
theorem Th21:
  for R being left_zeroed right_zeroed add-associative
add-cancelable distributive non empty doubleLoopStr, a,b being Element of R,
  n being Element of NAT holds (a * n) * b = a * (n * b)
proof
  let R be left_zeroed right_zeroed distributive add-cancelable
add-associative non empty doubleLoopStr, a,b be Element of R, n be Element of
  NAT;
  thus (a * n) * b = (n * a) * b by Th17
    .= n * (a * b) by Th19
    .= (a * b) * n by Th17
    .= a * (n * b) by Th20;
end;
