let K be Field; :: thesis: for a being Element of K
for p, q being FinSequence of K st p is first-line-of-circulant & q is first-line-of-circulant & len p = len q holds
(a * (LCirc p)) + (a * (LCirc q)) = LCirc (a * (p + q))
let a be Element of K; :: thesis: for p, q being FinSequence of K st p is first-line-of-circulant & q is first-line-of-circulant & len p = len q holds
(a * (LCirc p)) + (a * (LCirc q)) = LCirc (a * (p + q))
let p, q be FinSequence of K; :: thesis: ( p is first-line-of-circulant & q is first-line-of-circulant & len p = len q implies (a * (LCirc p)) + (a * (LCirc q)) = LCirc (a * (p + q)) )
assume that
A1: ( p is first-line-of-circulant & q is first-line-of-circulant ) and
A2: len p = len q ; :: thesis: (a * (LCirc p)) + (a * (LCirc q)) = LCirc (a * (p + q))
A3: ( len (LCirc p) = len p & width (LCirc p) = len p ) by MATRIX_1:24;
( len (LCirc q) = len p & width (LCirc q) = len p ) by A2, MATRIX_1:24;
then (a * (LCirc p)) + (a * (LCirc q)) = a * ((LCirc p) + (LCirc q)) by A3, MATRIX_5:20
.= a * (LCirc (p + q)) by A1, A2, Th34
.= LCirc (a * (p + q)) by A1, A2, Th33, Th42 ;
hence (a * (LCirc p)) + (a * (LCirc q)) = LCirc (a * (p + q)) ; :: thesis: verum