let X be non empty TopSpace; :: thesis: for X1, X2 being non empty SubSpace of X st X = X1 union X2 & X1 misses X2 holds
( X1,X2 are_weakly_separated iff ( X1 is closed SubSpace of X & X2 is closed SubSpace of X ) )
let X1, X2 be non empty SubSpace of X; :: thesis: ( X = X1 union X2 & X1 misses X2 implies ( X1,X2 are_weakly_separated iff ( X1 is closed SubSpace of X & X2 is closed SubSpace of X ) ) )
assume A1: X = X1 union X2 ; :: thesis: ( not X1 misses X2 or ( X1,X2 are_weakly_separated iff ( X1 is closed SubSpace of X & X2 is closed SubSpace of X ) ) )
assume A2: X1 misses X2 ; :: thesis: ( X1,X2 are_weakly_separated iff ( X1 is closed SubSpace of X & X2 is closed SubSpace of X ) )
thus ( X1,X2 are_weakly_separated implies ( X1 is closed SubSpace of X & X2 is closed SubSpace of X ) ) :: thesis: ( X1 is closed SubSpace of X & X2 is closed SubSpace of X implies X1,X2 are_weakly_separated )

proof

reconsider A2 = the carrier of X2 as Subset of X by Th1;
reconsider A1 = the carrier of X1 as Subset of X by Th1;
assume X1,X2 are_weakly_separated ; :: thesis: ( X1 is closed SubSpace of X & X2 is closed SubSpace of X )
then X1,X2 are_separated by A2, Th78;
then A3: A1,A2 are_separated ;
A1 \/ A2 = [#] X by A1, Def2;
then ( A1 is closed & A2 is closed ) by A3, CONNSP_1:4;
hence ( X1 is closed SubSpace of X & X2 is closed SubSpace of X ) by Th11; :: thesis: verum

end;

thus ( X1 is closed SubSpace of X & X2 is closed SubSpace of X implies X1,X2 are_weakly_separated ) by Th80; :: thesis: verum