reserve X,Y for set;
reserve R for domRing-like commutative Ring;
reserve c for Element of R;

theorem
  for R being non empty multLoopStr
  for a,b being Element of R holds
  a divides a * b & (R is commutative implies b divides a * b)
proof
  let R be non empty multLoopStr;
  let a,b be Element of R;
  thus a divides a * b;
  assume
A1: R is commutative;
  take a;
  thus thesis by A1,GROUP_1:def 12;
end;
