reserve X for non empty 1-sorted;
reserve A, A1, A2, B1, B2 for Subset of X;
reserve X for non empty TopSpace;
reserve A, A1, A2, B1, B2 for Subset of X;
reserve X for non empty 1-sorted;
reserve A, A1, A2, B1, B2 for Subset of X;
reserve X for non empty TopSpace,
  A1, A2 for Subset of X;

theorem Th21:
  A1,A2 are_weakly_separated iff (A1 \/ A2) \ A1,(A1 \/ A2) \ A2 are_separated
proof
A1: (A1 \/ A2) \ A1 = A2 \ A1 & (A1 \/ A2) \ A2 = A1 \ A2 by XBOOLE_1:40;
  thus A1,A2 are_weakly_separated implies (A1 \/ A2) \ A1,(A1 \/ A2) \ A2
  are_separated
  by A1;
  assume (A1 \/ A2) \ A1,(A1 \/ A2) \ A2 are_separated;
  hence thesis by A1;
end;
