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(E,n) implies cell(Gauge(E,n),i,width Gauge(E,n)) c= UBD E
proof
  let C be compact non vertical non horizontal Subset of TOP-REAL 2;
  assume
A1: i <= len Gauge(E,n);
  width Gauge(E,n) = len Gauge(E,n) by JORDAN8:def 1;
  then cell(Gauge(E,n),i,width Gauge(E,n)) misses E by A1,JORDAN8:15;
  then
A2: cell(Gauge(E,n),i,width Gauge(E,n)) c= E` by SUBSET_1:23;
  cell(Gauge(E,n),i,width Gauge(E,n)) is connected & cell(Gauge(E,n),i,
  width Gauge(E,n)) is non empty by A1,Th24,Th25;
  then consider W being Subset of TOP-REAL 2 such that
A3: W is_a_component_of E` and
A4: cell(Gauge(E,n),i,width Gauge(E,n)) c= W by A2,GOBOARD9:3;
  W is not bounded by A1,A4,Th27,RLTOPSP1:42;
  then W is_outside_component_of E by A3,JORDAN2C:def 3;
  then W c= UBD E by JORDAN2C:23;
  hence thesis by A4;
end;
