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

theorem
  M1 is Nilpotent & M2 is Nilpotent & M1*M2=-M2*M1 implies 
  M1+M2 is Nilpotent
proof
  assume that
A1: M1 is Nilpotent & M2 is Nilpotent and
A2: M1*M2=-M2*M1;
A4: M1*M1=0.(K,n) & M2*M2=0.(K,n) by A1;
A5: len (M1*M1)=n & width (M1*M1)=n by MATRIX_0:24;
A6: len M1=n & width M1=n by MATRIX_0:24;
A7: len (M1*M1+M2*M1)=n & width (M1*M1+M2*M1)=n by MATRIX_0:24;
A8: len (M2*M1)=n & width (M2*M1)=n by MATRIX_0:24;
A9: len (M1*M2)=n & width (M1*M2)=n by MATRIX_0:24;
A10: len M2=n & width M2=n by MATRIX_0:24;
  len (M1+M2)=n & width (M1+M2)=n by MATRIX_0:24;
  then (M1+M2)*(M1+M2)=(M1+M2)*M1+(M1+M2)*M2 by A6,A10,MATRIX_4:62
    .=M1*M1+M2*M1+(M1+M2)*M2 by A6,A10,MATRIX_4:63
    .=M1*M1+M2*M1+(M1*M2+M2*M2) by A6,A10,MATRIX_4:63
    .=M1*M1+M2*M1+M1*M2+M2*M2 by A9,A7,MATRIX_3:3
    .=(M1*M1+(M2*M1+(-(M2*M1))))+M2*M2 by A2,A5,A8,MATRIX_3:3
    .=M1*M1+0.(K,n,n)+M2*M2 by A8,MATRIX_4:2
    .=0.(K,n)+0.(K,n) by A4,MATRIX_3:4
    .=0.(K,n,n) by MATRIX_3:4
    .=0.(K,n);
  hence thesis;
end;
