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 Th51:
  d = d9 + 1 implies for A being Cell of d9,G holds card(star A) = 2
proof
  assume
A1: d = d9 + 1;
  then
A2: d9 < d by NAT_1:13;
  let A be Cell of d9,G;
  consider l,r,i0 such that
A3: A = cell(l,r) and
A4: l.i0 = r.i0 and
A5: l.i0 in G.i0 and
A6: for i st i <> i0 holds l.i < r.i & [l.i,r.i] is Gap of G.i by A1,Th38;
A7: now
    let i;
    i = i0 or i <> i0;
    hence l.i <= r.i by A4,A6;
  end;
  ex B1,B2 being set st B1 in star A & B2 in star A & B1 <> B2 &
  for B being set st B in star A holds B = B1 or B = B2
  proof
    ex l1,r1 st [l1.i0,r1.i0] is Gap of G.i0 & r1.i0 = l.i0 &
    ((l1.i0 < r1.i0 & for i st i <> i0 holds l1.i = l.i & r1.i = r.i) or
    for i holds r1.i < l1.i & [l1.i,r1.i] is Gap of G.i)
    proof
      consider l1i0 being Element of REAL such that
A8:   [l1i0,l.i0] is Gap of G.i0 by A5,Th16;
      per cases by A8,Th13;
      suppose
A9:     l1i0 < l.i0;
        defpred P[Element of Seg d,Element of REAL] means
     ($1 = i0 implies $2 = l1i0) &
        ($1 <> i0 implies $2 = l.$1);
A10:    for i ex li being Element of REAL st P[i,li]
        proof
          let i;
A11:         l.i in REAL by XREAL_0:def 1;
          i = i0 or i <> i0;
          hence thesis by A11;
        end;
        consider l1 being Function of Seg d,REAL such that
A12:    for i holds P[i,l1.i] from FUNCT_2:sch 3(A10);
        reconsider l1 as Element of REAL d by Def3;
        take l1,r;
        thus thesis by A4,A8,A9,A12;
      end;
      suppose
A13:    l.i0 < l1i0;
        consider l1,r1 such that
        cell(l1,r1) = infinite-cell(G) and
A14:    for i holds r1.i < l1.i & [l1.i,r1.i] is Gap of G.i by Def10;
        take l1,r1;
A15:    r1.i0 < l1.i0 by A14;
        [l1.i0,r1.i0] is Gap of G.i0 by A14;
        hence thesis by A8,A13,A14,A15,Th19;
      end;
    end;
    then consider l1,r1 such that
A16: [l1.i0,r1.i0] is Gap of G.i0 and
A17: r1.i0 = l.i0 and
A18: (l1.i0 < r1.i0 & for i st i <> i0 holds l1.i = l.i & r1.i = r.i)
    or for i holds r1.i < l1.i & [l1.i,r1.i] is Gap of G.i;
A19: now
      let i;
A20:  i
      <> i0 & l1.i = l.i & r1.i = r.i implies [l1.i,r1.i] is Gap of G.i by A6;
      i = i0 or i <> i0;
      hence [l1.i,r1.i] is Gap of G.i by A16,A18,A20;
    end;
A21: (for i holds l1.i < r1.i) or for i holds r1.i < l1.i
    proof
      per cases by A18;
      suppose
A22:    l1.i0 < r1.i0 & for i st i <> i0 holds l1.i = l.i & r1.i = r.i;
        now
          let i;
A23:      i <> i0 & l1.i = l.i & r1.i = r.i implies l1.i < r1.i by A6;
          i = i0 or i <> i0;
          hence l1.i < r1.i by A22,A23;
        end;
        hence thesis;
      end;
      suppose for i holds r1.i < l1.i & [l1.i,r1.i] is Gap of G.i;
        hence thesis;
      end;
    end;
    then reconsider B1 = cell(l1,r1) as Cell of d,G by A19,Th37;
    ex l2,r2 st [l2.i0,r2.i0] is Gap of G.i0 & l2.i0 = l.i0 &
    ((l2.i0 < r2.i0 & for i st i <> i0 holds l2.i = l.i & r2.i = r.i) or
    for i holds r2.i < l2.i & [l2.i,r2.i] is Gap of G.i)
    proof
      consider r2i0 being Element of REAL such that
