reserve X,x,y,z for set;
reserve n,m,k,k9,d9 for Nat;
reserve d for non zero Nat;
reserve i,i0,i1 for Element of Seg d;
reserve l,r,l9,r9,l99,r99,x,x9,l1,r1,l2,r2 for Element of REAL d;
reserve Gi for non trivial finite Subset of REAL;
reserve li,ri,li9,ri9,xi,xi9 for Real;
reserve G for Grating of d;

theorem Th15:
  xi in Gi implies ex ri being Element of REAL st [xi,ri] is Gap of Gi
proof
  assume
A1: xi in Gi;
  defpred P[Element of REAL] means $1 > xi;
  set Gi9 = { f(ri9) where ri9 is Element of REAL : f(ri9) in Gi & P[ri9]};
A2: Gi9 c= Gi from FRAENKEL:sch 17;
  then reconsider Gi9 as finite Subset of REAL by XBOOLE_1:1;
  per cases;
  suppose
A3: Gi9 is empty;
A4: now
      let xi9;
      assume that
A5:   xi9 in Gi and
A6:   xi9 > xi;
      xi9 in Gi9 by A5,A6;
      hence contradiction by A3;
    end;
    consider li being Element of REAL such that
A7: li in Gi and
A8: for xi9 st xi9 in Gi holds li <= xi9 by Th10;
    take li;
A9: now
      assume
A10:  li = xi;
      for xi9 being object holds xi9 in Gi iff xi9 = xi
      proof
        let xi9 be object;
        hereby
          assume
A11:      xi9 in Gi;
          then reconsider xi99 = xi9 as Element of REAL;
A12:      li <= xi99 by A8,A11;
          xi99 <= xi by A4,A11;
          hence xi9 = xi by A10,A12,XXREAL_0:1;
        end;
        thus thesis by A1;
      end;
      hence Gi = {xi} by TARSKI:def 1;
      hence contradiction;
    end;
    li <= xi by A1,A8;
    then
A13: li < xi by A9,XXREAL_0:1;
    for xi9 st xi9 in Gi holds not (xi < xi9 or xi9 < li) by A4,A8;
    hence thesis by A1,A7,A13,Th13;
  end;
  suppose Gi9 is non empty;
    then reconsider Gi9 as non empty finite Subset of REAL;
    consider ri being Element of REAL such that
A14: ri in Gi9 and
A15: for ri9 st ri9 in Gi9 holds ri9 >= ri by Th10;
    take ri;
    now
      thus xi in Gi by A1;
      thus ri in Gi by A2,A14;
      ex ri9 being Element of REAL st ri9 = ri & ri9 in Gi & xi < ri9 by A14;
      hence xi < ri;
      hereby
        let xi9;
        assume xi9 in Gi;
        then xi9 <= xi or xi9 in Gi9;
        hence not (xi < xi9 & xi9 < ri) by A15;
      end;
    end;
    hence thesis by Th13;
  end;
end;
