let K be Ring; :: thesis: for V being LeftMod of K
for W1, W2 being Subspace of V holds
( W1 c= W2 iff [#] W1 c= [#] W2 )
let V be LeftMod of K; :: thesis: for W1, W2 being Subspace of V holds
( W1 c= W2 iff [#] W1 c= [#] W2 )
let W1, W2 be Subspace of V; :: thesis: ( W1 c= W2 iff [#] W1 c= [#] W2 )
thus ( W1 c= W2 implies [#] W1 c= [#] W2 ) by Th36; :: thesis: ( [#] W1 c= [#] W2 implies W1 c= W2 )
assume [#] W1 c= [#] W2 ; :: thesis: W1 c= W2
then for a being Vector of V st a in W1 holds
a in W2 ;
hence W1 c= W2 by Th37; :: thesis: verum