let X be non empty TopSpace; :: thesis: for Y, X1, X2 being non empty SubSpace of X st X1 meets Y & X2 meets Y & X1,X2 are_separated holds
( X1 meet Y,X2 meet Y are_separated & Y meet X1,Y meet X2 are_separated )
let Y, X1, X2 be non empty SubSpace of X; :: thesis: ( X1 meets Y & X2 meets Y & X1,X2 are_separated implies ( X1 meet Y,X2 meet Y are_separated & Y meet X1,Y meet X2 are_separated ) )
assume that
A1: X1 meets Y and
A2: X2 meets Y ; :: thesis: ( not X1,X2 are_separated or ( X1 meet Y,X2 meet Y are_separated & Y meet X1,Y meet X2 are_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;
assume X1,X2 are_separated ; :: thesis: ( X1 meet Y,X2 meet Y are_separated & Y meet X1,Y meet X2 are_separated )
then A3: A1,A2 are_separated ;

now :: thesis: for D1, D2 being Subset of X st D1 = the carrier of (X1 meet Y) & D2 = the carrier of (X2 meet Y) holds
D1,D2 are_separated

let D1, D2 be Subset of X; :: thesis: ( D1 = the carrier of (X1 meet Y) & D2 = the carrier of (X2 meet Y) implies D1,D2 are_separated )
assume ( D1 = the carrier of (X1 meet Y) & D2 = the carrier of (X2 meet Y) ) ; :: thesis: D1,D2 are_separated
then ( A1 /\ C = D1 & A2 /\ C = D2 ) by A1, A2, Def4;
hence D1,D2 are_separated by A3, Th39; :: thesis: verum

end;

hence X1 meet Y,X2 meet Y are_separated ; :: thesis: Y meet X1,Y meet X2 are_separated
then X1 meet Y,Y meet X2 are_separated by A2, Th26;
hence Y meet X1,Y meet X2 are_separated by A1, Th26; :: thesis: verum