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 Th35:
  j <= width Gauge(C,n) implies cell(Gauge(C,n),0,j) c= UBD C
proof
  assume
A1: j <= width Gauge(C,n);
A2: C` is non empty by JORDAN2C:66;
A3: len Gauge(C,n) = width Gauge(C,n) by JORDAN8:def 1;
  then
A4: cell(Gauge(C,n),0,j) is non empty by A1,JORDAN1A:24;
  cell(Gauge(C,n),0,j) misses C by A3,A1,JORDAN8:18;
  then
A5: cell(Gauge(C,n),0,j) c= C` by SUBSET_1:23;
  cell(Gauge(C,n),0,j) is connected by A3,A1,JORDAN1A:25;
  then consider W being Subset of TOP-REAL 2 such that
A6: W is_a_component_of C` and
A7: cell(Gauge(C,n),0,j) c= W by A5,A2,A4,GOBOARD9:3;
  W is not bounded by A1,A7,Th33,RLTOPSP1:42;
  then W is_outside_component_of C by A6,JORDAN2C:def 3;
  then W c= UBD C by JORDAN2C:23;
  hence thesis by A7;
end;
