
theorem
  for G being _Graph, W1, W2 being Walk of G st W2 is_odd_substring_of W1, 0
  holds W1.findFirstVertex(W2) <= W1.findLastVertex(W2)
proof
  let G be _Graph;
  let W1, W2 be Walk of G;
  assume A1: W2 is_odd_substring_of W1, 0;
  then consider k1 being even Nat such that
    A2: W1.findFirstVertex(W2) = k1+1 and
    A3: for n being Nat st 1 <= n & n <= len W2 holds W1.(k1+n) = W2.n and
    A4: for l being even Nat st
        for n being Nat st 1 <= n & n <= len W2 holds W1.(l+n) = W2.n
      holds k1 <= l by Def3;
  consider k2 being even Nat such that
    A5: W1.findLastVertex(W2) = k2+len W2 and
    A6: for n being Nat st 1 <= n & n <= len W2 holds W1.(k2+n) = W2.n and
    A7: for l being even Nat st
        for n being Nat st 1 <= n & n <= len W2 holds W1.(l+n) = W2.n
      holds k2 <= l by A1, Def4;
  k1 = k2
  proof
    A9: k2 <= k1 by A3, A7;
    k1 <= k2 by A4, A6;
    hence thesis by A9, XXREAL_0:1;
  end;
  hence thesis by A2, A5, ABIAN:12, XREAL_1:6;
end;
