let n be Element of NAT ; for K being Field
for a, b being Element of K
for M1, M2 being Matrix of n,K st M1 is line_circulant & M2 is line_circulant holds
(a * M1) + (b * M2) is line_circulant
let K be Field; for a, b being Element of K
for M1, M2 being Matrix of n,K st M1 is line_circulant & M2 is line_circulant holds
(a * M1) + (b * M2) is line_circulant
let a, b be Element of K; for M1, M2 being Matrix of n,K st M1 is line_circulant & M2 is line_circulant holds
(a * M1) + (b * M2) is line_circulant
let M1, M2 be Matrix of n,K; ( M1 is line_circulant & M2 is line_circulant implies (a * M1) + (b * M2) is line_circulant )
assume
( M1 is line_circulant & M2 is line_circulant )
; (a * M1) + (b * M2) is line_circulant
then
( a * M1 is line_circulant & b * M2 is line_circulant )
by Th6;
hence
(a * M1) + (b * M2) is line_circulant
by Th7; verum