let X be non empty TopSpace; :: thesis: for X1, X2 being non empty open SubSpace of X holds
( X1 misses X2 iff X1,X2 are_separated )
let X1, X2 be non empty open SubSpace of X; :: thesis: ( X1 misses X2 iff X1,X2 are_separated )
reconsider A2 = the carrier of X2 as Subset of X by Th1;
reconsider A1 = the carrier of X1 as Subset of X by Th1;
A1: ( A1 is open & A2 is open ) by Th16;
thus ( X1 misses X2 implies X1,X2 are_separated ) :: thesis: ( X1,X2 are_separated implies X1 misses X2 )

proof

assume X1 misses X2 ; :: thesis: X1,X2 are_separated
then A1 misses A2 by Def3;
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 A1, Th41;
hence X1,X2 are_separated by Def8; :: thesis: verum

end;

assume X1,X2 are_separated ; :: thesis: X1 misses X2
then A1,A2 are_separated by Def8;
then A1 misses A2 by CONNSP_1:1;
hence X1 misses X2 by Def3; :: thesis: verum