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 a being Element of GF

for v being Element of V

for W being Subspace of V st v in W holds

a * v in v + W

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 a being Element of GF

for v being Element of V

for W being Subspace of V st v in W holds

a * v in v + W

let a be Element of GF; :: thesis: for v being Element of V

for W being Subspace of V st v in W holds

a * v in v + W

let v be Element of V; :: thesis: for W being Subspace of V st v in W holds

a * v in v + W

let W be Subspace of V; :: thesis: ( v in W implies a * v in v + W )

assume v in W ; :: thesis: a * v in v + W

then ( v + W = the carrier of W & a * v in W ) by Lm4, Th21;

hence a * v in v + W ; :: thesis: verum

for a being Element of GF

for v being Element of V

for W being Subspace of V st v in W holds

a * v in v + W

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 a being Element of GF

for v being Element of V

for W being Subspace of V st v in W holds

a * v in v + W

let a be Element of GF; :: thesis: for v being Element of V

for W being Subspace of V st v in W holds

a * v in v + W

let v be Element of V; :: thesis: for W being Subspace of V st v in W holds

a * v in v + W

let W be Subspace of V; :: thesis: ( v in W implies a * v in v + W )

assume v in W ; :: thesis: a * v in v + W

then ( v + W = the carrier of W & a * v in W ) by Lm4, Th21;

hence a * v in v + W ; :: thesis: verum