let R be Ring; :: thesis: for V being RightMod of
for v, u being Vector of
for W being Submodule of V
for w1, w2 being Vector of st w1 = v & w2 = u holds
w1 + w2 = v + u
let V be RightMod of ; :: thesis: for v, u being Vector of
for W being Submodule of V
for w1, w2 being Vector of st w1 = v & w2 = u holds
w1 + w2 = v + u
let v, u be Vector of ; :: thesis: for W being Submodule of V
for w1, w2 being Vector of st w1 = v & w2 = u holds
w1 + w2 = v + u
let W be Submodule of V; :: thesis: for w1, w2 being Vector of st w1 = v & w2 = u holds
w1 + w2 = v + u
let w1, w2 be Vector of ; :: 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_1:72; :: thesis: verum