
theorem Th23:
  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 i < len 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;
  SpanStart(C,n) in BDD C by A1,Th6;
  then
A7: E-bound C >= E-bound BDD C by JORDAN1C:7;
A8: 1 <= j by A4,MATRIX_0:32;
  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
A9: E-bound L~Span(C,n) >= Gauge(C,n)*(i,j)`1 by A5,PSCOMP_1:24;
A10: BDD C c= Cl BDD C by PRE_TOPC:18;
A11: BDD C is bounded by JORDAN2C:106;
  then
A12: Cl BDD C is compact by TOPREAL6:79;
  SpanStart(C,n) in BDD C by A1,Th6;
  then
A13: E-bound BDD C = E-bound Cl BDD C by A11,TOPREAL6:86;
  L~Span(C,n) c= BDD C by A1,Th21;
  then E-bound L~Span(C,n) <= E-bound Cl BDD C by A12,A10,PSCOMP_1:67
,XBOOLE_1:1;
  then
A14: E-bound BDD C >= Gauge(C,n)*(i,j)`1 by A13,A9,XXREAL_0:2;
A15: len Gauge(C,n) >= 4 by JORDAN8:10;
  then len Gauge(C,n) >= 1+1 by XXREAL_0:2;
  then
A16: len Gauge(C,n)-1 >= 1 by XREAL_1:19;
A17: len Gauge(C,n)-'1 >= 1 by A16,XREAL_0:def 2;
A18: j <= width Gauge(C,n) by A4,MATRIX_0:32;
  len Gauge(C,n) = width Gauge(C,n) by JORDAN8:def 1;
  then Gauge(C,n)*(len Gauge(C,n)-'1,j)`1 = E-bound C by A8,A18,JORDAN8:12;
  then
A19: Gauge(C,n)*(len Gauge(C,n)-'1,j)`1 >= Gauge(C,n)*(i,j)`1 by A7,A14,
XXREAL_0:2;
  i <= len Gauge(C,n) by A4,MATRIX_0:32;
  then i <= len Gauge(C,n)-'1 by A8,A18,A17,A19,GOBOARD5:3;
  then i < len Gauge(C,n)-'1+1 by NAT_1:13;
  hence thesis by A15,XREAL_1:235,XXREAL_0:2;
end;
