reserve 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,
  i, j, m, n for Nat,
  p for Point of TOP-REAL 2;

theorem Th36:
  j <= len Gauge(E,n) implies cell(Gauge(E,n),len Gauge(E,n),j) c= UBD E
proof
A1: E` is non empty by JORDAN2C:66;
  assume
A2: j <= len Gauge(E,n);
  then cell(Gauge(E,n),len Gauge(E,n),j) misses E by JORDAN8:16;
  then
A3: cell(Gauge(E,n),len Gauge(E,n),j) c= E` by SUBSET_1:23;
A4: width Gauge(E,n) = len Gauge(E,n) by JORDAN8:def 1;
  then
A5: cell(Gauge(E,n),len Gauge(E,n),j) is non empty by A2,JORDAN1A:24;
  cell(Gauge(E,n),len Gauge(E,n),j) is connected by A4,A2,JORDAN1A:25;
  then consider W being Subset of TOP-REAL 2 such that
A6: W is_a_component_of E` and
A7: cell(Gauge(E,n),len Gauge(E,n),j) c= W by A3,A1,A5,GOBOARD9:3;
  W is not bounded by A4,A2,A7,Th34,RLTOPSP1:42;
  then W is_outside_component_of E by A6,JORDAN2C:def 3;
  then W c= UBD E by JORDAN2C:23;
  hence thesis by A7;
end;
