
theorem Th25:
  for C be Simple_closed_curve for i,j,k,n be Nat st n
  is_sufficiently_large_for C & 1 <= k & k <= len Span(C,n) & [i,j] in Indices
  Gauge(C,n) & Span(C,n)/.k = Gauge(C,n)*(i,j) holds j < width Gauge(C,n)
proof
  let C be Simple_closed_curve;
  let i,j,k,n be Nat;
  assume that
A1: n is_sufficiently_large_for C and
A2: 1 <= k and
A3: k <= len Span(C,n) and
A4: [i,j] in Indices Gauge(C,n) and
A5: Span(C,n)/.k = Gauge(C,n)*(i,j);
A6: len Span(C,n) > 4 by GOBOARD7:34;
  k in dom Span(C,n) by A2,A3,FINSEQ_3:25;
  then Span(C,n)/.k in L~Span(C,n) by A6,GOBOARD1:1,XXREAL_0:2;
  then
A7: N-bound L~Span(C,n) >= Gauge(C,n)*(i,j)`2 by A5,PSCOMP_1:24;
A8: BDD C c= Cl BDD C by PRE_TOPC:18;
A9: BDD C is bounded by JORDAN2C:106;
  then
A10: Cl BDD C is compact by TOPREAL6:79;
  SpanStart(C,n) in BDD C by A1,Th6;
  then
A11: N-bound BDD C = N-bound Cl BDD C by A9,TOPREAL6:87;
  L~Span(C,n) c= BDD C by A1,Th21;
  then N-bound L~Span(C,n) <= N-bound Cl BDD C by A10,A8,PSCOMP_1:66,XBOOLE_1:1
;
  then
A12: N-bound BDD C >= Gauge(C,n)*(i,j)`2 by A11,A7,XXREAL_0:2;
A13: len Gauge(C,n) = width Gauge(C,n) by JORDAN8:def 1;
A14: len Gauge(C,n) >= 4 by JORDAN8:10;
  then len Gauge(C,n) >= 1+1 by XXREAL_0:2;
  then
A15: len Gauge(C,n)-1 >= 1 by XREAL_1:19;
  SpanStart(C,n) in BDD C by A1,Th6;
  then
A16: N-bound C >= N-bound BDD C by JORDAN1C:9;
A17: i <= len Gauge(C,n) by A4,MATRIX_0:32;
A18: 1 <= i by A4,MATRIX_0:32;
  then Gauge(C,n)*(i,len Gauge(C,n)-'1)`2 = N-bound C by A17,JORDAN8:14;
  then
A19: Gauge(C,n)*(i,len Gauge(C,n)-'1)`2 >= Gauge(C,n)*(i,j)`2 by A16,A12,
XXREAL_0:2;
A20: len Gauge(C,n)-'1 >= 1 by A15,XREAL_0:def 2;
  j <= width Gauge(C,n) by A4,MATRIX_0:32;
  then j <= len Gauge(C,n)-'1 by A18,A17,A20,A19,GOBOARD5:4;
  then j < len Gauge(C,n)-'1+1 by NAT_1:13;
  hence thesis by A13,A14,XREAL_1:235,XXREAL_0:2;
end;
