let n be Element of NAT ; :: thesis: for K being Field
for a, b, c being Element of K
for M1, M2, M3 being Matrix of n,K st M1 is line_circulant & M2 is line_circulant & M3 is line_circulant holds
((a * M1) + (b * M2)) + (c * M3) is line_circulant
let K be Field; :: thesis: for a, b, c being Element of K
for M1, M2, M3 being Matrix of n,K st M1 is line_circulant & M2 is line_circulant & M3 is line_circulant holds
((a * M1) + (b * M2)) + (c * M3) is line_circulant
let a, b, c be Element of K; :: thesis: for M1, M2, M3 being Matrix of n,K st M1 is line_circulant & M2 is line_circulant & M3 is line_circulant holds
((a * M1) + (b * M2)) + (c * M3) is line_circulant
let M1, M2, M3 be Matrix of n,K; :: thesis: ( M1 is line_circulant & M2 is line_circulant & M3 is line_circulant implies ((a * M1) + (b * M2)) + (c * M3) is line_circulant )
assume ( M1 is line_circulant & M2 is line_circulant & M3 is line_circulant ) ; :: thesis: ((a * M1) + (b * M2)) + (c * M3) is line_circulant
then ( a * M1 is line_circulant & b * M2 is line_circulant & c * M3 is line_circulant ) by Th6;
hence ((a * M1) + (b * M2)) + (c * M3) is line_circulant by Th8; :: thesis: verum