let n be Nat; for K being Field
for M1, M2, M3 being Matrix of n,K st M1 is invertible & M2 * M1 = M3 * M1 holds
M2 = M3
let K be Field; for M1, M2, M3 being Matrix of n,K st M1 is invertible & M2 * M1 = M3 * M1 holds
M2 = M3
let M1, M2, M3 be Matrix of n,K; ( M1 is invertible & M2 * M1 = M3 * M1 implies M2 = M3 )
assume that
A1:
M1 is invertible
and
A2:
M2 * M1 = M3 * M1
; M2 = M3
A3:
M1 ~ is_reverse_of M1
by A1, MATRIX_6:def 4;
A4:
width M2 = n
by MATRIX_0:24;
A5:
( width M1 = n & len M1 = n )
by MATRIX_0:24;
A6:
width M3 = n
by MATRIX_0:24;
A7:
len (M1 ~) = n
by MATRIX_0:24;
M2 =
M2 * (1. (K,n))
by MATRIX_3:19
.=
M2 * (M1 * (M1 ~))
by A3, MATRIX_6:def 2
.=
(M3 * M1) * (M1 ~)
by A2, A5, A4, A7, MATRIX_3:33
.=
M3 * (M1 * (M1 ~))
by A5, A6, A7, MATRIX_3:33
.=
M3 * (1. (K,n))
by A3, MATRIX_6:def 2
.=
M3
by MATRIX_3:19
;
hence
M2 = M3
; verum