reserve i,n for Nat,
  K for Field,
  M1,M2,M3,M4 for Matrix of n,K;

theorem
  M1 is Idempotent & M2 is Idempotent & M1 commutes_with M2 & M1*
  M2=0.(K,n) implies M1+M2 is Idempotent
proof
  assume that
A2: M1 is Idempotent & M2 is Idempotent and
A3: M1 commutes_with M2 and
A4: M1*M2=0.(K,n);
A5: M1*M2=0.(K,n,n) by A4;
A6: M1*M1=M1 & M2*M2=M2 by A2;
  (M1+M2)*(M1+M2)=M1*M1+0.(K,n)+0.(K,n)+M2*M2 by A3,A4,MATRIX_6:35
    .=M1*M1+0.(K,n)+M2*M2 by A5,MATRIX_3:4
    .=M1+M2 by A6,A5,MATRIX_3:4;
  hence thesis;
end;
