let n be Element of NAT ; :: thesis: 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; :: thesis: 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; :: thesis: 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; :: thesis: ( 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 ) ; :: thesis: (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; :: thesis: verum