let n be Nat; :: thesis: for K being Field
for M1, M2 being Matrix of n,K st M1 is Idempotent & M2 is Idempotent & M2 * M1 = 1. K,n holds
M1 * M2 is Idempotent
let K be Field; :: thesis: for M1, M2 being Matrix of n,K st M1 is Idempotent & M2 is Idempotent & M2 * M1 = 1. K,n holds
M1 * M2 is Idempotent
let M1, M2 be Matrix of n,K; :: thesis: ( M1 is Idempotent & M2 is Idempotent & M2 * M1 = 1. K,n implies M1 * M2 is Idempotent )
assume A1: ( M1 is Idempotent & M2 is Idempotent & M2 * M1 = 1. K,n ) ; :: thesis: M1 * M2 is Idempotent
A2: ( len M1 = n & width M1 = n & len M2 = n & width M2 = n ) by MATRIX_1:25;
( len (M1 * M2) = n & width (M1 * M2) = n ) by MATRIX_1:25;
then (M1 * M2) * (M1 * M2) = ((M1 * M2) * M1) * M2 by A2, MATRIX_3:35
.= (M1 * (1. K,n)) * M2 by A1, A2, MATRIX_3:35
.= M1 * M2 by MATRIX_3:21 ;
hence M1 * M2 is Idempotent by Def1; :: thesis: verum