let G be _Graph; for W being Walk of G
for m, n being Element of NAT st W is Trail-like holds
W .cut (m,n) is Trail-like
let W be Walk of G; for m, n being Element of NAT st W is Trail-like holds
W .cut (m,n) is Trail-like
let m, n be Element of NAT ; ( W is Trail-like implies W .cut (m,n) is Trail-like )
assume A1:
W is Trail-like
; W .cut (m,n) is Trail-like
now W .cut (m,n) is Trail-like per cases
( ( m is odd & n is odd & m <= n & n <= len W ) or not m is odd or not n is odd or not m <= n or not n <= len W )
;
suppose A2:
(
m is
odd &
n is
odd &
m <= n &
n <= len W )
;
W .cut (m,n) is Trail-like now for x, y being even Element of NAT st 1 <= x & x < y & y <= len (W .cut (m,n)) holds
(W .cut (m,n)) . x <> (W .cut (m,n)) . yreconsider m9 =
m as
odd Element of
NAT by A2;
let x,
y be
even Element of
NAT ;
( 1 <= x & x < y & y <= len (W .cut (m,n)) implies (W .cut (m,n)) . x <> (W .cut (m,n)) . y )assume that A3:
1
<= x
and A4:
x < y
and A5:
y <= len (W .cut (m,n))
;
(W .cut (m,n)) . x <> (W .cut (m,n)) . yreconsider xaa1 =
x - 1 as
odd Element of
NAT by A3, INT_1:5;
reconsider yaa1 =
y - 1 as
odd Element of
NAT by A3, A4, INT_1:5, XXREAL_0:2;
x - 1
< y - 1
by A4, XREAL_1:14;
then A6:
xaa1 + m < yaa1 + m
by XREAL_1:8;
x <= len (W .cut (m,n))
by A4, A5, XXREAL_0:2;
then
x - 1
< (len (W .cut (m,n))) - 0
by XREAL_1:15;
then A7:
(W .cut (m,n)) . (xaa1 + 1) = W . (m + xaa1)
by A2, Lm15;
A8:
y - 1
< (len (W .cut (m,n))) - 0
by A5, XREAL_1:15;
then A9:
(W .cut (m,n)) . (yaa1 + 1) = W . (m + yaa1)
by A2, Lm15;
m + yaa1 in dom W
by A2, A8, Lm15;
then A10:
m + yaa1 <= len W
by FINSEQ_3:25;
1
<= m + xaa1
by ABIAN:12, NAT_1:12;
then
W . (m9 + xaa1) <> W . (m9 + yaa1)
by A1, A10, A6, Lm57;
hence
(W .cut (m,n)) . x <> (W .cut (m,n)) . y
by A7, A9;
verum end; hence
W .cut (
m,
n) is
Trail-like
by Lm57;
verum
hence
W .cut (m,n) is Trail-like
; verum