reserve E for compact non vertical non horizontal Subset of TOP-REAL 2,
  C for compact connected non vertical non horizontal Subset of TOP-REAL 2,
  G for Go-board,
  i, j, m, n for Nat,
  p for Point of TOP-REAL 2;

theorem Th26:
  for C be Simple_closed_curve for i be Nat st 1 < i &
  i < len Gauge(C,n) holds LSeg(Gauge(C,n)*(i,1),Gauge(C,n)*(i,len Gauge(C,n)))
  meets Lower_Arc C
proof
  let C be Simple_closed_curve;
  let i be Nat;
  assume that
A1: 1 < i and
A2: i < len Gauge(C,n);
  set r = (Gauge(C,n)*(i,2))`1;
  4 <= len Gauge(C,n) by JORDAN8:10;
  then
A3: 1+1 <= len Gauge(C,n) by XXREAL_0:2;
  then 1 <= len Gauge(C,n)-1 by XREAL_1:19;
  then
A4: 1 <= len Gauge(C,n)-'1 by XREAL_0:def 2;
A5: len Gauge(C,n) = width Gauge(C,n) by JORDAN8:def 1;
  then
A6: Gauge(C,n)*(i,2) in LSeg(Gauge(C,n)*(i,1),Gauge(C,n)*(i,len Gauge(C,n))
  ) by A1,A2,A3,JORDAN1A:16;
A7: len Gauge(C,n)-'1 <= len Gauge(C,n) by NAT_D:35;
  then
A8: Gauge(C,n)*(i,len Gauge(C,n)-'1) in LSeg(Gauge(C,n)*(i,1),Gauge(C,n)*(i
  ,len Gauge(C,n))) by A1,A2,A5,A4,JORDAN1A:16;
A9: r = (Gauge(C,n)*(i,1))`1 by A1,A2,A5,A3,GOBOARD5:2
    .= (Gauge(C,n)*(i,len Gauge(C,n)-'1))`1 by A1,A2,A5,A4,A7,GOBOARD5:2;
  1+1 <= i by A1,NAT_1:13;
  then (Gauge(C,n)*(2,2))`1 <= r by A2,A5,A3,SPRECT_3:13;
  then
A10: W-bound C <= r by A3,JORDAN8:11;
  i+1 <= len Gauge(C,n) by A2,NAT_1:13;
  then i <= len Gauge(C,n)-1 by XREAL_1:19;
  then i <= len Gauge(C,n)-'1 by XREAL_0:def 2;
  then
  r <= Gauge(C,n)*(len Gauge(C,n)-'1,len Gauge(C,n)-'1)`1 by A1,A5,A4,A7,A9,
SPRECT_3:13;
  then
A11: r <= E-bound C by A4,JORDAN8:12,NAT_D:35;
A12: Gauge(C,n)*(i,len Gauge(C,n)-'1) = |[(Gauge(C,n)*(i,len Gauge(C,n)-'1))
  `1, (Gauge(C,n)*(i,len Gauge(C,n)-'1))`2]| by EUCLID:53
    .= |[(Gauge(C,n)*(i,len Gauge(C,n)-'1))`1,N-bound C]| by A1,A2,JORDAN8:14;
  Gauge(C,n)*(i,2) = |[(Gauge(C,n)*(i,2))`1,(Gauge(C,n)*(i,2))`2]| by EUCLID:53
    .= |[(Gauge(C,n)*(i,2))`1,S-bound C]| by A1,A2,JORDAN8:13;
  then
  LSeg(Gauge(C,n)*(i,2),Gauge(C,n)*(i,len Gauge(C,n)-'1)) meets Lower_Arc
  C by A12,A9,A10,A11,JORDAN6:70;
  hence thesis by A6,A8,TOPREAL1:6,XBOOLE_1:63;
end;
