
theorem
  for G being _Graph, W being Walk of G, u,e,v being object
  st e Joins u,v,G & e in W.edges()
  holds G.walkOf(u,e,v) is_odd_substring_of W, 0
    or G.walkOf(v,e,u) is_odd_substring_of W, 0
proof
  let G be _Graph, W be Walk of G;
  let u,e,v be object;
  assume that
    A1: e Joins u,v,G and
    A2: e in W.edges();
  consider n being odd Element of NAT such that
    A3: n < len W & W.(n+1) = e by A2, GLIB_001:100;
  n+2 <= len W by A3, GLIB_001:1;
  then A4: G.walkOf(W.n,e,W.(n+2)) is_odd_substring_of W, 0 by A3, Th31;
  e Joins W.n,W.(n+2),G by A3, GLIB_001:def 3;
  then per cases by A1, GLIB_000:15;
  suppose W.n = u & W.(n+2) = v;
    hence thesis by A4;
  end;
  suppose W.n = v & W.(n+2) = u;
    hence thesis by A4;
  end;
end;
