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

proof

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

now :: thesis: for D, C being Subset of X st D = the carrier of (X1 union X2) & C = the carrier of Y holds
D,C are_weakly_separated

let D, C be Subset of X; :: thesis: ( D = the carrier of (X1 union X2) & C = the carrier of Y implies D,C are_weakly_separated )
assume that
A2: D = the carrier of (X1 union X2) and
A3: C = the carrier of Y ; :: thesis: D,C are_weakly_separated
A1 \/ A2 = D by A2, Def2;
hence D,C are_weakly_separated by A1, A3, Th53; :: thesis: verum

end;

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

end;

hence ( Y,X1 are_weakly_separated & Y,X2 are_weakly_separated implies Y,X1 union X2 are_weakly_separated ) ; :: thesis: verum