let K be Field; :: thesis: for a being Element of K
for p, q being FinSequence of K st p is first-line-of-anti-circular & q is first-line-of-anti-circular & len p = len q & len p > 0 holds
(a * (ACirc p)) + (a * (ACirc q)) = ACirc (a * (p + q))
let a be Element of K; :: thesis: for p, q being FinSequence of K st p is first-line-of-anti-circular & q is first-line-of-anti-circular & len p = len q & len p > 0 holds
(a * (ACirc p)) + (a * (ACirc q)) = ACirc (a * (p + q))
let p, q be FinSequence of K; :: thesis: ( p is first-line-of-anti-circular & q is first-line-of-anti-circular & len p = len q & len p > 0 implies (a * (ACirc p)) + (a * (ACirc q)) = ACirc (a * (p + q)) )
set n = len p;
assume A1: ( p is first-line-of-anti-circular & q is first-line-of-anti-circular & len p = len q & len p > 0 ) ; :: thesis: (a * (ACirc p)) + (a * (ACirc q)) = ACirc (a * (p + q))
then A3: ( a * p is first-line-of-anti-circular & a * q is first-line-of-anti-circular & p + q is first-line-of-anti-circular ) by Th60, Th62;
A5: ( len (ACirc q) = len p & len (ACirc p) = len p & width (ACirc q) = len p & width (ACirc p) = len p ) by A1, MATRIX_1:25;
(a * (ACirc p)) + (a * (ACirc q)) = a * ((ACirc p) + (ACirc q)) by A1, A5, MATRIX_5:36
.= a * (ACirc (p + q)) by A1, Th61
.= ACirc (a * (p + q)) by A3, Th63 ;
hence (a * (ACirc p)) + (a * (ACirc q)) = ACirc (a * (p + q)) ; :: thesis: verum