let X be TopSpace; :: thesis: for A1, A2 being Subset of X st A1 is open & A2 is open holds
A1,A2 are_weakly_separated
let A1, A2 be Subset of X; :: thesis: ( A1 is open & A2 is open implies A1,A2 are_weakly_separated )
assume that
A1: A1 is open and
A2: A2 is open ; :: thesis: A1,A2 are_weakly_separated
A2 \ A1 misses A1 by XBOOLE_1:79;
then Cl (A2 \ A1) misses A1 by A1, Th36;
then A3: A1 \ A2 misses Cl (A2 \ A1) by XBOOLE_1:36, XBOOLE_1:63;
A2 misses A1 \ A2 by XBOOLE_1:79;
then A2 misses Cl (A1 \ A2) by A2, Th36;
then Cl (A1 \ A2) misses A2 \ A1 by XBOOLE_1:36, XBOOLE_1:63;
then A1 \ A2,A2 \ A1 are_separated by A3, CONNSP_1:def 1;
hence A1,A2 are_weakly_separated ; :: thesis: verum