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

theorem Th14:
  for a,b,c,d being Element of R holds
    b <> 0.R & d <> 0.R & b divides a & d divides c implies
      (a/b) * (c/d) = (a * c) / (b * d)
proof
  let a,b,c,d be Element of R;
  assume that
A1: b <> 0.R & d <> 0.R and
A2: b divides a & d divides c;
A3: (b * d) divides (a * c) by A2,Th3;
  set z = (a * c)/(b * d);
  set y = c/d;
  set x = a/b;
A4: (b * d) <> 0.R by A1,VECTSP_2:def 1;
  x * b = a & y * d = c by A1,A2,Def4;
  then z * (b * d) = (x * b) * (y * d) by A3,A4,Def4
    .= x * (b * (y * d)) by GROUP_1:def 3
    .= x * (y * (b * d)) by GROUP_1:def 3
    .= (x * y) * (b * d) by GROUP_1:def 3;
  hence thesis by A4,Th1;
end;
