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

theorem Th27:
  for Amp being AmpleSet of R holds Amp is multiplicative implies
  for x,y being Element of Amp holds y divides x & y <> 0.R implies x/y in Amp
proof
  let Amp be AmpleSet of R;
  assume
A1: Amp is multiplicative;
  let x,y be Element of Amp;
  assume that
A2: y divides x and
A3: y <> 0.R;
  now
    per cases;
    case
A4:   x <> 0.R;
      set d = x/y;
      consider d9 being Element of Amp such that
A5:   d9 is_associated_to d by Th22;
      consider u being Element of R such that
A6:   u is unital and
A7:   d * u = d9 by A5,Th18;
      x = y * d by A2,A3,Def4;
      then
A8:   u * x = y * d9 by A7,GROUP_1:def 3;
A9:   x is_associated_to u * x
      proof
        u divides 1.R by A6;
        then consider e being Element of R such that
A10:    u * e = 1.R;
A11:    x divides u * x;
        (u * x) * e = (e * u) * x by GROUP_1:def 3
          .= x by A10;
        then u * x divides x;
        hence thesis by A11;
      end;
A12:  y * d9 in Amp by A1;
      1.R * x = x
        .= u * x by A8,A12,A9,Th22;
      then u = 1.R by A4,Th1;
      then d9 = d by A7;
      hence thesis;
    end;
    case
A13:  x = 0.R;
      set d = x/y;
      x = y * d by A2,A3,Def4;
      then
A14:  d = 0.R by A3,A13,VECTSP_2:def 1;
      0.R is Element of Amp by Th24;
      hence thesis by A14;
    end;
  end;
  hence thesis;
end;
