reserve x for object,
  a,b for Real,
  k,k1,i1,j1,w for Nat,
  m,m1,n,n1 for Integer;

theorem Th1:
  x in RAT implies ex m,n st n > 0 & x = m/n
proof
  assume
A1: x in RAT;
  per cases;
  suppose
    x = 0;
    then x = 0/1;
    hence thesis;
  end;
  suppose
A2: x <> 0;
    consider m,n such that
A3: x = m/n by A1,Def1;
    per cases;
    suppose n = 0;
      hence thesis by A2,A3,XCMPLX_1:49;
    end;
    suppose
A4:   n > 0;
      take m,n;
      thus n > 0 by A4;
      thus thesis by A3;
    end;
    suppose
A5:   n < 0;
      take -m,-n;
      thus -n > 0 by A5,XREAL_1:58;
      thus thesis by A3,XCMPLX_1:191;
    end;
  end;
end;
