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 ;

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 ;

hence M1 = 1. (K,n) by A1; :: thesis: verum

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 ;

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 ;

hence M1 = 1. (K,n) by A1; :: thesis: verum