
theorem Th36:
  for G being _Graph, W being Walk of G
  holds 1 = W.findFirstVertex(W) & W.findLastVertex(W) = len W
proof
  let G be _Graph, W be Walk of G;
  A2: W is_odd_substring_of W, 0 by Th15, ABIAN:12;
  set n1 = W.findFirstVertex(W);
  set n2 = W.findLastVertex(W);
  consider k1 being even Nat such that
    A3: n1 = k1+1 and
    (for n being Nat st 1 <= n & n <= len W holds W.(k1+n) = W.n) and
    A5: for l being even Nat
      st for n being Nat st 1 <= n & n <= len W holds W.(l+n) = W.n
      holds k1 <= l by A2, Def3;
  A6: for n being Nat st 1 <= n & n <= len W holds W.(0+n) = W.n;
  then k1 <= 0 by A5;
  then k1 = 0;
  hence 1 = n1 by A3;
  consider k2 being even Nat such that
    A7: n2 = k2+len W and
    (for n being Nat st 1 <= n & n <= len W holds W.(k2+n) = W.n) and
    A9: for l being even Nat
      st for n being Nat st 1 <= n & n <= len W holds W.(l+n) = W.n
      holds k2 <= l by A2, Def4;
  k2 <= 0 by A6, A9;
  then k2 = 0;
  hence n2 = len W by A7;
end;
