
theorem Th8:
  for C be Simple_closed_curve for n be Nat st n
  is_sufficiently_large_for C holds 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;
  set G = Gauge(C,n);
  set f = Span(C,n);
  assume not thesis;
  then consider c be object such that
A2: c in C and
A3: c in L~f by XBOOLE_0:3;
  L~f = union { LSeg(f,i) where i is Nat: 1 <= i & i+1 <= len f
  } by TOPREAL1:def 4;
  then consider Z be set such that
A4: c in Z and
A5: Z in { LSeg(f,i) where i is Nat: 1 <= i & i+1 <= len f }
  by A3,TARSKI:def 4;
  consider i be Nat such that
A6: Z = LSeg(f,i) and
A7: 1 <= i and
A8: i+1 <= len f by A5;
  f is_sequence_on G by A1,JORDAN13:def 1;
  then LSeg(f,i) = left_cell(f,i,G) /\ right_cell(f,i,G) by A7,A8,GOBRD13:29;
  then
A9: c in right_cell(f,i,G) by A4,A6,XBOOLE_0:def 4;
  right_cell(f,i,G) misses C by A1,A7,A8,Th7;
  hence contradiction by A2,A9,XBOOLE_0:3;
end;
