let n be Nat; :: thesis: 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; :: thesis: 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; :: thesis: ( M1 is invertible & M1 * M2 = M1 * M3 implies M2 = M3 )
assume that
A1: M1 is invertible and
A2: M1 * M2 = M1 * M3 ; :: thesis: 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 ; :: thesis: verum