A24:  [l.i0,r2i0] is Gap of G.i0 by A5,Th15;
      per cases by A24,Th13;
      suppose
A25:    l.i0 < r2i0;
        defpred P[Element of Seg d,Element of REAL] means
  ($1 = i0 implies $2 = r2i0) &
        ($1 <> i0 implies $2 = r.$1);
A26:    for i ex ri being Element of REAL st P[i,ri]
        proof
          let i;
A27:        r.i in REAL by XREAL_0:def 1;
          i = i0 or i <> i0;
          hence thesis by A27;
        end;
        consider r2 being Function of Seg d,REAL such that
A28:    for i holds P[i,r2.i] from FUNCT_2:sch 3(A26);
        reconsider r2 as Element of REAL d by Def3;
        take l,r2;
        thus thesis by A24,A25,A28;
      end;
      suppose
A29:    r2i0 < l.i0;
        consider l2,r2 such that
        cell(l2,r2) = infinite-cell(G) and
A30:    for i holds r2.i < l2.i & [l2.i,r2.i] is Gap of G.i by Def10;
        take l2,r2;
A31:    r2.i0 < l2.i0 by A30;
        [l2.i0,r2.i0] is Gap of G.i0 by A30;
        hence thesis by A24,A29,A30,A31,Th19;
      end;
    end;
    then consider l2,r2 such that
A32: [l2.i0,r2.i0] is Gap of G.i0 and
A33: l2.i0 = l.i0 and
A34: (l2.i0 < r2.i0 & for i st i <> i0 holds l2.i = l.i & r2.i = r.i)
    or for i holds r2.i < l2.i & [l2.i,r2.i] is Gap of G.i;
A35: now
      let i;
A36:  i
      <> i0 & l2.i = l.i & r2.i = r.i implies [l2.i,r2.i] is Gap of G.i by A6;
      i = i0 or i <> i0;
      hence [l2.i,r2.i] is Gap of G.i by A32,A34,A36;
    end;
    (for i holds l2.i < r2.i) or for i holds r2.i < l2.i
    proof
      per cases by A34;
      suppose
A37:    l2.i0 < r2.i0 & for i st i <> i0 holds l2.i = l.i & r2.i = r.i;
        now
          let i;
A38:      i <> i0 & l2.i = l.i & r2.i = r.i implies l2.i < r2.i by A6;
          i = i0 or i <> i0;
          hence l2.i < r2.i by A37,A38;
        end;
        hence thesis;
      end;
      suppose for i holds r2.i < l2.i & [l2.i,r2.i] is Gap of G.i;
        hence thesis;
      end;
    end;
    then reconsider B2 = cell(l2,r2) as Cell of d,G by A35,Th37;
    take B1,B2;
    A c= B1
    proof
      per cases by A18;
      suppose
A39:    l1.i0 < r1.i0 & for i st i <> i0 holds l1.i = l.i & r1.i = r.i;
A40:    now
          let i;
          i = i0 or i <> i0 & l1.i = l.i & r1.i = r.i by A39;
          hence l1.i <= r1.i by A6,A39;
        end;
        now
          let i;
          i = i0 or i <> i0 & l1.i = l.i & r1.i = r.i by A39;
          hence l1.i <= l.i & l.i <= r.i & r.i <= r1.i by A4,A17,A40;
        end;
        hence thesis by A3,A40,Th25;
      end;
      suppose for i holds r1.i < l1.i & [l1.i,r1.i] is Gap of G.i;
        hence thesis by A3,A4,A7,A17,Th27;
      end;
    end;
    hence B1 in star A by A1;
    A c= B2
    proof
      per cases by A34;
      suppose
A41:    l2.i0 < r2.i0 & for i st i <> i0 holds l2.i = l.i & r2.i = r.i;
A42:    now
          let i;
          i = i0 or i <> i0 & l2.i = l.i & r2.i = r.i by A41;
          hence l2.i <= r2.i by A6,A41;
        end;
        now
          let i;
          i = i0 or i <> i0 & l2.i = l.i & r2.i = r.i by A41;
          hence l2.i <= l.i & l.i <= r.i & r.i <= r2.i by A4,A33,A42;
        end;
