
theorem Th13:
  for C be Simple_closed_curve for n be Nat st n
  is_sufficiently_large_for C holds BDD L~Span(C,n) c= BDD C
proof
  let C be Simple_closed_curve;
  let n be Nat;
  assume that
A1: n is_sufficiently_large_for C and
A2: not BDD L~Span(C,n) c= BDD C;
  set f = Span(C,n);
A3: Cl BDD L~f = Cl RightComp f by GOBRD14:37
    .= (RightComp f) \/ L~f by GOBRD14:21;
  len f >= 1 by GOBOARD7:34,XXREAL_0:2;
  then
A4: 1 in dom f by FINSEQ_3:25;
  len f > 4 by GOBOARD7:34;
  then f/.1 in L~f by A4,GOBOARD1:1,XXREAL_0:2;
  then SpanStart(C,n) in L~f by A1,JORDAN13:def 1;
  then
A5: SpanStart(C,n) in Cl BDD L~f by A3,XBOOLE_0:def 3;
  SpanStart(C,n) in BDD C by A1,Th6;
  then
A6: BDD L~f meets BDD C by A5,PRE_TOPC:def 7;
  BDD C misses UBD C by JORDAN2C:24;
  then
A7: BDD C,UBD C are_separated by TSEP_1:37;
  BDD L~f misses UBD L~f by JORDAN2C:24;
  then C misses BDD L~f by A1,Th12,XBOOLE_1:63;
  then
A8: BDD L~f c= C` by SUBSET_1:23;
  BDD C \/ UBD C = C` by JORDAN2C:27;
  then BDD L~f c= UBD C by A2,A8,A7,CONNSP_1:16;
  then BDD C meets UBD C by A6,XBOOLE_1:63;
  hence contradiction by JORDAN2C:24;
end;
