
theorem Th31:
  for C be Subset of TOP-REAL 2 for n,m,i be Nat st m
<= n & 1 < i & i+1 < len Gauge(C,m) holds 2|^(n-'m)*(i-2)+2+1 < len Gauge(C,n)
proof
  let C be Subset of TOP-REAL 2;
  let n,m,i be Nat;
  assume that
A1: m <= n and
A2: 1 < i and
A3: i+1 < len Gauge(C,m);
  1+1 <= i by A2,NAT_1:13;
  then reconsider i2 = i-2 as Element of NAT by INT_1:5;
A4: 2|^(n-'m) > 0 by NEWTON:83;
  len Gauge(C,m) = 2|^m + (2+1) by JORDAN8:def 1
    .= 2|^m + 2+1;
  then i < 2|^m + 2 by A3,XREAL_1:7;
  then i2 < 2|^m by XREAL_1:19;
  then 2|^(n-'m)*i2 < 2|^(n-'m)*2|^m by A4,XREAL_1:68;
  then 2|^(n-'m)*i2 < 2|^(n-'m+m) by NEWTON:8;
  then 2|^(n-'m)*i2 < 2|^n by A1,XREAL_1:235;
  then 2|^(n-'m)*(i2)+2 < 2|^n + 2 by XREAL_1:8;
  then 2|^(n-'m)*(i-2)+2+1 < 2|^n + 2+1 by XREAL_1:8;
  then 2|^(n-'m)*(i-2)+2+1 < 2|^n + (1+2);
  hence thesis by JORDAN8:def 1;
end;
