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

theorem Th3:
  for R being commutative associative non empty multLoopStr
  for a,b,c,d being Element of R st
    (b divides a & d divides c) holds (b * d) divides (a * c)
proof
  let R be commutative associative non empty multLoopStr;
  let a,b,c,d be Element of R;
  assume that
A1: b divides a and
A2: d divides c;
  consider x being Element of R such that
A3: b * x = a by A1;
  consider y being Element of R such that
A4: d * y = c by A2;
  (b * d) * (y * x) = ((b * d) * y) * x by GROUP_1:def 3
    .= (b * c) * x by A4,GROUP_1:def 3
    .= a * c by A3,GROUP_1:def 3;
  hence thesis;
end;
