reserve a, b, i, k, m, n for Nat,
  r for Real,
  D for non empty Subset of TOP-REAL 2,
  C for compact connected non vertical non horizontal Subset of TOP-REAL 2;

theorem
  ex i being Nat st 1 <= i & i < len Cage(C,n) & N-max C in
  right_cell(Cage(C,n),i)
proof
A1: Cage(C,n) is_sequence_on Gauge(C,n) by JORDAN9:def 1;
  consider i be Nat such that
A2: 1 <= i and
A3: i < len Cage(C,n) and
A4: N-max C in right_cell(Cage(C,n),i,Gauge(C,n)) by Th7;
  take i;
  thus 1 <= i & i < len Cage(C,n) by A2,A3;
  i+1 <= len Cage(C,n) by A3,NAT_1:13;
  then right_cell(Cage(C,n),i,Gauge(C,n)) c= right_cell(Cage(C,n),i) by A2,A1,
GOBRD13:33;
  hence thesis by A4;
end;
