
theorem Th35:
  for C be Simple_closed_curve for n,m be Nat st n
  is_sufficiently_large_for C & n <= m holds RightComp(Span(C,n)) c= RightComp(
  Span(C,m))
proof
  let C be Simple_closed_curve;
  let n,m be Nat;
  assume that
A1: n is_sufficiently_large_for C and
A2: n <= m;
A3: L~Span(C,n) misses RightComp(Span(C,n)) by SPRECT_3:25;
A4: RightComp Span(C,n) misses LeftComp Span(C,n) by GOBRD14:14;
  Cl LeftComp Span(C,n) = (LeftComp Span(C,n)) \/ L~Span(C,n) by GOBRD14:22;
  then Cl LeftComp(Span(C,n)) misses RightComp(Span(C,n)) by A3,A4,XBOOLE_1:70;
  then L~Span(C,m) misses RightComp(Span(C,n)) by A1,A2,Th34,XBOOLE_1:63;
  then
A5: RightComp(Span(C,n)) c= (L~Span(C,m))` by SUBSET_1:23;
A6: RightComp(Span(C,m)) is_a_component_of (L~Span(C,m))` by GOBOARD9:def 2;
  RightComp(Span(C,n)) meets RightComp(Span(C,m)) by A1,A2,Th32;
  hence thesis by A6,A5,GOBOARD9:4;
end;
