let n be Element of NAT ; 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; 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; 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; ( 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
; ((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) - (b * M2) is line_circulant
by A1, Th13;
hence
((a * M1) - (b * M2)) - (c * M3) is line_circulant
by A3, Th7; verum