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

theorem Th2:
  for R being associative non empty multLoopStr
  for a,b,c being Element of R holds
    a divides b & b divides c implies a divides c
proof
  let R be associative non empty multLoopStr;
  let A,B,C be Element of R;
  now
    assume that
A1: A divides B and
A2: B divides C;
    consider D being Element of R such that
A3: A * D = B by A1;
    consider E being Element of R such that
A4: B * E = C by A2;
    A * (D * E) = C by A3,A4,GROUP_1:def 3;
    hence thesis;
  end;
  hence thesis;
end;
