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 Th105:
  W.edges() = W.reverse().edges()
proof
  now
    let e be object;
    hereby
      assume e in W.edges();
      then consider n being even Element of NAT such that
A1:   1 <= n and
A2:   n <= len W and
A3:   W.n = e by Lm46;
A4:   n in dom W by A1,A2,FINSEQ_3:25;
      then
A5:   (len W - n + 1) in dom W.reverse() by Th23;
      then reconsider rn = len W - n + 1 as even Element of NAT;
A6:   1 <= rn by A5,FINSEQ_3:25;
A7:   rn <= len W.reverse() by A5,FINSEQ_3:25;
      e = W.reverse().(len W - n + 1) by A3,A4,Th23;
      hence e in W.reverse().edges() by A6,A7,Lm46;
    end;
    assume e in W.reverse().edges();
    then consider n being even Element of NAT such that
A8: 1 <= n and
A9: n <= len W.reverse() and
A10: W.reverse().n = e by Lm46;
A11: n in dom W.reverse() by A8,A9,FINSEQ_3:25;
    then
A12: (len W.reverse() - n + 1) in dom W.reverse().reverse() by Th23;
    then reconsider rn = len W.reverse() - n + 1 as even Element of NAT;
    e = W.reverse().reverse().(len W.reverse() - n + 1) by A10,A11,Th23;
    then
A13: e = W.rn;
    rn in dom W by A12;
    then
A14: rn <= len W by FINSEQ_3:25;
    1 <= rn by A12,FINSEQ_3:25;
    hence e in W.edges() by A13,A14,Lm46;
  end;
  hence thesis by TARSKI:2;
end;
