let X be non empty TopSpace; :: thesis: for Y, X1, X2 being non empty SubSpace of X st X1 meets X2 holds
( ( X1,Y are_weakly_separated & X2,Y are_weakly_separated implies X1 meet X2,Y are_weakly_separated ) & ( Y,X1 are_weakly_separated & Y,X2 are_weakly_separated implies Y,X1 meet X2 are_weakly_separated ) )
let Y, X1, X2 be non empty SubSpace of X; :: thesis: ( X1 meets X2 implies ( ( X1,Y are_weakly_separated & X2,Y are_weakly_separated implies X1 meet X2,Y are_weakly_separated ) & ( Y,X1 are_weakly_separated & Y,X2 are_weakly_separated implies Y,X1 meet X2 are_weakly_separated ) ) )
assume A1: X1 meets X2 ; :: thesis: ( ( X1,Y are_weakly_separated & X2,Y are_weakly_separated implies X1 meet X2,Y are_weakly_separated ) & ( Y,X1 are_weakly_separated & Y,X2 are_weakly_separated implies Y,X1 meet 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;
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 meet X2,Y are_weakly_separated ) :: thesis: ( Y,X1 are_weakly_separated & Y,X2 are_weakly_separated implies Y,X1 meet X2 are_weakly_separated )

proof

assume ( X1,Y are_weakly_separated & X2,Y are_weakly_separated ) ; :: thesis: X1 meet X2,Y are_weakly_separated
then A2: ( A1,C are_weakly_separated & A2,C are_weakly_separated ) by Def9;

now

let D, C be Subset of X; :: thesis: ( D = the carrier of (X1 meet X2) & C = the carrier of Y implies D,C are_weakly_separated )
assume that
A3: D = the carrier of (X1 meet X2) and
A4: C = the carrier of Y ; :: thesis: D,C are_weakly_separated
A1 /\ A2 = D by A1, A3, Def5;
hence D,C are_weakly_separated by A2, A4, Th57; :: thesis: verum

end;

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

end;

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