
theorem Th28:
  for C be compact non vertical non horizontal Subset of TOP-REAL
  2 for n,m be Nat st n <= m & n is_sufficiently_large_for C holds m
  is_sufficiently_large_for C
proof
  let C be compact non vertical non horizontal Subset of TOP-REAL 2;
  let n,m be Nat;
  assume that
A1: n <= m and
A2: n is_sufficiently_large_for C;
  consider j be Nat such that
A3: j < width Gauge(C,n) and
A4: cell(Gauge(C,n),X-SpanStart(C,n)-'1,j) c= BDD C by A2,JORDAN1H:def 3;
  set iin = X-SpanStart(C,n);
  set iim = X-SpanStart(C,m);
  n >= 1 by A2,JORDAN1H:51;
  then
A5: iim = 2|^(m-'n)*(iin-2)+2 by A1,Th27;
A6: j > 1
  proof
A7: X-SpanStart(C,n)-'1 <= len Gauge(C,n) by JORDAN1H:50;
    assume
A8: j <= 1;
    per cases by A8,NAT_1:25;
    suppose
A9:   j = 0;
      width Gauge(C,n) >= 0;
      then
A10:  cell(Gauge(C,n),X-SpanStart(C,n)-'1,0) is non empty by A7,JORDAN1A:24;
      cell(Gauge(C,n),X-SpanStart(C,n)-'1,0) c= UBD C by A7,JORDAN1A:49;
      hence contradiction by A4,A9,A10,JORDAN2C:24,XBOOLE_1:68;
    end;
    suppose
A11:  j = 1;
      set i1 = X-SpanStart(C,n);
A12:  i1-'1 <= i1 by NAT_D:44;
      i1 < len Gauge(C,n) by JORDAN1H:49;
      then
A13:  i1-'1 < len Gauge(C,n) by A12,XXREAL_0:2;
      BDD C c= C` by JORDAN2C:25;
      then
A14:  cell(Gauge(C,n),X-SpanStart(C,n)-'1,1) c= C` by A4,A11;
      UBD C is_outside_component_of C by JORDAN2C:68;
      then
A15:  UBD C is_a_component_of C` by JORDAN2C:def 3;
      width Gauge(C,n) <> 0 by MATRIX_0:def 10;
      then
A16:  0+1 <= width Gauge(C,n) by NAT_1:14;
      then
A17:  cell(Gauge(C,n),X-SpanStart(C,n)-'1,1) is non empty by A7,JORDAN1A:24;
A18:  1 <= i1-'1 by JORDAN1H:50;
      0 < width Gauge(C,n) by A16;
      then
      cell(Gauge(C,n),i1-'1,0)/\cell(Gauge(C,n),i1-'1,0+1) = LSeg(Gauge(C
      ,n)*(i1-'1,0+1),Gauge(C,n)*(i1-'1+1,0+1)) by A13,A18,GOBOARD5:26;
      then
A19:  cell(Gauge(C,n),i1-'1,0) meets cell(Gauge(C,n),i1-'1,0+1) by
XBOOLE_0:def 7;
      cell(Gauge(C,n),X-SpanStart(C,n)-'1,0) c= UBD C by A7,JORDAN1A:49;
      then cell(Gauge(C,n),X-SpanStart(C,n)-'1,1) c= UBD C by A16,A13,A19,A15
,A14,GOBOARD9:4,JORDAN1A:25;
      hence contradiction by A4,A11,A17,JORDAN2C:24,XBOOLE_1:68;
    end;
  end;
  then 2|^(m-'n)*(j-2)+2 > 1 by A1,A3,JORDAN1A:32;
  then reconsider j1 = 2|^(m-'n)*(j-2)+2 as Element of NAT by INT_1:3;
  iin > 2 by JORDAN1H:49;
  then
A20: iin >= 2+1 by NAT_1:13;
A21: j+1 < width Gauge(C,n)
  proof
