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 union X2
let X1, X2 be non empty SubSpace of X; :: thesis: ( X1 meets X2 implies X1 meet X2 is SubSpace of X1 union X2 )
assume X1 meets X2 ; :: thesis: X1 meet X2 is SubSpace of X1 union X2
then A1: X1 meet X2 is SubSpace of X1 by Th27;
X1 is SubSpace of X1 union X2 by Th22;
hence X1 meet X2 is SubSpace of X1 union X2 by A1, Th7; :: thesis: verum