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 A1: ( v = w1 & u = w2 ) ; :: thesis: w1 + w2 = v + u

w1 + w2 = ( the addF of V || the carrier of W) . [w1,w2] by Def2;

hence w1 + w2 = v + u by A1, FUNCT_1:49; :: 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 A1: ( v = w1 & u = w2 ) ; :: thesis: w1 + w2 = v + u

w1 + w2 = ( the addF of V || the carrier of W) . [w1,w2] by Def2;

hence w1 + w2 = v + u by A1, FUNCT_1:49; :: thesis: verum