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
  for W1 being Walk of G1, W2 being Walk of G2 st W1 = W2 holds for n
  being Element of NAT holds W1.vertexAt(n) = W2.vertexAt(n)
proof
  let W1 be Walk of G1, W2 be Walk of G2;
  assume
A1: W1 = W2;
  let n be Element of NAT;
  now
    per cases;
    suppose
A2:   n is odd & n <= len W1;
      hence W1.vertexAt(n) = W2.n by A1,Def8
        .= W2.vertexAt(n) by A1,A2,Def8;
    end;
    suppose
A3:   not (n is odd & n <= len W1);
      hence W1.vertexAt(n) = W1.first() by Def8
        .= W2.first() by A1
        .= W2.vertexAt(n) by A1,A3,Def8;
    end;
  end;
  hence thesis;
end;
