let C be Simple_closed_curve; :: thesis: for i, j, k, n being 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 > 1
let i, j, k, n be Nat; :: thesis: ( 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) implies j > 1 )
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) ; :: thesis: j > 1
A6: len (Span (C,n)) > 4 by GOBOARD7:34;
SpanStart (C,n) in BDD C by A1, Th6;
then A7: S-bound C <= S-bound (BDD C) by JORDAN1C:8;
A8: i <= len (Gauge (C,n)) 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: S-bound (L~ (Span (C,n))) <= ((Gauge (C,n)) * (i,j)) `2 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: S-bound (BDD C) = S-bound (Cl (BDD C)) by A11, TOPREAL6:88;
L~ (Span (C,n)) c= BDD C by A1, Th21;
then S-bound (L~ (Span (C,n))) >= S-bound (Cl (BDD C)) by A12, A10, PSCOMP_1:68, XBOOLE_1:1;
then A14: S-bound (BDD C) <= ((Gauge (C,n)) * (i,j)) `2 by A13, A9, XXREAL_0:2;
len (Gauge (C,n)) >= 4 by JORDAN8:10;
then A15: len (Gauge (C,n)) >= 2 by XXREAL_0:2;
A16: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def 1;
A17: 1 <= i by A4, MATRIX_0:32;
then ((Gauge (C,n)) * (i,2)) `2 = S-bound C by A8, JORDAN8:13;
then A18: ((Gauge (C,n)) * (i,2)) `2 <= ((Gauge (C,n)) * (i,j)) `2 by A7, A14, XXREAL_0:2;
1 <= j by A4, MATRIX_0:32;
then j >= 1 + 1 by A17, A8, A16, A15, A18, GOBOARD5:4;
hence j > 1 by NAT_1:13; :: thesis: verum