let n be Nat; :: thesis: for K being Field
for M2, M1 being Matrix of n,K st M2 is invertible & M1 is_congruent_Matrix_of M2 & n > 0 holds
M1 is invertible
let K be Field; :: thesis: for M2, M1 being Matrix of n,K st M2 is invertible & M1 is_congruent_Matrix_of M2 & n > 0 holds
M1 is invertible
let M2, M1 be Matrix of n,K; :: thesis: ( M2 is invertible & M1 is_congruent_Matrix_of M2 & n > 0 implies M1 is invertible )
assume A1: ( M2 is invertible & M1 is_congruent_Matrix_of M2 & n > 0 ) ; :: thesis: M1 is invertible
then consider M4 being Matrix of n,K such that
A2: ( M4 is invertible & M1 = ((M4 @ ) * M2) * M4 ) by Def6;
A3: ( width M4 = n & len M4 = n & width (M4 ~ ) = n & len (M4 ~ ) = n & len M4 > 0 ) by A1, MATRIX_1:25;
A4: M4 ~ is_reverse_of M4 by A2, MATRIX_6:def 4;
then ( (M4 ~ ) * M4 = M4 * (M4 ~ ) & (M4 ~ ) * M4 = 1. K,n ) by MATRIX_6:def 2;
then A5: ( ((M4 ~ ) * M4) @ = (M4 @ ) * ((M4 ~ ) @ ) & (M4 * (M4 ~ )) @ = ((M4 ~ ) @ ) * (M4 @ ) & ((M4 ~ ) * M4) @ = 1. K,n ) by A3, MATRIX_3:24, MATRIX_6:10;
set M5 = M4 @ ;
set M6 = (M4 ~ ) @ ;
( (M4 @ ) * ((M4 ~ ) @ ) = ((M4 ~ ) @ ) * (M4 @ ) & (M4 @ ) * ((M4 ~ ) @ ) = 1. K,n ) by A4, A5, MATRIX_6:def 2;
then M4 @ is_reverse_of (M4 ~ ) @ by MATRIX_6:def 2;
then M4 @ is invertible by MATRIX_6:def 3;
then (M4 @ ) * M2 is invertible by A1, MATRIX_6:37;
hence M1 is invertible by A2, MATRIX_6:37; :: thesis: verum