let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ; for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for u, v being Element of V
for W being Subspace of V
for w1, w2 being Element of W st w1 = v & w2 = u holds
w1 - w2 = v - u
let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF; for u, v being Element of V
for W being Subspace of V
for w1, w2 being Element of W st w1 = v & w2 = u holds
w1 - w2 = v - u
let u, v be Element of V; for W being Subspace of V
for w1, w2 being Element of W st w1 = v & w2 = u holds
w1 - w2 = v - u
let W be Subspace of V; for w1, w2 being Element of W st w1 = v & w2 = u holds
w1 - w2 = v - u
let w1, w2 be Element 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