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

proof

assume ( X1,Y are_separated & X2,Y are_separated ) ; :: thesis: X1 union X2,Y are_separated
then A1: ( A1,C are_separated & A2,C are_separated ) by Def8;

now

let D, C be Subset of X; :: thesis: ( D = the carrier of (X1 union X2) & C = the carrier of Y implies D,C are_separated )
assume A2: ( D = the carrier of (X1 union X2) & C = the carrier of Y ) ; :: thesis: D,C are_separated
then A1 \/ A2 = D by Def2;
hence D,C are_separated by A1, A2, Th45; :: thesis: verum

end;

hence X1 union X2,Y are_separated by Def8; :: thesis: verum

end;

assume A3: X1 union X2,Y are_separated ; :: thesis: ( X1,Y are_separated & X2,Y are_separated )
( X1 is SubSpace of X1 union X2 & X2 is SubSpace of X1 union X2 & Y is SubSpace of Y ) by Th2, Th22;
hence ( X1,Y are_separated & X2,Y are_separated ) by A3, Th77; :: thesis: verum