
theorem Th14:
  for C be Simple_closed_curve for n be Nat st n
  is_sufficiently_large_for C holds UBD C c= UBD L~Span(C,n)
proof
  let C be Simple_closed_curve;
  let n be Nat;
  assume
A1: n is_sufficiently_large_for C;
  let x be object;
A2: BDD C misses UBD C by JORDAN2C:24;
  assume
A3: x in UBD C;
  then reconsider p = x as Point of TOP-REAL 2;
A4: Cl BDD L~Span(C,n) c= Cl BDD C by A1,Th13,PRE_TOPC:19;
A5: now
    assume x in L~Span(C,n);
    then p in (RightComp Span(C,n)) \/ L~Span(C,n) by XBOOLE_0:def 3;
    then p in Cl RightComp Span(C,n) by GOBRD14:21;
    then p in Cl BDD L~Span(C,n) by GOBRD14:37;
    hence contradiction by A4,A2,A3,PRE_TOPC:def 7;
  end;
  BDD L~Span(C,n) c= BDD C by A1,Th13;
  then not x in BDD L~Span(C,n) by A2,A3,XBOOLE_0:3;
  then not x in RightComp Span(C,n) by GOBRD14:37;
  then p in LeftComp Span(C,n) by A5,GOBRD14:17;
  hence thesis by GOBRD14:36;
end;
