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 that
A1: w1 = v and
A2: w2 = u ; :: thesis: w1 - w2 = v - u
- w2 = - u by A2, Th15;
hence w1 - w2 = v - u by A1, Th13; :: thesis: verum