
theorem Th20:
  for C be Simple_closed_curve for n be Nat st n
  is_sufficiently_large_for C holds UBD C misses L~Span(C,n)
proof
  let C be Simple_closed_curve;
  let n be Nat;
  assume
A1: n is_sufficiently_large_for C;
  assume UBD C meets L~Span(C,n);
  then consider x be object such that
A2: x in UBD C and
A3: x in L~Span(C,n) by XBOOLE_0:3;
  UBD C = union {B where B is Subset of TOP-REAL 2: B
  is_outside_component_of C} by JORDAN2C:def 5;
  then consider Z be set such that
A4: x in Z and
A5: Z in {B where B is Subset of TOP-REAL 2: B is_outside_component_of C
  } by A2,TARSKI:def 4;
  consider B be Subset of TOP-REAL 2 such that
A6: Z = B and
A7: B is_outside_component_of C by A5;
  B misses L~Span(C,n) by A1,A7,Th19;
  hence thesis by A3,A4,A6,XBOOLE_0:3;
end;
