let n be Nat; for C being Simple_closed_curve
for i, j, k being Nat st 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (C,n)) holds
LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Lower_Arc C
let C be Simple_closed_curve; for i, j, k being Nat st 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (C,n)) holds
LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Lower_Arc C
let i, j, k be Nat; ( 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (C,n)) implies LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Lower_Arc C )
assume that
A1:
1 < j
and
A2:
j <= k
and
A3:
k < len (Gauge (C,n))
and
A4:
1 <= i
and
A5:
i <= width (Gauge (C,n))
and
A6:
(Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n))
and
A7:
(Gauge (C,n)) * (j,i) in L~ (Lower_Seq (C,n))
; LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Lower_Arc C
consider j1, k1 being Nat such that
A8:
j <= j1
and
A9:
j1 <= k1
and
A10:
k1 <= k
and
A11:
(LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))}
and
A12:
(LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))}
by A1, A2, A3, A4, A5, A6, A7, Th14;
A13:
k1 < len (Gauge (C,n))
by A3, A10, XXREAL_0:2;
1 < j1
by A1, A8, XXREAL_0:2;
then
LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i))) meets Lower_Arc C
by A4, A5, A9, A11, A12, A13, Th28;
hence
LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Lower_Arc C
by A1, A3, A4, A5, A8, A9, A10, Th6, XBOOLE_1:63; verum