let R be Ring; :: thesis: for a being Scalar of R
for V being RightMod of R
for v being Vector of V
for W being Submodule of V st v in W holds
v * a in W
let a be Scalar of R; :: thesis: for V being RightMod of R
for v being Vector of V
for W being Submodule of V st v in W holds
v * a in W
let V be RightMod of R; :: thesis: for v being Vector of V
for W being Submodule of V st v in W holds
v * a in W
let v be Vector of V; :: thesis: for W being Submodule of V st v in W holds
v * a in W
let W be Submodule of V; :: thesis: ( v in W implies v * a in W )
reconsider VW = the carrier of W as Subset of V by Def2;
assume v in W ; :: thesis: v * a in W
then A1: v in the carrier of W ;
VW is linearly-closed by Lm1;
then v * a in the carrier of W by A1;
hence v * a in W ; :: thesis: verum