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, v being set st W1 = W2
  holds W1.find(v) = W2.find(v) & W1.rfind(v) = W2.rfind(v)
proof
  let W1 be Walk of G1, W2 be Walk of G2, v be set;
  assume
A1: W1 = W2;
  now
    per cases;
    suppose
A2:   v in W1.vertices();
      then
A3:   W2.(W1.find(v)) = v by A1,Def19;
A4:   v in W2.vertices() by A1,A2,Th74;
A5:   for n being odd Nat st n <= len W2 & W2.n = v holds W1.find(v) <= n
      by A1,A2,Def19;
      W1.find(v) <= len W2 by A1,A2,Def19;
      hence W1.find(v) = W2.find(v) by A4,A3,A5,Def19;
A6:   W2.(W1.rfind(v)) = v by A1,A2,Def21;
A7:   for n being odd Element of NAT st n <= len W2 & W2.n = v holds n <=
      W1 .rfind(v) by A1,A2,Def21;
      W1.rfind(v) <= len W2 by A1,A2,Def21;
      hence W1.rfind(v) = W2.rfind(v) by A4,A6,A7,Def21;
    end;
    suppose
A8:   not v in W1.vertices();
      then
A9:   not v in W2.vertices() by A1,Th74;
      thus W1.find(v) = len W2 by A1,A8,Def19
        .= W2.find(v) by A9,Def19;
      thus W1.rfind(v) = len W2 by A1,A8,Def21
        .= W2.rfind(v) by A9,Def21;
    end;
  end;
  hence thesis;
end;
