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 Th51:
  for B1, B2 being Subset of X st B1 c= A2 & B2 c= A1 holds A1,A2
  are_weakly_separated implies A1 \/ B1,A2 \/ B2 are_weakly_separated
proof
  let B1, B2 be Subset of X such that
A1: B1 c= A2 and
A2: B2 c= A1;
  A2 c= A2 \/ B2 by XBOOLE_1:7;
  then B1 c= A2 \/ B2 by A1,XBOOLE_1:1;
  then
A3: B1 \ (A2 \/ B2) = {} by XBOOLE_1:37;
  A1 c= A1 \/ B1 by XBOOLE_1:7;
  then B2 c= A1 \/ B1 by A2,XBOOLE_1:1;
  then
A4: B2 \ (A1 \/ B1) = {} by XBOOLE_1:37;
  (A2 \/ B2) \ (A1 \/ B1) = (A2 \ (A1 \/ B1)) \/ (B2 \ (A1 \/ B1)) by
XBOOLE_1:42;
  then
A5: (A2 \/ B2) \ (A1 \/ B1) c= A2 \ A1 by A4,XBOOLE_1:7,34;
  (A1 \/ B1) \ (A2 \/ B2) = (A1 \ (A2 \/ B2)) \/ (B1 \ (A2 \/ B2)) by
XBOOLE_1:42;
  then
A6: (A1 \/ B1) \ (A2 \/ B2) c= A1 \ A2 by A3,XBOOLE_1:7,34;
  assume A1,A2 are_weakly_separated;
  then A1 \ A2,A2 \ A1 are_separated;
  then (A1 \/ B1) \ (A2 \/ B2),(A2 \/ B2) \ (A1 \/ B1) are_separated by A6,A5,
CONNSP_1:7;
  hence thesis;
end;
