let n be Nat; :: thesis: for K being Field
for M1 being Matrix of n,K st M1 is Self_Reversible & M1 is Idempotent holds
M1 = 1. K,n
let K be Field; :: thesis: for M1 being Matrix of n,K st M1 is Self_Reversible & M1 is Idempotent holds
M1 = 1. K,n
let M1 be Matrix of n,K; :: thesis: ( M1 is Self_Reversible & M1 is Idempotent implies M1 = 1. K,n )
assume A1: ( M1 is Self_Reversible & M1 is Idempotent ) ; :: thesis: M1 = 1. K,n
then M1 is invertible by Def4;
then M1 ~ is_reverse_of M1 by MATRIX_6:def 4;
then 1. K,n = M1 * (M1 ~ ) by MATRIX_6:def 2
.= M1 * M1 by A1, Def4 ;
hence M1 = 1. K,n by A1, Def1; :: thesis: verum