
theorem
  for C be Simple_closed_curve for n,m be Nat st n
is_sufficiently_large_for C & n <= m holds LeftComp(Span(C,m)) c= LeftComp(Span
  (C,n))
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: (Cl RightComp(Span(C,n)))` = LeftComp(Span(C,n)) by JORDAN1H:43;
A4: (Cl RightComp(Span(C,m)))` = LeftComp(Span(C,m)) by JORDAN1H:43;
  Cl RightComp(Span(C,n)) c= Cl RightComp(Span(C,m)) by A1,A2,Th35,PRE_TOPC:19;
  hence thesis by A3,A4,SUBSET_1:12;
end;
