let K be Field; :: thesis: for a being Element of K
for p, q being FinSequence of K st p is first-col-of-circulant & q is first-col-of-circulant & len p = len q & len p > 0 holds
(a * (CCirc p)) + (a * (CCirc q)) = CCirc (a * (p + q))
let a be Element of K; :: thesis: for p, q being FinSequence of K st p is first-col-of-circulant & q is first-col-of-circulant & len p = len q & len p > 0 holds
(a * (CCirc p)) + (a * (CCirc q)) = CCirc (a * (p + q))
let p, q be FinSequence of K; :: thesis: ( p is first-col-of-circulant & q is first-col-of-circulant & len p = len q & len p > 0 implies (a * (CCirc p)) + (a * (CCirc q)) = CCirc (a * (p + q)) )
assume that
A1: ( p is first-col-of-circulant & q is first-col-of-circulant ) and
A2: len p = len q ; :: thesis: ( not len p > 0 or (a * (CCirc p)) + (a * (CCirc q)) = CCirc (a * (p + q)) )
A3: ( len (CCirc p) = len p & width (CCirc p) = len p ) by MATRIX_0:24;
( len (CCirc q) = len p & width (CCirc q) = len p ) by A2, MATRIX_0:24;
then (a * (CCirc p)) + (a * (CCirc q)) = a * ((CCirc p) + (CCirc q)) by A3, MATRIX_5:20
.= a * (CCirc (p + q)) by A1, A2, Th38
.= CCirc (a * (p + q)) by A1, A2, Th37, Th47 ;
hence ( not len p > 0 or (a * (CCirc p)) + (a * (CCirc q)) = CCirc (a * (p + q)) ) ; :: thesis: verum