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

theorem Th5:
  n is_sufficiently_large_for C implies Y-SpanStart(C,n) <= 2|^(n-'
  ApproxIndex C)*(Y-InitStart C-'2)+2
proof
  set m = ApproxIndex C;
A1: X-SpanStart(C,m) > 2 by JORDAN1H:49;
  set j0 = 2|^(n-'m)*(Y-InitStart C-'2)+2;
  set i1 = X-SpanStart(C,n);
  assume
A2: n is_sufficiently_large_for C;
  then
A3: n >=ApproxIndex C by Def1;
A4: 1 < Y-InitStart C by Th2;
  then 1+1 <= Y-InitStart C by NAT_1:13;
  then
A5: Y-InitStart C-'2 = Y-InitStart C-2 by XREAL_1:233;
A6: Y-InitStart C+1 < width Gauge(C,m) by Th3;
A7: now
    m-'1 <= m by NAT_D:44;
    then
A8: 2|^(m-'1) <= 2|^m by PREPOWER:93;
    len Gauge(C,m) = 2|^m + 3 by JORDAN8:def 1;
    then
A9: X-SpanStart(C,m) < len Gauge(C,m) by A8,XREAL_1:8;
A10: 2+1 <= X-SpanStart(C,m) by A1,NAT_1:13;
A11: i1 = 2|^(n-'m)*(X-SpanStart(C,m)-2)+2 by A2,Th4;
    let j;
    assume that
A12: j0 <= j and
A13: j <= 2|^(n-'m)*(Y-InitStart C-'2)+2;
A14: cell(Gauge(C,m),X-SpanStart(C,m)-'1,Y-InitStart C) c= BDD C by Def2;
    j = j0 by A12,A13,XXREAL_0:1;
    then cell(Gauge(C,n),i1-'1,j) c= cell(Gauge(C,m),X-SpanStart(C,m)-'1,
    Y-InitStart C) by A3,A6,A5,A10,A9,A11,Th2,JORDAN1A:48;
    hence cell(Gauge(C,n),i1-'1,j) c= BDD C by A14;
  end;
  Y-InitStart C < Y-InitStart C +1 by XREAL_1:29;
  then Y-InitStart C < width Gauge(C,m) by Th3,XXREAL_0:2;
  then j0 <= width Gauge(C,n) by A3,A4,A5,JORDAN1A:32;
  hence thesis by A2,A7,Def3;
end;
