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

proof

assume W1 is Subspace of W2 ; :: thesis: W1 + W2 = W2
then the carrier of W1 c= the carrier of W2 by RUSUB_1:def 1;
hence W1 + W2 = W2 by Lm3; :: thesis: verum

end;

thus ( W1 + W2 = W2 implies W1 is Subspace of W2 ) by Th7; :: thesis: verum