let X be non empty TopSpace; :: thesis: for X1, X2 being non empty SubSpace of X holds
( ( X1 misses X2 & X1,X2 are_weakly_separated ) iff X1,X2 are_separated )
let X1, X2 be non empty SubSpace of X; :: thesis: ( ( X1 misses X2 & X1,X2 are_weakly_separated ) iff X1,X2 are_separated )
reconsider A1 = the carrier of X1 as Subset of X by Th1;
reconsider A2 = the carrier of X2 as Subset of X by Th1;
thus ( X1 misses X2 & X1,X2 are_weakly_separated implies X1,X2 are_separated ) :: thesis: ( X1,X2 are_separated implies ( X1 misses X2 & X1,X2 are_weakly_separated ) )

proof

assume ( X1 misses X2 & X1,X2 are_weakly_separated ) ; :: thesis: X1,X2 are_separated
then ( A1 misses A2 & A1,A2 are_weakly_separated ) by Def3, Def9;
then for A1, A2 being Subset of X st A1 = the carrier of X1 & A2 = the carrier of X2 holds
A1,A2 are_separated by Th51;
hence X1,X2 are_separated by Def8; :: thesis: verum

end;

assume X1,X2 are_separated ; :: thesis: ( X1 misses X2 & X1,X2 are_weakly_separated )
then A1: A1,A2 are_separated by Def8;
then ( A1 misses A2 & A1,A2 are_weakly_separated ) by Th51;
hence X1 misses X2 by Def3; :: thesis: X1,X2 are_weakly_separated
for A1, A2 being Subset of X st A1 = the carrier of X1 & A2 = the carrier of X2 holds
A1,A2 are_weakly_separated by A1, Th51;
hence X1,X2 are_weakly_separated by Def9; :: thesis: verum