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 col_circulant & M2 is col_circulant holds
(a * M1) - (b * M2) is col_circulant
let K be Field; for a, b being Element of K
for M1, M2 being Matrix of n,K st M1 is col_circulant & M2 is col_circulant holds
(a * M1) - (b * M2) is col_circulant
let a, b be Element of K; for M1, M2 being Matrix of n,K st M1 is col_circulant & M2 is col_circulant holds
(a * M1) - (b * M2) is col_circulant
let M1, M2 be Matrix of n,K; ( M1 is col_circulant & M2 is col_circulant implies (a * M1) - (b * M2) is col_circulant )
assume that
A1:
M1 is col_circulant
and
A2:
M2 is col_circulant
; (a * M1) - (b * M2) is col_circulant
b * M2 is col_circulant
by A2, Th20;
then A3:
- (b * M2) is col_circulant
by Th25;
a * M1 is col_circulant
by A1, Th20;
hence
(a * M1) - (b * M2) is col_circulant
by A3, Th21; verum