let x be Element of L; :: according to VECTSP_1:def 13 :: thesis: for b₁, b₂ being Element of the carrier of (Formal-Series L) holds x * (b₁ + b₂) = (x * b₁) + (x * b₂)
let v, w be Element of (Formal-Series L); :: thesis: x * (v + w) = (x * v) + (x * w)
reconsider p = v, q = w as sequence of L by Def2;
reconsider x9 = x as Element of L ;
reconsider k = v + w as Element of (Formal-Series L) ;
A1: x * v = x * p by Def2;
reconsider r = k as sequence of L by Def2;
A2: x * w = x * q by Def2;
x * k = x * r by Def2;
hence x * (v + w) = x * (p + q) by Def2
.= (x9 * p) + (x9 * q) by Th6
.= (x * v) + (x * w) by A1, A2, Def2 ;
:: thesis: verum

let x, y be Element of L; :: according to VECTSP_1:def 14 :: thesis: for b₁ being Element of the carrier of (Formal-Series L) holds (x + y) * b₁ = (x * b₁) + (y * b₁)
let v be Element of (Formal-Series L); :: thesis: (x + y) * v = (x * v) + (y * v)
reconsider p = v as sequence of L by Def2;
reconsider x9 = x, y9 = y as Element of L ;
A3: x * v = x * p by Def2;
A4: y * v = y * p by Def2;
thus (x + y) * v = (x9 + y9) * p by Def2
.= (x9 * p) + (y9 * p) by Th7
.= (x * v) + (y * v) by A3, A4, Def2 ; :: thesis: verum

let x, y be Element of L; :: according to VECTSP_1:def 15 :: thesis: for b₁ being Element of the carrier of (Formal-Series L) holds (x * y) * b₁ = x * (y * b₁)
let v be Element of (Formal-Series L); :: thesis: (x * y) * v = x * (y * v)
reconsider p = v as sequence of L by Def2;
reconsider x9 = x, y9 = y as Element of L ;
A5: y * v = y * p by Def2;
thus (x * y) * v = (x9 * y9) * p by Def2
.= x9 * (y9 * p) by Th8
.= x * (y * v) by A5, Def2 ; :: thesis: verum