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 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 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 implies (a * (ACirc p)) + (a * (ACirc q)) = ACirc (a * (p + q)) )
assume that
A1: ( p is first-line-of-anti-circular & q is first-line-of-anti-circular ) and
A2: len p = len q ; :: thesis: (a * (ACirc p)) + (a * (ACirc q)) = ACirc (a * (p + q))
A3: ( len (ACirc p) = len p & width (ACirc p) = len p ) by MATRIX_1:24;
( len (ACirc q) = len p & width (ACirc q) = len p ) by A2, MATRIX_1:24;
then (a * (ACirc p)) + (a * (ACirc q)) = a * ((ACirc p) + (ACirc q)) by A3, MATRIX_5:20
.= a * (ACirc (p + q)) by A1, A2, Th61
.= ACirc (a * (p + q)) by A1, A2, Th60, Th63 ;
hence (a * (ACirc p)) + (a * (ACirc q)) = ACirc (a * (p + q)) ; :: thesis: verum