    assume j+1 >= width Gauge(C,n);
    then
A22: j+1 > width Gauge(C,n) or j+1 = width Gauge(C,n) by XXREAL_0:1;
A23: X-SpanStart(C,n)-'1 <= len Gauge(C,n) by JORDAN1H:50;
    per cases by A3,A22,NAT_1:13;
    suppose
A24:  j = width Gauge(C,n);
A25:  cell(Gauge(C,n),X-SpanStart(C,n)-'1,width Gauge(C,n)) c= UBD C by A23,
JORDAN1A:50;
      cell(Gauge(C,n),X-SpanStart(C,n)-'1,j) is non empty by A23,A24,
JORDAN1A:24;
      hence contradiction by A4,A24,A25,JORDAN2C:24,XBOOLE_1:68;
    end;
    suppose
      j + 1 = width Gauge(C,n);
      then
A26:  cell(Gauge(C,n),X-SpanStart(C,n)-'1,width Gauge(C,n)-'1) c= BDD C
      by A4,NAT_D:34;
      BDD C c= C` by JORDAN2C:25;
      then
A27:  cell(Gauge(C,n),X-SpanStart(C,n)-'1,width Gauge(C,n)-'1) c= C` by A26;
A28:  cell(Gauge(C,n),X-SpanStart(C,n)-'1,width Gauge(C,n)) c= UBD C by A23,
JORDAN1A:50;
      set i1 = X-SpanStart(C,n);
A29:  i1-'1 <= i1 by NAT_D:44;
      i1 < len Gauge(C,n) by JORDAN1H:49;
      then
A30:  i1-'1 < len Gauge(C,n) by A29,XXREAL_0:2;
      UBD C is_outside_component_of C by JORDAN2C:68;
      then
A31:  UBD C is_a_component_of C` by JORDAN2C:def 3;
      width Gauge(C,n)-'1 <= width Gauge(C,n) by NAT_D:44;
      then
A32:  cell(Gauge(C,n),X-SpanStart(C,n)-'1,width Gauge(C,n)-'1) is non
      empty by A23,JORDAN1A:24;
A33:  width Gauge(C,n)-1 < width Gauge(C,n) by XREAL_1:146;
A34:  1 <= i1-'1 by JORDAN1H:50;
A35:  width Gauge(C,n)<>0 by MATRIX_0:def 10;
      then width Gauge(C,n)-'1+1 = width Gauge(C,n) by NAT_1:14,XREAL_1:235;
      then cell(Gauge(C,n),i1-'1,width Gauge(C,n))/\ cell(Gauge(C,n),i1-'1,
width Gauge(C,n)-'1) = LSeg(Gauge(C,n)*(i1-'1,width Gauge(C,n)), Gauge(C,n)*(i1
      -'1+1,width Gauge(C,n))) by A30,A33,A34,GOBOARD5:26;
      then
A36:  cell(Gauge(C,n),i1-'1,width Gauge(C,n)) meets cell(Gauge(C,n),i1-'1
      ,width Gauge(C,n)-'1) by XBOOLE_0:def 7;
      width Gauge(C,n)-'1 < width Gauge(C,n) by A35,A33,NAT_1:14,XREAL_1:233;
      then cell(Gauge(C,n),X-SpanStart(C,n)-'1,width Gauge(C,n)-'1) c= UBD C
      by A28,A30,A36,A31,A27,GOBOARD9:4,JORDAN1A:25;
      hence contradiction by A26,A32,JORDAN2C:24,XBOOLE_1:68;
    end;
  end;
  iin < len Gauge(C,n) by JORDAN1H:49;
  then cell(Gauge(C,m),iim-'1,j1) c= cell(Gauge(C,n),iin-'1,j) by A1,A5,A20,A6
,A21,JORDAN1A:48;
  then
A37: cell(Gauge(C,m),X-SpanStart(C,m)-'1,j1) c= BDD C by A4;
  j1 < width Gauge(C,m) by A1,A3,A6,JORDAN1A:32;
  hence thesis by A37,JORDAN1H:def 3;
end;
