
theorem Th6:
  for C be Simple_closed_curve for n be Nat st n
  is_sufficiently_large_for C holds SpanStart(C,n) in BDD C
proof
  let C be Simple_closed_curve;
  let n be Nat;
A1: 1 <= X-SpanStart(C,n)-'1 by JORDAN1H:50;
A2: X-SpanStart(C,n)-'1 < len Gauge(C,n) by JORDAN1H:50;
  assume
A3: n is_sufficiently_large_for C;
  then
A4: cell(Gauge(C,n),X-SpanStart(C,n)-'1,Y-SpanStart(C,n)) c= BDD C by
JORDAN11:6;
A5: Y-SpanStart(C,n) <= width Gauge(C,n) by A3,JORDAN11:7;
  1 < Y-SpanStart(C,n) by A3,JORDAN11:7;
  then LSeg(Gauge(C,n)*(X-SpanStart(C,n)-'1,Y-SpanStart(C,n)), Gauge(C,n)*(
X-SpanStart(C,n)-'1+1,Y-SpanStart(C,n))) c= cell(Gauge(C,n),X-SpanStart(C,n)-'1
  ,Y-SpanStart(C,n)) by A1,A2,A5,GOBOARD5:22;
  then
A6: LSeg(Gauge(C,n)*(X-SpanStart(C,n)-'1,Y-SpanStart(C,n)), Gauge(C,n)*(
  X-SpanStart(C,n)-'1+1,Y-SpanStart(C,n))) c= BDD C by A4;
A7: 2 < X-SpanStart(C,n) by JORDAN1H:49;
  Gauge(C,n)*(X-SpanStart(C,n)-'1+1,Y-SpanStart(C,n)) in LSeg(Gauge(C,n)*
  (X-SpanStart(C,n)-'1,Y-SpanStart(C,n)), Gauge(C,n)*(X-SpanStart(C,n)-'1+1,
  Y-SpanStart(C,n))) by RLTOPSP1:68;
  then Gauge(C,n)*(X-SpanStart(C,n)-'1+1,Y-SpanStart(C,n)) in BDD C by A6;
  hence thesis by A7,XREAL_1:235,XXREAL_0:2;
end;
