theorem Th53:
  for B being Subset of X holds A1,B are_weakly_separated & A2,B
  are_weakly_separated implies A1 \/ A2,B are_weakly_separated
proof
  let B be Subset of X;
  thus A1,B are_weakly_separated & A2,B are_weakly_separated implies A1 \/ A2,
  B are_weakly_separated
  proof
    assume that
A1: A1,B are_weakly_separated and
A2: A2,B are_weakly_separated;
    A2 \ B,B \ A2 are_separated by A2;
    then
A3: A2 \ B,(B \ A1) /\ (B \ A2) are_separated by Th40;
    A1 \ B,B \ A1 are_separated by A1;
    then A1 \ B,(B \ A1) /\ (B \ A2) are_separated by Th40;
    then (A1 \ B) \/ (A2 \ B),(B \ A1) /\ (B \ A2) are_separated by A3,Th41;
    then (A1 \/ A2) \ B,(B \ A1) /\ (B \ A2) are_separated by XBOOLE_1:42;
    then (A1 \/ A2) \ B,B \ (A1 \/ A2) are_separated by XBOOLE_1:53;
    hence thesis;
  end;
end;
