let G be _Graph; :: thesis: for W being Walk of G
for x being set holds
( x in W .vertices() iff ex n being odd Element of NAT st
( n <= len W & W . n = x ) )
let W be Walk of G; :: thesis: for x being set holds
( x in W .vertices() iff ex n being odd Element of NAT st
( n <= len W & W . n = x ) )
let x be set ; :: thesis: ( x in W .vertices() iff ex n being odd Element of NAT st
( n <= len W & W . n = x ) )
set vs = W .vertexSeq() ;

hereby :: thesis: ( ex n being odd Element of NAT st
( n <= len W & W . n = x ) implies x in W .vertices() )

assume x in W .vertices() ; :: thesis: ex n being odd Element of NAT st
( n <= len W & W . n = x )
then consider i being Nat such that
A1: i in dom (W .vertexSeq()) and
A2: (W .vertexSeq()) . i = x by FINSEQ_2:10;
set n1 = 2 * i;
reconsider n1 = 2 * i as even Nat ;
set n = n1 - 1;
A3: 1 <= i by A1, FINSEQ_3:25;
then 1 <= i + i by NAT_1:12;
then reconsider n = n1 - 1 as odd Element of NAT by INT_1:5;
take n = n; :: thesis: ( n <= len W & W . n = x )
A4: i <= len (W .vertexSeq()) by A1, FINSEQ_3:25;
then i * 2 <= (len (W .vertexSeq())) * 2 by XREAL_1:64;
then i * 2 <= (len W) + 1 by Def14;
then n1 - 1 <= ((len W) + 1) - 1 by XREAL_1:13;
hence n <= len W ; :: thesis: W . n = x
thus W . n = x by A2, A3, A4, Def14; :: thesis: verum

end;

assume ex n being odd Element of NAT st
( n <= len W & W . n = x ) ; :: thesis: x in W .vertices()
then consider n being odd Element of NAT such that
A5: n <= len W and
A6: W . n = x ;
set n1 = n + 1;
reconsider n1 = n + 1 as even Element of NAT ;
set i = n1 div 2;
A7: 2 divides n1 by PEPIN:22;
then A8: 2 * (n1 div 2) = n1 by NAT_D:3;
reconsider i = n1 div 2 as Element of NAT ;
1 <= n by ABIAN:12;
then 1 + 1 <= n1 by XREAL_1:7;
then 2 * 1 <= 2 * i by A7, NAT_D:3;
then A9: 1 <= i by XREAL_1:68;
n1 <= (len W) + 1 by A5, XREAL_1:7;
then 2 * i <= 2 * (len (W .vertexSeq())) by A8, Def14;
then A10: i <= len (W .vertexSeq()) by XREAL_1:68;
then A11: i in dom (W .vertexSeq()) by A9, FINSEQ_3:25;
(W .vertexSeq()) . i = W . ((2 * i) - 1) by A9, A10, Def14
.= x by A6, A8 ;
hence x in W .vertices() by A11, FUNCT_1:def 3; :: thesis: verum