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 v + 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 v + 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 v + W
let v be Vector of V; :: thesis: for W being Submodule of V st v in W holds
v * a in v + W
let W be Submodule of V; :: thesis: ( v in W implies v * a in v + W )
assume v in W ; :: thesis: v * a in v + W
then ( v + W = the carrier of W & v * a in W ) by Lm3, Th21;
hence v * a in v + W ; :: thesis: verum