
theorem Th19:
  for C be Simple_closed_curve for P be Subset of TOP-REAL 2 for n
  be Nat st n is_sufficiently_large_for C holds P
  is_outside_component_of C implies P misses L~Span(C,n)
proof
  let C be Simple_closed_curve;
  let P be Subset of TOP-REAL 2;
  let n be Nat;
  assume that
A1: n is_sufficiently_large_for C and
A2: P is_outside_component_of C and
A3: P meets L~Span(C,n);
A4: UBD C c= LeftComp Span(C,n) by A1,Th16;
  consider x be object such that
A5: x in P and
A6: x in L~Span(C,n) by A3,XBOOLE_0:3;
  P c= UBD C by A2,JORDAN2C:23;
  then x in UBD C by A5;
  hence contradiction by A6,A4,GOBRD14:17;
end;