        hence thesis by A3,A42,Th25;
      end;
      suppose for i holds r2.i < l2.i & [l2.i,r2.i] is Gap of G.i;
        hence thesis by A3,A7,A33,Th27;
      end;
    end;
    hence B2 in star A by A1;
A43: l1 <> l2 by A17,A18,A33;
    (for i holds l1.i <= r1.i) or for i holds r1.i < l1.i by A21;
    hence B1 <> B2 by A43,Th28;
    let B be set;
    assume
A44: B in star A;
    then reconsider B as Cell of d,G by A1;
    consider l9,r9 such that
A45: B = cell(l9,r9) and
A46: for i holds [l9.i,r9.i] is Gap of G.i and
A47: (for i holds l9.i < r9.i) or for i holds r9.i < l9.i by Th36;
A48: [l9.i0,r9.i0] is Gap of G.i0 by A46;
A49: A c= B by A44,Th47;
    per cases by A47;
    suppose
A50:  for i holds l9.i < r9.i;
A51:  now
        let i;
        assume
A52:    i <> i0;
        l9.i < r9.i by A50;
        then l.i = l9.i & r.i = r9.i or
        l.i = l9.i & r.i = l9.i or l.i = r9.i & r.i = r9.i
        by A2,A3,A45,A49,Th42;
        hence l9.i = l.i & r9.i = r.i by A6,A52;
      end;
      thus thesis
      proof
A53:    l9.i0 < r9.i0 by A50;
        per cases by A2,A3,A4,A45,A49,A53,Th42;
        suppose
A54:      l.i0 = r9.i0 & r.i0 = r9.i0;
          then
A55:      l9.i0 = l1.i0 by A16,A17,A48,Th18;
          reconsider l9,r9,l1,r1 as Function of Seg d,REAL by Def3;
A56:      now
            let i;
A57:        l1.i0 < l.i0 by A50,A54,A55;
            then i = i0 or i <> i0 & l9.i = l.i & l1.i = l.i by A17,A18,A51;
            hence l9.i = l1.i by A16,A17,A48,A54,Th18;
            i = i0 or i <> i0 & r9.i = r.i & r1.i = r.i by A17,A18,A51,A57;
            hence r9.i = r1.i by A17,A54;
          end;
          then l9 = l1 by FUNCT_2:63;
          hence thesis by A45,A56,FUNCT_2:63;
        end;
        suppose
A58:      l.i0 = l9.i0 & r.i0 = l9.i0;
          then
A59:      r9.i0 = r2.i0 by A32,A33,A48,Th17;
          reconsider l9,r9,l2,r2 as Function of Seg d,REAL by Def3;
A60:      now
            let i;
A61:        l.i0 < r2.i0 by A50,A58,A59;
            then i = i0 or i <> i0 & r9.i = r.i & r2.i = r.i by A33,A34,A51;
            hence r9.i = r2.i by A32,A33,A48,A58,Th17;
            i = i0 or i <> i0 & l9.i = l.i & l2.i = l.i by A33,A34,A51,A61;
            hence l9.i = l2.i by A33,A58;
          end;
          then l9 = l2 by FUNCT_2:63;
          hence thesis by A45,A60,FUNCT_2:63;
        end;
      end;
    end;
    suppose
A62:  for i holds r9.i < l9.i;
      consider i1 such that
A63:  l.i1 = l9.i1 & r.i1 = l9.i1 or l.i1 = r9.i1 & r.i1 = r9.i1
      by A2,A3,A45,A49,Th43;
A64:  i0 = i1 by A6,A63;
      thus thesis
      proof
        per cases by A63,A64;
        suppose
A65:      l.i0 = r9.i0 & r.i0 = r9.i0;
          then l9.i0 = l1.i0 by A16,A17,A48,Th18;
          then B1 = infinite-cell(G) by A17,A18,A62,A65,Th45;
          hence thesis by A45,A62,Th45;
        end;
        suppose
A66:      l.i0 = l9.i0 & r.i0 = l9.i0;
          then r9.i0 = r2.i0 by A32,A33,A48,Th17;
          then B2 = infinite-cell(G) by A33,A34,A62,A66,Th45;
          hence thesis by A45,A62,Th45;
        end;
      end;
    end;
  end;
  hence thesis by Th5;
end;
