let G be _Graph; :: thesis: for W being Walk of G
for m, n being odd Element of NAT st m <= n & n <= len W & W . m = W . n holds
for x being Element of NAT st m <= x & x <= len (W .remove (m,n)) holds
( (W .remove (m,n)) . x = W . ((x - m) + n) & (x - m) + n is Element of NAT & (x - m) + n <= len W )

let W be Walk of G; :: thesis: for m, n being odd Element of NAT st m <= n & n <= len W & W . m = W . n holds
for x being Element of NAT st m <= x & x <= len (W .remove (m,n)) holds
( (W .remove (m,n)) . x = W . ((x - m) + n) & (x - m) + n is Element of NAT & (x - m) + n <= len W )

let m, n be odd Element of NAT ; :: thesis: ( m <= n & n <= len W & W . m = W . n implies for x being Element of NAT st m <= x & x <= len (W .remove (m,n)) holds
( (W .remove (m,n)) . x = W . ((x - m) + n) & (x - m) + n is Element of NAT & (x - m) + n <= len W ) )

set W2 = W .remove (m,n);
set WA = W .cut (1,m);
set WB = W .cut (n,(len W));
assume that
A1: m <= n and
A2: n <= len W and
A3: W . m = W . n ; :: thesis: for x being Element of NAT st m <= x & x <= len (W .remove (m,n)) holds
( (W .remove (m,n)) . x = W . ((x - m) + n) & (x - m) + n is Element of NAT & (x - m) + n <= len W )

A4: (W .cut (1,m)) .last() = (W .cut (n,(len W))) .first() by A1, A2, A3, Lm28;
let x be Element of NAT ; :: thesis: ( m <= x & x <= len (W .remove (m,n)) implies ( (W .remove (m,n)) . x = W . ((x - m) + n) & (x - m) + n is Element of NAT & (x - m) + n <= len W ) )
assume that
A5: m <= x and
A6: x <= len (W .remove (m,n)) ; :: thesis: ( (W .remove (m,n)) . x = W . ((x - m) + n) & (x - m) + n is Element of NAT & (x - m) + n <= len W )
A7: len (W .cut (1,m)) = m by ;
then consider a being Nat such that
A8: (len (W .cut (1,m))) + a = x by ;
reconsider a = a as Element of NAT by ORDINAL1:def 12;
(len (W .remove (m,n))) + n = (len W) + m by A1, A2, A3, Lm24;
then (m + a) + n <= m + (len W) by ;
then A9: ((a + n) + m) - m <= ((len W) + m) - m by XREAL_1:13;
(len (W .cut (n,(len W)))) + n = (len W) + 1 by ;
then (a + n) + 1 <= (len (W .cut (n,(len W)))) + n by ;
then ((a + 1) + n) - n <= ((len (W .cut (n,(len W)))) + n) - n by XREAL_1:13;
then A10: (a + 1) - 1 < ((len (W .cut (n,(len W)))) + 1) - 1 by NAT_1:13;
W .remove (m,n) = (W .cut (1,m)) .append (W .cut (n,(len W))) by A1, A2, A3, Def12;
then (W .remove (m,n)) . x = (W .cut (n,(len W))) . (a + 1) by A4, A8, A10, Lm13
.= W . (a + n) by ;
hence (W .remove (m,n)) . x = W . ((x - m) + n) by A7, A8; :: thesis: ( (x - m) + n is Element of NAT & (x - m) + n <= len W )
a + n is Element of NAT ;
hence (x - m) + n is Element of NAT by A7, A8; :: thesis: (x - m) + n <= len W
thus (x - m) + n <= len W by A7, A8, A9; :: thesis: verum