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

theorem
  M1 is Nilpotent & M1 commutes_with M2 implies M1*M2 is Nilpotent
proof
  assume that
A1: M1 is Nilpotent and
A2: M1 commutes_with M2;
A4: len M1=n & width M1=n by MATRIX_0:24;
A5: width M2=n by MATRIX_0:24;
A6: width (M2*M1)=n by MATRIX_0:24;
A7: len M2=n by MATRIX_0:24;
  (M1*M2)*(M1*M2)=(M2*M1)*(M1*M2) by A2,MATRIX_6:def 1
    .=(M2*M1*M1)*M2 by A4,A7,A6,MATRIX_3:33
    .=(M2*(M1*M1))*M2 by A4,A5,MATRIX_3:33
    .=(M2*(0.(K,n)))*M2 by A1
    .=(0.(K,n,n))*M2 by A7,A5,MATRIX_6:2
    .=0.(K,n) by A7,A5,MATRIX_6:1;
  hence thesis;
end;
