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 that
A1: ( M1 is line_circulant & M2 is line_circulant ) and
A2: M3 is line_circulant ; :: thesis: ((a * M1) + (b * M2)) - (c * M3) is line_circulant
c * M3 is line_circulant by A2, Th6;
then A3: - (c * M3) is line_circulant by Th11;
( a * M1 is line_circulant & b * M2 is line_circulant ) by A1, Th6;
then (a * M1) + (b * M2) is line_circulant by Th7;
hence ((a * M1) + (b * M2)) - (c * M3) is line_circulant by A3, Th7; :: thesis: verum