reserve x, y, y1, y2 for object;
reserve V for Z_Module;
reserve W, W1, W2 for Submodule of V;
reserve u, v for VECTOR of V;
reserve i, j, k, n for Element of NAT;

theorem LMThFRat31X:
  for n being Integer st n <> 0 & n <> -1 & n <> -2
  holds not n/(n+1) in INT
  proof
    let n be Integer;
    assume AS: n <> 0 & n <> -1 & n <> -2;
    consider m be Nat such that
    A0: n = m or n = - m by INT_1:2;
    per cases by A0;
    suppose n = m;
      hence not n/(n+1) in INT by AS,NAT_1:16,NAT_D:33;
    end;
    suppose D1: n = -m;
      then
      D2: n/(n+1) = (-m)/(-(m-1))
      .= m /(m-1) by XCMPLX_1:191
      .= (m-1)/(m-1) + 1/(m-1);
      D32: m <> 0 & m <> 1 & m <> 2 by D1,AS;
      then 1 <= m by NAT_1:14;
      then 1 < m by AS,D1,XXREAL_0:1;
      then 1+1 <= m by NAT_1:13;
      then 2 < m by AS,D1,XXREAL_0:1;
      then 2+1 <= m by NAT_1:13;
      then D3: 2+1 -1 <= m-1 by XREAL_1:9;
      then
      D31: 1 < m -1 & m-1 <> 0 by XXREAL_0:2;
      D4: n/(n+1) = 1 + 1/(m-1) by D2,D3,XCMPLX_1:60;
      thus not n/(n+1) in INT
      proof
        assume n/(n+1) in INT;
        then reconsider k = n/(n+1) as Integer;
        D5: k-1 = 1/(m-1) by D4;
        reconsider j = m-1 as Nat by D32;
        not 1/j is Integer by D31,NAT_D:33;
        hence contradiction by D5;
      end;
    end;
  end;
