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

for w being Element of W st w = v holds

a * w = a * v

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

for w being Element of W st w = v holds

a * w = a * v

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

for W being Subspace of V

for w being Element of W st w = v holds

a * w = a * v

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

for w being Element of W st w = v holds

a * w = a * v

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

a * w = a * v

let w be Element of W; :: thesis: ( w = v implies a * w = a * v )

assume A1: w = v ; :: thesis: a * w = a * v

a * w = ( the lmult of V | [: the carrier of GF, the carrier of W:]) . [a,w] by Def2;

hence a * w = a * v by A1, FUNCT_1:49; :: thesis: verum

for a being Element of GF

for v being Element of V

for W being Subspace of V

for w being Element of W st w = v holds

a * w = a * v

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

for w being Element of W st w = v holds

a * w = a * v

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

for W being Subspace of V

for w being Element of W st w = v holds

a * w = a * v

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

for w being Element of W st w = v holds

a * w = a * v

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

a * w = a * v

let w be Element of W; :: thesis: ( w = v implies a * w = a * v )

assume A1: w = v ; :: thesis: a * w = a * v

a * w = ( the lmult of V | [: the carrier of GF, the carrier of W:]) . [a,w] by Def2;

hence a * w = a * v by A1, FUNCT_1:49; :: thesis: verum