let X be non empty TopSpace; :: thesis: for A1, A2, C1, C2 being Subset of X st C1 c= A1 & C2 c= A2 & C1 /\ C2 = A1 /\ A2 & A1,A2 are_weakly_separated holds
C1,C2 are_weakly_separated
let A1, A2, C1, C2 be Subset of X; :: thesis: ( C1 c= A1 & C2 c= A2 & C1 /\ C2 = A1 /\ A2 & A1,A2 are_weakly_separated implies C1,C2 are_weakly_separated )
assume ( C1 c= A1 & C2 c= A2 ) ; :: thesis: ( not C1 /\ C2 = A1 /\ A2 or not A1,A2 are_weakly_separated or C1,C2 are_weakly_separated )
then A1: ( C1 \ (C1 /\ C2) c= A1 \ (C1 /\ C2) & C2 \ (C1 /\ C2) c= A2 \ (C1 /\ C2) ) by XBOOLE_1:33;
assume A2: C1 /\ C2 = A1 /\ A2 ; :: thesis: ( not A1,A2 are_weakly_separated or C1,C2 are_weakly_separated )
assume A1,A2 are_weakly_separated ; :: thesis: C1,C2 are_weakly_separated
then A1 \ (C1 /\ C2),A2 \ (C1 /\ C2) are_separated by A2, Th23;
then C1 \ (C1 /\ C2),C2 \ (C1 /\ C2) are_separated by A1, CONNSP_1:7;
hence C1,C2 are_weakly_separated by Th23; :: thesis: verum