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 Th38:
  d = d9 + 1 implies for A being Subset of REAL d holds A in cells(d9,G) iff
  ex l,r,i0 st A = cell(l,r) & l.i0 = r.i0 & l.i0 in G.i0 &
  for i st i <> i0 holds l.i < r.i & [l.i,r.i] is Gap of G.i
proof
  assume
A1: d = d9 + 1;
  then
A2: d9 < d by NAT_1:13;
  let A be Subset of REAL d;
  hereby
    assume A in cells(d9,G);
    then consider l,r such that
A3: A = cell(l,r) and
A4: (ex X being Subset of Seg d st card X = d9 & for i holds (i in X &
l.i < r.i & [l.i,r.i] is Gap of G.i) or (not i in X & l.i = r.i & l.i in G.i))
    or (d9 = d & for i holds r.i < l.i & [l.i,r.i] is Gap of G.i)
    by A2,Th29;
    take l,r;
    consider X being Subset of Seg d such that
A5: card X = d9 and
A6: for i holds i in X & l.i < r.i & [l.i,r.i] is Gap of G.i or not i
    in X & l.i = r.i & l.i in G.i
    by A1,A4;
    card(Seg d \ X) = card Seg d - card X by CARD_2:44
      .= d - d9 by A5,FINSEQ_1:57
      .= 1 by A1;
    then consider i0 being object such that
A7: Seg d \ X = {i0} by CARD_2:42;
    i0 in Seg d \ X by A7,TARSKI:def 1;
    then reconsider i0 as Element of Seg d by XBOOLE_0:def 5;
    take i0;
A8: now
      let i;
      i in Seg d \ X iff i = i0 by A7,TARSKI:def 1;
      hence i in X iff i <> i0 by XBOOLE_0:def 5;
    end;
    thus A = cell(l,r) by A3;
    not i0 in X by A8;
    hence l.i0 = r.i0 & l.i0 in G.i0 by A6;
    let i;
    assume i <> i0;
    then i in X by A8;
    hence l.i < r.i & [l.i,r.i] is Gap of G.i by A6;
  end;
  given l,r,i0 such that
A9: A = cell(l,r) and
A10: l.i0 = r.i0 and
A11: l.i0 in G.i0 and
A12: for i st i <> i0 holds l.i < r.i & [l.i,r.i] is Gap of G.i;
  reconsider X = Seg d \ {i0} as Subset of Seg d by XBOOLE_1:36;
  card X = d9 & for i holds i in X & l.i < r.i & [l.i,r.i] is Gap of G.i or
  not i in X & l.i = r.i & l.i in G.i
  proof
    thus card X = card Seg d - card {i0} by CARD_2:44
      .= d - card {i0} by FINSEQ_1:57
      .= d - 1 by CARD_1:30
      .= d9 by A1;
    let i;
    i in {i0} iff i = i0 by TARSKI:def 1;
    hence thesis by A10,A11,A12,XBOOLE_0:def 5;
  end;
  hence thesis by A2,A9,Th29;
end;
