  hence thesis;
end;
