reserve i,j,k,n for Nat,
  C for being_simple_closed_curve Subset of TOP-REAL 2;

theorem
  n is_sufficiently_large_for C implies cell(Gauge(C,n),X-SpanStart(C,n)
  -'1,Y-SpanStart(C,n)-'1) meets C
proof
  set i1 = X-SpanStart(C,n);
A1: Y-SpanStart(C,n)-1 < Y-SpanStart(C,n) by XREAL_1:146;
  assume
A2: n is_sufficiently_large_for C;
  then
A3: 1 < Y-SpanStart(C,n) by Th7;
  assume
A4: cell(Gauge(C,n),X-SpanStart(C,n)-'1,Y-SpanStart(C,n)-'1) misses C;
A5: for k st Y-SpanStart(C,n)-'1 <= k & k <= 2|^(n-'ApproxIndex C)*(
  Y-InitStart C-'2)+2 holds cell(Gauge(C,n),i1-'1,k) c= BDD C
  proof
    let k such that
A6: Y-SpanStart(C,n)-'1 <= k and
A7: k <= 2|^(n-'ApproxIndex C)*(Y-InitStart C-'2)+2;
    per cases by A6,XXREAL_0:1;
    suppose
A8:   Y-SpanStart(C,n)-'1 = k;
      1 < Y-SpanStart(C,n) by A2,Th7;
      then
A9:   k+1 = Y-SpanStart(C,n) by A8,XREAL_1:235;
A10:  cell(Gauge(C,n),i1-'1,k) c= C` by A4,A8,SUBSET_1:23;
A11:  k < k+1 by XREAL_1:29;
      Y-SpanStart(C,n) <= width Gauge(C,n) by A2,Th7;
      then
A12:  k < width Gauge(C,n) by A9,A11,XXREAL_0:2;
      set W = {B where B is Subset of TOP-REAL 2: B is_inside_component_of C};
A13:  i1-'1 <= i1 by NAT_D:44;
      i1 < len Gauge(C,n) by JORDAN1H:49;
      then
A14:  i1-'1 < len Gauge(C,n) by A13,XXREAL_0:2;
A15:  BDD C = union W by JORDAN2C:def 4;
      1+1 < X-SpanStart(C,n) by JORDAN1H:49;
      then 1 <= i1-1 by XREAL_1:19;
      then 1 <= i1-'1 by NAT_D:39;
      then
      cell(Gauge(C,n),i1-'1,k) /\ cell(Gauge(C,n),i1-'1,k+1) = LSeg(Gauge
      (C,n)*(i1-'1,k+1),Gauge(C,n)*(i1-'1+1,k+1)) by A14,A12,GOBOARD5:26;
      then cell(Gauge(C,n),i1-'1,k) meets cell(Gauge(C,n),i1-'1,k+1);
      then cell(Gauge(C,n),i1-'1,k) meets BDD C by A2,A9,Th6,XBOOLE_1:63;
      then consider e being set such that
A16:  e in W and
A17:  cell(Gauge(C,n),i1-'1,k) meets e by A15,ZFMISC_1:80;
      consider B being Subset of TOP-REAL 2 such that
A18:  e = B and
A19:  B is_inside_component_of C by A16;
A20:  B c= BDD C by A15,A16,A18,ZFMISC_1:74;
      B is_a_component_of C` by A19,JORDAN2C:def 2;
      then cell(Gauge(C,n),i1-'1,k) c= B by A10,A14,A12,A17,A18,GOBOARD9:4
,JORDAN1A:25;
      hence thesis by A20;
    end;
    suppose
      Y-SpanStart(C,n)-'1 < k;
      then Y-SpanStart(C,n) < k+1 by NAT_D:55;
      then Y-SpanStart(C,n) <= k by NAT_1:13;
      hence thesis by A2,A7,Def3;
    end;
  end;
  Y-SpanStart(C,n) <= width Gauge(C,n) by A2,Def3;
  then Y-SpanStart(C,n)-'1 <= width Gauge(C,n) by NAT_D:44;
  then Y-SpanStart(C,n)-'1 >=Y-SpanStart(C,n) by A2,A5,Def3;
  hence contradiction by A3,A1,XREAL_1:233;
end;
