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 Th23:
  A1,A2 are_weakly_separated iff A1 \ (A1 /\ A2),A2 \ (A1 /\ A2) are_separated
proof
A1: A2 \ (A1 /\ A2) = A2 \ A1 by XBOOLE_1:47;
A2: A1 \ (A1 /\ A2) = A1 \ A2 by XBOOLE_1:47;
  thus A1,A2 are_weakly_separated implies A1 \ (A1 /\ A2),A2 \ (A1 /\ A2)
  are_separated
  by A2,XBOOLE_1:47;
  assume A1 \ (A1 /\ A2),A2 \ (A1 /\ A2) are_separated;
  hence thesis by A2,A1;
end;
