let n be Nat; :: thesis: for K being Field
for M1, M2 being Matrix of n,K st M1 is Idempotent & M2 is Idempotent & M1 commutes_with M2 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 & M1 commutes_with M2 holds
(M1 @) * (M2 @) is Idempotent
let M1, M2 be Matrix of n,K; :: thesis: ( M1 is Idempotent & M2 is Idempotent & M1 commutes_with M2 implies (M1 @) * (M2 @) is Idempotent )
assume that
A1: M1 is Idempotent and
A2: M2 is Idempotent and
A3: M1 commutes_with M2 ; :: thesis: (M1 @) * (M2 @) is Idempotent
set M3 = (M1 @) * (M2 @);

per cases ( n > 0 or n = 0 ) by NAT_1:3;

suppose A4: n > 0 ; :: thesis: (M1 @) * (M2 @) is Idempotent

A5: M1 * M1 = M1 by A1;
A6: len M1 = n by MATRIX_0:24;
A7: len (M1 @) = n by MATRIX_0:24;
A8: width (M2 @) = n by MATRIX_0:24;
A9: width M2 = n by MATRIX_0:24;
A10: len M2 = n by MATRIX_0:24;
A11: width M1 = n by MATRIX_0:24;
A12: ( width (M1 @) = n & len (M2 @) = n ) by MATRIX_0:24;
width ((M1 @) * (M2 @)) = n by MATRIX_0:24;
then ((M1 @) * (M2 @)) * ((M1 @) * (M2 @)) = (((M1 @) * (M2 @)) * (M1 @)) * (M2 @) by A7, A12, MATRIX_3:33
.= ((M1 @) * ((M2 @) * (M1 @))) * (M2 @) by A7, A12, A8, MATRIX_3:33
.= ((M1 @) * ((M1 * M2) @)) * (M2 @) by A4, A11, A10, A9, MATRIX_3:22
.= ((M1 @) * ((M2 * M1) @)) * (M2 @) by A3, MATRIX_6:def 1
.= ((M1 @) * ((M1 @) * (M2 @))) * (M2 @) by A4, A6, A11, A9, MATRIX_3:22
.= (((M1 @) * (M1 @)) * (M2 @)) * (M2 @) by A7, A12, MATRIX_3:33
.= (((M1 * M1) @) * (M2 @)) * (M2 @) by A4, A6, A11, MATRIX_3:22
.= (M1 @) * ((M2 @) * (M2 @)) by A5, A12, A8, MATRIX_3:33
.= (M1 @) * ((M2 * M2) @) by A4, A10, A9, MATRIX_3:22
.= (M1 @) * (M2 @) by A2 ;
hence (M1 @) * (M2 @) is Idempotent ; :: thesis: verum

end;

suppose n = 0 ; :: thesis: (M1 @) * (M2 @) is Idempotent

then ((M1 @) * (M2 @)) * ((M1 @) * (M2 @)) = (M1 @) * (M2 @) by MATRIX_0:45;
hence (M1 @) * (M2 @) is Idempotent ; :: thesis: verum

end;

end;