let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ; :: thesis: 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; :: thesis: 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; :: thesis: 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; :: thesis: for w1, w2 being Element of W st w1 = v & w2 = u holds

w1 - w2 = v - u

let w1, w2 be Element 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

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; :: thesis: 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; :: thesis: 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; :: thesis: for w1, w2 being Element of W st w1 = v & w2 = u holds

w1 - w2 = v - u

let w1, w2 be Element 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