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 Th40; :: thesis: ( [#] W1 c= [#] W2 implies W1 c= W2 )
assume A1: [#] W1 c= [#] W2 ; :: thesis: W1 c= W2

now

let a be Vector of V; :: thesis: ( a in W1 implies a in W2 )
assume a in W1 ; :: thesis: a in W2
then a in [#] W1 by STRUCT_0:def 5;
hence a in W2 by A1, STRUCT_0:def 5; :: thesis: verum

end;

hence W1 c= W2 by Th41; :: thesis: verum