let X be non empty TopSpace; :: thesis: for X1, X2 being non empty SubSpace of X holds X1 is SubSpace of X1 union X2
let X1, X2 be non empty SubSpace of X; :: thesis: X1 is SubSpace of X1 union X2
set A1 = the carrier of X1;
set A2 = the carrier of X2;
set A = the carrier of (X1 union X2);
the carrier of (X1 union X2) = the carrier of X1 \/ the carrier of X2 by Def2;
then the carrier of X1 c= the carrier of (X1 union X2) by XBOOLE_1:7;
hence X1 is SubSpace of X1 union X2 by Th4; :: thesis: verum