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 * M2 = - (M2 * M1) 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 * M2 = - (M2 * M1) holds
M1 + M2 is Idempotent
let M1, M2 be Matrix of n,K; :: thesis: ( M1 is Idempotent & M2 is Idempotent & M1 * M2 = - (M2 * M1) implies M1 + M2 is Idempotent )
assume A1: ( M1 is Idempotent & M2 is Idempotent & M1 * M2 = - (M2 * M1) ) ; :: thesis: M1 + M2 is Idempotent
then A2: ( M1 * M1 = M1 & M2 * M2 = M2 ) by Def1;

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

suppose A3: n > 0 ; :: thesis: M1 + M2 is Idempotent

A4: ( len M1 = n & width M1 = n & len M2 = n & width M2 = n ) by MATRIX_1:25;
A5: ( len (M1 + M2) = n & width (M1 + M2) = n ) by MATRIX_1:25;
A6: ( len (M1 * M1) = n & width (M1 * M1) = n & len (M1 * M2) = n & width (M1 * M2) = n ) by MATRIX_1:25;
A7: ( len (M2 * M1) = n & width (M2 * M1) = n & len (M2 * M2) = n & width (M2 * M2) = n & len (- (M2 * M1)) = n & width (- (M2 * M1)) = n ) by MATRIX_1:25;
A8: ( len ((M1 * M1) + (M2 * M1)) = n & width ((M1 * M1) + (M2 * M1)) = n & len ((M1 * M2) + (M2 * M2)) = n & width ((M1 * M2) + (M2 * M2)) = n ) by MATRIX_1:25;
(M1 + M2) * (M1 + M2) = ((M1 + M2) * M1) + ((M1 + M2) * M2) by A3, A4, A5, MATRIX_4:62
.= ((M1 * M1) + (M2 * M1)) + ((M1 + M2) * M2) by A3, A4, MATRIX_4:63
.= ((M1 * M1) + (M2 * M1)) + ((M1 * M2) + (M2 * M2)) by A3, A4, MATRIX_4:63
.= (((M1 * M1) + (M2 * M1)) + (M1 * M2)) + (M2 * M2) by A6, A7, A8, MATRIX_3:5
.= ((M1 * M1) + ((M2 * M1) + (- (M2 * M1)))) + (M2 * M2) by A1, A6, A7, MATRIX_3:5
.= ((M1 * M1) + (0. K,n,n)) + (M2 * M2) by A7, MATRIX_4:2
.= M1 + M2 by A2, MATRIX_3:6 ;
hence M1 + M2 is Idempotent by Def1; :: thesis: verum

end;

suppose n = 0 ; :: thesis: M1 + M2 is Idempotent

then (M1 + M2) * (M1 + M2) = M1 + M2 by MATRIX_1:36;
hence M1 + M2 is Idempotent by Def1; :: thesis: verum

end;

end;