reserve p, q for Point of TOP-REAL 2,
  r for Real,
  h for non constant standard special_circular_sequence,
  g for FinSequence of TOP-REAL 2,
  f for non empty FinSequence of TOP-REAL 2,
  I, i1, i, j, k for Nat;

theorem Th33:
  p in L~h & 1 <= I & I <= len GoB h implies (GoB h)*(I,1)`2 <= p `2
proof
  assume that
A1: p in L~h and
A2: 1 <= I and
A3: I <= len GoB h;
  consider i such that
A4: 1<=i and
A5: i+1<=len h and
A6: p in LSeg(h/.i,h/.(i+1)) by A1,SPPOL_2:14;
  i<=i+1 by NAT_1:11;
  then i<=len h by A5,XXREAL_0:2;
  then
A7: (GoB h)*(I,1)`2<=(h/.i)`2 by A2,A3,A4,Th6;
  1<=i+1 by NAT_1:11;
  then
A8: (GoB h)*(I,1)`2<=(h/.(i+1))`2 by A2,A3,A5,Th6;
  now
    per cases;
    case
      (h/.i)`2<=(h/.(i+1))`2;
      then (h/.i)`2<=p`2 by A6,TOPREAL1:4;
      hence thesis by A7,XXREAL_0:2;
    end;
    case
      (h/.i)`2>(h/.(i+1))`2;
      then (h/.(i+1))`2<=p`2 by A6,TOPREAL1:4;
      hence thesis by A8,XXREAL_0:2;
    end;
  end;
  hence thesis;
end;
