reserve i, i1, i2, j, j1, j2, k, m, n, t for Nat,
  D for non empty Subset of TOP-REAL 2,
  E for compact non vertical non horizontal Subset of TOP-REAL 2,
  C for compact connected non vertical non horizontal Subset of TOP-REAL 2,
  G for Go-board,
  p, q, x for Point of TOP-REAL 2,
  r, s for Real;

theorem
  for C being compact non vertical non horizontal Subset of TOP-REAL 2
  holds i <= len Gauge(C,n) implies cell(Gauge(C,n),i,0) c= UBD C
proof
  let C be compact non vertical non horizontal Subset of TOP-REAL 2;
  assume
A1: i <= len Gauge(C,n);
  then cell(Gauge(C,n),i,0) misses C by JORDAN8:17;
  then
A2: cell(Gauge(C,n),i,0) c= C` by SUBSET_1:23;
  0 <= width Gauge(C,n);
  then
  cell(Gauge(C,n),i,0) is connected & cell(Gauge(C,n),i,0) is non empty by A1
,Th24,Th25;
  then consider W being Subset of TOP-REAL 2 such that
A3: W is_a_component_of C` and
A4: cell(Gauge(C,n),i,0) c= W by A2,GOBOARD9:3;
  W is not bounded by A1,A4,Th26,RLTOPSP1:42;
  then W is_outside_component_of C by A3,JORDAN2C:def 3;
  then W c= UBD C by JORDAN2C:23;
  hence thesis by A4;
end;
