let X be TopSpace; :: thesis: for A1, A2 being Subset of X st A1 is closed & A2 is closed holds
A1,A2 are_weakly_separated
let A1, A2 be Subset of X; :: thesis: ( A1 is closed & A2 is closed implies A1,A2 are_weakly_separated )
assume ( A1 is closed & A2 is closed ) ; :: thesis: A1,A2 are_weakly_separated
then ( Cl (A1 \ A2) c= A1 & Cl (A2 \ A1) c= A2 ) by TOPS_1:31, XBOOLE_1:36;
then ( (Cl (A1 \ A2)) \ A1 = {} & (Cl (A2 \ A1)) \ A2 = {} ) by XBOOLE_1:37;
then ( Cl (A1 \ A2) misses A2 \ A1 & A1 \ A2 misses Cl (A2 \ A1) ) by Lm1;
then A1 \ A2,A2 \ A1 are_separated by CONNSP_1:def 1;
hence A1,A2 are_weakly_separated by Def7; :: thesis: verum