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 Th30:
  for C be compact connected non vertical non horizontal Subset of
TOP-REAL 2 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 L~Cage(C,n)
proof
  let C be compact connected non vertical non horizontal Subset of TOP-REAL 2;
  let i be Nat;
  assume that
A1: 1 <= i and
A2: i <= len Gauge(C,n);
A3: Gauge(C,n)*(i,1) = |[(Gauge(C,n)*(i,1))`1,(Gauge(C,n)*(i,1))`2]| by
EUCLID:53
    .= |[(Gauge(C,n)*(i,1))`1,S-bound L~Cage(C,n)]| by A1,A2,JORDAN1A:72;
A4: Gauge(C,n)*(i,len Gauge(C,n)) = |[(Gauge(C,n)*(i,len Gauge(C,n)))`1, (
  Gauge(C,n)*(i,len Gauge(C,n)))`2]| by EUCLID:53
    .= |[(Gauge(C,n)*(i,len Gauge(C,n)))`1, N-bound L~Cage(C,n)]| by A1,A2,
JORDAN1A:70;
  set r = (Gauge(C,n)*(i,1))`1;
  4 <= len Gauge(C,n) by JORDAN8:10;
  then
A5: 1 <= len Gauge(C,n) by XXREAL_0:2;
A6: len Gauge(C,n) = width Gauge(C,n) by JORDAN8:def 1;
  then (Gauge(C,n)*(1,1))`1 <= r by A1,A2,A5,SPRECT_3:13;
  then
A7: W-bound L~Cage(C,n) <= r by A5,JORDAN1A:73;
  r <= Gauge(C,n)*(len Gauge(C,n),1)`1 by A1,A2,A6,A5,SPRECT_3:13;
  then
A8: r <= E-bound L~Cage(C,n) by A5,JORDAN1A:71;
  r = (Gauge(C,n)*(i,len Gauge(C,n)))`1 by A1,A2,A6,A5,GOBOARD5:2;
  hence thesis by A3,A4,A7,A8,JORDAN6:70;
end;
