let X be non empty TopSpace; :: thesis: for X1, X2 being non empty SubSpace of X holds
( X1 is SubSpace of X2 iff X1 union X2 = TopStruct(# the carrier of X2, the topology of X2 #) )
let X1, X2 be non empty SubSpace of X; :: thesis: ( X1 is SubSpace of X2 iff X1 union X2 = TopStruct(# the carrier of X2, the topology of X2 #) )
set A1 = the carrier of X1;
set A2 = the carrier of X2;
thus ( X1 is SubSpace of X2 iff X1 union X2 = TopStruct(# the carrier of X2, the topology of X2 #) ) :: thesis: verum

proof

thus ( X1 is SubSpace of X2 implies X1 union X2 = TopStruct(# the carrier of X2, the topology of X2 #) ) :: thesis: ( X1 union X2 = TopStruct(# the carrier of X2, the topology of X2 #) implies X1 is SubSpace of X2 )

proof

assume X1 is SubSpace of X2 ; :: thesis: X1 union X2 = TopStruct(# the carrier of X2, the topology of X2 #)
then A1: the carrier of X1 \/ the carrier of X2 = the carrier of TopStruct(# the carrier of X2, the topology of X2 #) by BORSUK_1:1, XBOOLE_1:12;
TopStruct(# the carrier of X2, the topology of X2 #) is strict SubSpace of X by Lm3;
hence X1 union X2 = TopStruct(# the carrier of X2, the topology of X2 #) by A1, Def2; :: thesis: verum

end;

assume X1 union X2 = TopStruct(# the carrier of X2, the topology of X2 #) ; :: thesis: X1 is SubSpace of X2
then the carrier of X1 \/ the carrier of X2 = the carrier of X2 by Def2;
then the carrier of X1 c= the carrier of X2 by XBOOLE_1:7;
hence X1 is SubSpace of X2 by Th4; :: thesis: verum

end;