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

theorem
  for a,b,c being Element of R holds
    c <> 0.R & c divides a & c divides b implies
      (a/c = b/c iff a = b)
proof
  let a,b,c be Element of R;
  assume
A1: c <> 0.R;
  assume that
A2: c divides a and
A3: c divides b;
  now
    assume (a/c) = (b/c);
    consider e being Element of R such that
A4: e = (b/c);
    e * c = b by A1,A3,A4,Def4;
    hence a/c = b/c implies a = b by A1,A2,A4,Def4;
  end;
  hence thesis;
end;
