
theorem Th10:
  for C be Simple_closed_curve for n be Nat st n
  is_sufficiently_large_for C holds C misses RightComp Span(C,n)
proof
  let C be Simple_closed_curve;
  let n be Nat;
  set f = Span(C,n);
  RightComp f is_a_component_of (L~f)` by GOBOARD9:def 2;
  then
A1: ex L be Subset of (TOP-REAL 2)|(L~f)` st L = RightComp f & L
  is a_component by CONNSP_1:def 6;
  LeftComp f is_a_component_of (L~f)` by GOBOARD9:def 1;
  then
A2: ex R be Subset of (TOP-REAL 2)|(L~f)` st R = LeftComp f & R
  is a_component by CONNSP_1:def 6;
  assume
A3: n is_sufficiently_large_for C;
  then
A4: C misses L~f by Th8;
  C c= the carrier of (TOP-REAL 2)|(L~f)`
  proof
    let c be object;
    assume
A5: c in C;
    then not c in L~f by A4,XBOOLE_0:3;
    then c in (L~f)` by A5,SUBSET_1:29;
    hence thesis by PRE_TOPC:8;
  end;
  then reconsider C1 = C as Subset of (TOP-REAL 2)|(L~f)`;
  assume
A6: C meets RightComp f;
A7: C1 is connected by CONNSP_1:23;
  C meets LeftComp f by A3,Th9;
  hence contradiction by A6,A1,A2,A7,JORDAN2C:92,SPRECT_4:6;
end;
