reserve i,j,n for Nat,
  K for Field,
  a for Element of K,
  M,M1,M2,M3,M4 for Matrix of n,K;
reserve A for Matrix of K;

theorem
  for R being Ring, M1,M2 being Matrix of n,R
  holds M1*M2=0.(R,n,n) & M1 is invertible implies M1 commutes_with M2
proof
  let R be Ring;
  let M1,M2 be Matrix of n,R;
  assume that
A2: M1*M2=0.(R,n,n) and
A3: M1 is invertible;
A4: M1~ is_reverse_of M1 by A3,Def4;
A5: len M2 = n by MATRIX_0:24;
A6: len M1=n & width M1=n by MATRIX_0:24;
A7: len (M1~) =n by MATRIX_0:24;
A8: width (M1~) = n by MATRIX_0:24;
  M2=(1.(R,n))*M2 by MATRIX_3:18
    .=(M1~*M1)*M2 by A4
    .=M1~*(0.(R,n,n)) by A2,A6,A5,A8,MATRIX_3:33
    .=0.(R,n,n) by A7,A8,Th2;
  hence thesis by Th3;
end;
