let X be non empty TopSpace; :: thesis: for X1, X2, Y being non empty SubSpace of X st X1,X2 are_weakly_separated holds
X1 union Y,X2 union Y are_weakly_separated
let X1, X2, Y be non empty SubSpace of X; :: thesis: ( X1,X2 are_weakly_separated implies X1 union Y,X2 union Y are_weakly_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;
reconsider C = the carrier of Y as Subset of X by Th1;
assume X1,X2 are_weakly_separated ; :: thesis: X1 union Y,X2 union Y are_weakly_separated
then A1: A1,A2 are_weakly_separated by Def9;

now

let D1, D2 be Subset of X; :: thesis: ( D1 = the carrier of (X1 union Y) & D2 = the carrier of (X2 union Y) implies D1,D2 are_weakly_separated )
assume ( D1 = the carrier of (X1 union Y) & D2 = the carrier of (X2 union Y) ) ; :: thesis: D1,D2 are_weakly_separated
then ( A1 \/ C = D1 & A2 \/ C = D2 ) by Def2;
hence D1,D2 are_weakly_separated by A1, Th55; :: thesis: verum

end;

hence X1 union Y,X2 union Y are_weakly_separated by Def9; :: thesis: verum