let F be Field; :: thesis: for V, W being VectSp of F
for X being Subset of V st V is Subspace of W holds
X is Subset of W
let V, W be VectSp of F; :: thesis: for X being Subset of V st V is Subspace of W holds
X is Subset of W
let X be Subset of V; :: thesis: ( V is Subspace of W implies X is Subset of W )
assume V is Subspace of W ; :: thesis: X is Subset of W
then A1: [#] V c= [#] W by VECTSP_4:def 2;
X c= [#] W

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in X or x in [#] W )
assume x in X ; :: thesis: x in [#] W
then x in [#] V ;
hence x in [#] W by A1; :: thesis: verum

end;

hence X is Subset of W ; :: thesis: verum