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