let R be Ring; 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; 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; 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; for w1, w2 being Vector of W st w1 = v & w2 = u holds
w1 - w2 = v - u
let w1, w2 be Vector of W; ( w1 = v & w2 = u implies w1 - w2 = v - u )
assume that
A1:
w1 = v
and
A2:
w2 = u
; w1 - w2 = v - u
- w2 = - u
by A2, Th15;
hence
w1 - w2 = v - u
by A1, Th13; verum