let X be non empty TopSpace; :: thesis: for A1, A2 being Subset of X holds
( A1,A2 are_weakly_separated iff A1 \ (A1 /\ A2),A2 \ (A1 /\ A2) are_separated )
let A1, A2 be Subset of X; :: thesis: ( A1,A2 are_weakly_separated iff A1 \ (A1 /\ A2),A2 \ (A1 /\ A2) are_separated )
A1: ( A1 \ (A1 /\ A2) = A1 \ A2 & A2 \ (A1 /\ A2) = A2 \ A1 ) by XBOOLE_1:47;
thus ( A1,A2 are_weakly_separated implies A1 \ (A1 /\ A2),A2 \ (A1 /\ A2) are_separated ) :: thesis: ( A1 \ (A1 /\ A2),A2 \ (A1 /\ A2) are_separated implies A1,A2 are_weakly_separated )

proof

assume A1 \ A2,A2 \ A1 are_separated ; :: according to TSEP_1:def 7 :: thesis: A1 \ (A1 /\ A2),A2 \ (A1 /\ A2) are_separated
hence A1 \ (A1 /\ A2),A2 \ (A1 /\ A2) are_separated by A1; :: thesis: verum

end;

assume A1 \ (A1 /\ A2),A2 \ (A1 /\ A2) are_separated ; :: thesis: A1,A2 are_weakly_separated
hence A1,A2 are_weakly_separated by A1, TSEP_1:def 7; :: thesis: verum