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