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, b being Element of GF

for L being Linear_Combination of V holds (a + b) * L = (a * L) + (b * L)

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 L being Linear_Combination of V holds (a + b) * L = (a * L) + (b * L)

let a, b be Element of GF; :: thesis: for L being Linear_Combination of V holds (a + b) * L = (a * L) + (b * L)

let L be Linear_Combination of V; :: thesis: (a + b) * L = (a * L) + (b * L)

let v be Element of V; :: according to VECTSP_6:def 7 :: thesis: ((a + b) * L) . v = ((a * L) + (b * L)) . v

thus ((a + b) * L) . v = (a + b) * (L . v) by Def9

.= (a * (L . v)) + (b * (L . v)) by VECTSP_1:def 7

.= ((a * L) . v) + (b * (L . v)) by Def9

.= ((a * L) . v) + ((b * L) . v) by Def9

.= ((a * L) + (b * L)) . v by Th22 ; :: thesis: verum

for a, b being Element of GF

for L being Linear_Combination of V holds (a + b) * L = (a * L) + (b * L)

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 L being Linear_Combination of V holds (a + b) * L = (a * L) + (b * L)

let a, b be Element of GF; :: thesis: for L being Linear_Combination of V holds (a + b) * L = (a * L) + (b * L)

let L be Linear_Combination of V; :: thesis: (a + b) * L = (a * L) + (b * L)

let v be Element of V; :: according to VECTSP_6:def 7 :: thesis: ((a + b) * L) . v = ((a * L) + (b * L)) . v

thus ((a + b) * L) . v = (a + b) * (L . v) by Def9

.= (a * (L . v)) + (b * (L . v)) by VECTSP_1:def 7

.= ((a * L) . v) + (b * (L . v)) by Def9

.= ((a * L) . v) + ((b * L) . v) by Def9

.= ((a * L) + (b * L)) . v by Th22 ; :: thesis: verum