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 Th25:
  (for i holds l9.i <= r9.i) implies (cell(l,r) c= cell(l9,r9) iff
  for i holds l9.i <= l.i & l.i <= r.i & r.i <= r9.i)
proof
  assume
A1: for i holds l9.i <= r9.i;
  thus cell(l,r) c= cell(l9,r9) implies
  for i holds l9.i <= l.i & l.i <= r.i & r.i <= r9.i
  proof
    assume
A2: cell(l,r) c= cell(l9,r9);
    let i0;
    per cases;
    suppose
A3:   r.i0 < l.i0;
      defpred P[Element of Seg d,Element of REAL] means $2 > l.$1 & $2 > r9.$1;
A4:   for i ex xi being Element of REAL st P[i,xi] by Th2;
      consider x being Function of Seg d,REAL such that
A5:   for i holds P[i,x.i] from FUNCT_2:sch 3(A4);
      reconsider x as Element of REAL d by Def3;
      ex i st r.i < l.i & (x.i <= r.i or l.i <= x.i)
      proof
        take i0;
        thus thesis by A3,A5;
      end;
      then
A6:   x in cell(l,r);
      ex i st x.i < l9.i or r9.i < x.i
      proof
        take i0;
        thus thesis by A5;
      end;
      hence thesis by A1,A2,A6,Th21;
    end;
    suppose
A7:   l.i0 <= r.i0;
A8:   l in cell(l,r) by Th23;
      r in cell(l,r) by Th23;
      hence thesis by A1,A2,A7,A8,Th21;
    end;
  end;
  assume
A9: for i holds l9.i <= l.i & l.i <= r.i & r.i <= r9.i;
  let x be object;
  assume
A10: x in cell(l,r);
  then reconsider x as Element of REAL d;
  now
    let i;
A11: l9.i <= l.i by A9;
A12: l.i <= x.i by A9,A10,Th21;
A13: x.i <= r.i by A9,A10,Th21;
    r.i <= r9.i by A9;
    hence l9.i <= x.i & x.i <= r9.i by A11,A12,A13,XXREAL_0:2;
    hence l9.i <= r9.i by XXREAL_0:2;
  end;
  hence thesis;
end;
