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