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) + W = the carrier of 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) + W = the carrier of 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) + W = the carrier of W
let v be Vector of V; :: thesis: for W being Submodule of V st v in W holds
(v * a) + W = the carrier of W
let W be Submodule of V; :: thesis: ( v in W implies (v * a) + W = the carrier of W )
assume v in W ; :: thesis: (v * a) + W = the carrier of W
then v * a in W by Th29;
hence (v * a) + W = the carrier of W by Lm3; :: thesis: verum