let G be _Graph; :: thesis: for W being Walk of G holds W .vertices() = (W .reverse()) .vertices()
let W be Walk of G; :: thesis: W .vertices() = (W .reverse()) .vertices()

now :: thesis: for x being object holds
( ( x in W .vertices() implies x in (W .reverse()) .vertices() ) & ( x in (W .reverse()) .vertices() implies x in W .vertices() ) )

reconsider lenW = len W as odd Element of NAT ;
let x be object ; :: thesis: ( ( x in W .vertices() implies x in (W .reverse()) .vertices() ) & ( x in (W .reverse()) .vertices() implies x in W .vertices() ) )

hereby :: thesis: ( x in (W .reverse()) .vertices() implies x in W .vertices() )

reconsider lenW = len W as odd Element of NAT ;
assume x in W .vertices() ; :: thesis: x in (W .reverse()) .vertices()
then consider n being odd Element of NAT such that
A1: n <= len W and
A2: W . n = x by Lm45;
A3: (lenW - n) + 1 is odd Element of NAT by A1, FINSEQ_5:1;
1 <= n by ABIAN:12;
then A4: n in dom W by A1, FINSEQ_3:25;
then n in Seg (len W) by FINSEQ_1:def 3;
then (lenW - n) + 1 in Seg (len W) by FINSEQ_5:2;
then (lenW - n) + 1 in dom W by FINSEQ_1:def 3;
then (lenW - n) + 1 <= len W by FINSEQ_3:25;
then A5: (lenW - n) + 1 <= len (W .reverse()) by FINSEQ_5:def 3;
(W .reverse()) . (((len W) - n) + 1) = x by A2, A4, Th23;
hence x in (W .reverse()) .vertices() by A3, A5, Lm45; :: thesis: verum

end;

assume x in (W .reverse()) .vertices() ; :: thesis: x in W .vertices()
then consider n being odd Element of NAT such that
A6: n <= len (W .reverse()) and
A7: (W .reverse()) . n = x by Lm45;
A8: 1 <= n by ABIAN:12;
then n in dom (W .reverse()) by A6, FINSEQ_3:25;
then A9: W . (((len W) - n) + 1) = x by A7, FINSEQ_5:def 3;
A10: n <= len W by A6, FINSEQ_5:def 3;
then n in Seg (len W) by A8, FINSEQ_1:1;
then (lenW - n) + 1 in Seg (len W) by FINSEQ_5:2;
then A11: (lenW - n) + 1 <= len W by FINSEQ_1:1;
(lenW - n) + 1 is odd Element of NAT by A10, FINSEQ_5:1;
hence x in W .vertices() by A9, A11, Lm45; :: thesis: verum

end;

hence W .vertices() = (W .reverse()) .vertices() by TARSKI:2; :: thesis: verum