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

theorem Th15:
  for a,b,c being Element of R holds
    a <> 0.R & (a * b) divides (a * c) implies b divides c
proof
  let A,B,C be Element of R;
  assume that
A1: A <> 0.R and
A2: (A * B) divides (A * C);
  consider D being Element of R such that
A3: (A * B) * D = A * C by A2;
  A divides (A * C);
  then
A4: (A * C)/A = (A/A) * C by A1,Th11;
  A divides (A * (B * D));
  then
A5: (A * (B * D))/A = (A/A) * (B * D) by A1,Th11;
A6: A * (B * D) = A * C by A3,GROUP_1:def 3;
  B * D = 1.R * (B * D)
    .= (A/A) * C by A1,A6,A5,A4,Th9
    .= 1.R * C by A1,Th9
    .= C;
  hence thesis;
end;
