let n be Nat; for K being Field
for M1, M2, M3 being Matrix of n,K st M1 is invertible & M1 * M2 = M1 * M3 holds
M2 = M3
let K be Field; for M1, M2, M3 being Matrix of n,K st M1 is invertible & M1 * M2 = M1 * M3 holds
M2 = M3
let M1, M2, M3 be Matrix of n,K; ( M1 is invertible & M1 * M2 = M1 * M3 implies M2 = M3 )
assume that
A1:
M1 is invertible
and
A2:
M1 * M2 = M1 * M3
; M2 = M3
A3:
M1 ~ is_reverse_of M1
by A1, MATRIX_6:def 4;
A4:
len M2 = n
by MATRIX_0:24;
A5:
( width M1 = n & len M1 = n )
by MATRIX_0:24;
A6:
len M3 = n
by MATRIX_0:24;
A7:
width (M1 ~) = n
by MATRIX_0:24;
M2 =
(1. (K,n)) * M2
by MATRIX_3:18
.=
((M1 ~) * M1) * M2
by A3, MATRIX_6:def 2
.=
(M1 ~) * (M1 * M3)
by A2, A5, A4, A7, MATRIX_3:33
.=
((M1 ~) * M1) * M3
by A5, A6, A7, MATRIX_3:33
.=
(1. (K,n)) * M3
by A3, MATRIX_6:def 2
.=
M3
by MATRIX_3:18
;
hence
M2 = M3
; verum