let n be Nat; :: thesis: for K being Field
for M1, M2 being Matrix of n,K st M1 is invertible & M1 * M2 = 1. K,n holds
M1 is_reverse_of M2
let K be Field; :: thesis: for M1, M2 being Matrix of n,K st M1 is invertible & M1 * M2 = 1. K,n holds
M1 is_reverse_of M2
let M1, M2 be Matrix of n,K; :: thesis: ( M1 is invertible & M1 * M2 = 1. K,n implies M1 is_reverse_of M2 )
A1: ( width M1 = n & len M1 = n ) by MATRIX_1:25;
A2: ( len M2 = n & width (M1 ~ ) = n ) by MATRIX_1:25;
assume that
A3: M1 is invertible and
A4: M1 * M2 = 1. K,n ; :: thesis: M1 is_reverse_of M2
A5: M1 ~ is_reverse_of M1 by A3, Def4;
(M1 ~ ) * (M1 * M2) = M1 ~ by A4, MATRIX_3:21;
then ((M1 ~ ) * M1) * M2 = M1 ~ by A1, A2, MATRIX_3:35;
then (1. K,n) * M2 = M1 ~ by A5, Def2;
then M2 = M1 ~ by MATRIX_3:20;
hence M1 is_reverse_of M2 by A3, Def4; :: thesis: verum