
theorem Th34:
  for G being _Graph, W1, W2 being Walk of G st W2 is_odd_substring_of W1, 0
  holds W1.(W1.findFirstVertex(W2)) = W2.first() &
    W1.(W1.findLastVertex(W2)) = W2.last()
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
    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
    A4: W1.findLastVertex(W2) = k2+len W2 and
    A5: for n being Nat st 1 <= n & n <= len W2 holds W1.(k2+n) = W2.n and
    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;
  A6: 1 <= len W2 by ABIAN:12;
  then W1.(W1.findFirstVertex(W2)) = W2.1 by A2, A3;
  hence W1.(W1.findFirstVertex(W2)) = W2.first() by GLIB_001:def 6;
  W1.(W1.findLastVertex(W2)) = W2.len W2 by A4, A5, A6;
  hence W1.(W1.findLastVertex(W2)) = W2.last() by GLIB_001:def 7;
end;
