reserve n for Nat;

theorem
  for C be compact connected non vertical non horizontal Subset of
TOP-REAL 2 for n be Nat st n > 0 for i,j be Nat st 1 <= i
& i <= len Gauge(C,n) & 1 <= j & j <= width Gauge(C,n) & Gauge(C,n)*(i,j) in L~
Cage(C,n) holds LSeg(Gauge(C,n)*(i,1),Gauge(C,n)*(i,j)) meets Lower_Arc L~Cage(
  C,n)
proof
  let C be compact connected non vertical non horizontal Subset of TOP-REAL 2;
  let n be Nat;
  assume n > 0;
  then
A1: L~Lower_Seq(C,n) = Lower_Arc L~Cage(C,n) by Th56;
  let i,j be Nat;
  assume 1 <= i & i <= len Gauge(C,n) & 1 <= j & j <= width Gauge(C,n) &
  Gauge( C,n)*(i,j) in L~Cage(C,n);
  hence thesis by A1,Th46;
end;
