let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative commutative 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, b being Element of GF

for v being Element of V holds (a - b) * v = (a * v) - (b * 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, b being Element of GF

for v being Element of V holds (a - b) * v = (a * v) - (b * v)

let a, b be Element of GF; :: thesis: for v being Element of V holds (a - b) * v = (a * v) - (b * v)

let v be Element of V; :: thesis: (a - b) * v = (a * v) - (b * v)

thus (a - b) * v = (a * v) + ((- b) * v) by VECTSP_1:def 15

.= (a * v) - (b * v) by VECTSP_1:21 ; :: thesis: verum

for a, b being Element of GF

for v being Element of V holds (a - b) * v = (a * v) - (b * 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, b being Element of GF

for v being Element of V holds (a - b) * v = (a * v) - (b * v)

let a, b be Element of GF; :: thesis: for v being Element of V holds (a - b) * v = (a * v) - (b * v)

let v be Element of V; :: thesis: (a - b) * v = (a * v) - (b * v)

thus (a - b) * v = (a * v) + ((- b) * v) by VECTSP_1:def 15

.= (a * v) - (b * v) by VECTSP_1:21 ; :: thesis: verum