reserve G,G1,G2 for _Graph;
reserve W,W1,W2 for Walk of G;
reserve e,x,y,z for set;
reserve v for Vertex of G;
reserve n,m for Element of NAT;

theorem
  W.vertices() = W.reverse().vertices()
proof
  now
    reconsider lenW = len W as odd Element of NAT;
    let x be object;
    hereby
      reconsider lenW = len W as odd Element of NAT;
      assume x in W.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;
    end;
    assume x in W.reverse().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;
  end;
  hence thesis by TARSKI:2;
end;
