let X be non empty TopSpace; :: thesis: for X1, X2 being non empty SubSpace of X st X1 meets X2 holds
( X1 meet X2 is SubSpace of X1 & X1 meet X2 is SubSpace of X2 )
let X1, X2 be non empty SubSpace of X; :: thesis: ( X1 meets X2 implies ( X1 meet X2 is SubSpace of X1 & X1 meet X2 is SubSpace of X2 ) )
assume X1 meets X2 ; :: thesis: ( X1 meet X2 is SubSpace of X1 & X1 meet X2 is SubSpace of X2 )
then the carrier of (X1 meet X2) = the carrier of X1 /\ the carrier of X2 by Def4;
then ( the carrier of (X1 meet X2) c= the carrier of X1 & the carrier of (X1 meet X2) c= the carrier of X2 ) by XBOOLE_1:17;
hence ( X1 meet X2 is SubSpace of X1 & X1 meet X2 is SubSpace of X2 ) by Th4; :: thesis: verum