let V be RealUnitarySpace; :: thesis: for W2 being Subspace of V
for W1 being strict Subspace of V holds
( W1 is Subspace of W2 iff W1 /\ W2 = W1 )
let W2 be Subspace of V; :: thesis: for W1 being strict Subspace of V holds
( W1 is Subspace of W2 iff W1 /\ W2 = W1 )
let W1 be strict Subspace of V; :: thesis: ( W1 is Subspace of W2 iff W1 /\ W2 = W1 )
thus ( W1 is Subspace of W2 implies W1 /\ W2 = W1 ) :: thesis: ( W1 /\ W2 = W1 implies W1 is Subspace of W2 )

proof

assume W1 is Subspace of W2 ; :: thesis: W1 /\ W2 = W1
then A1: the carrier of W1 c= the carrier of W2 by RUSUB_1:def 1;
the carrier of (W1 /\ W2) = the carrier of W1 /\ the carrier of W2 by Def2;
hence W1 /\ W2 = W1 by A1, RUSUB_1:24, XBOOLE_1:28; :: thesis: verum

end;

thus ( W1 /\ W2 = W1 implies W1 is Subspace of W2 ) by Th16; :: thesis: verum