let R be Ring; :: thesis: for V being RightMod of R
for u, v being Vector of V
for W being Submodule of V
for w1, w2 being Vector of W st w1 = v & w2 = u holds
w1 + w2 = v + u
let V be RightMod of R; :: thesis: for u, v being Vector of V
for W being Submodule of V
for w1, w2 being Vector of W st w1 = v & w2 = u holds
w1 + w2 = v + u
let u, v be Vector of V; :: thesis: for W being Submodule of V
for w1, w2 being Vector of W st w1 = v & w2 = u holds
w1 + w2 = v + u
let W be Submodule of V; :: thesis: for w1, w2 being Vector of W st w1 = v & w2 = u holds
w1 + w2 = v + u
let w1, w2 be Vector of W; :: thesis: ( w1 = v & w2 = u implies w1 + w2 = v + u )
assume A1: ( v = w1 & u = w2 ) ; :: thesis: w1 + w2 = v + u
set IW = [: the carrier of W, the carrier of W:];
w1 + w2 = ( the addF of V | [: the carrier of W, the carrier of W:]) . [w1,w2] by Def2;
hence w1 + w2 = v + u by A1, FUNCT_1:49; :: thesis: verum