
theorem
  for G being _Graph, W being Walk of G, u,e,v being object
  st e Joins u,v,G & G.walkOf(u,e,v) is_odd_substring_of W, 0
  holds e in W.edges() & u in W.vertices() & v in W.vertices()
proof
  let G be _Graph, W be Walk of G, u,e,v be object;
  set W2 = G.walkOf(u,e,v);
  assume e Joins u,v,G & W2 is_odd_substring_of W, 0;
  then A1: e Joins u,v,G & W2 is_odd_substring_of W, 1 by Th18;
  then A2: len W2 = 3 by GLIB_001:14;
  consider i being odd Nat such that
    A4: 1<=i & i<=len W & mid(W,i,(i-'1)+len W2) = W2 by A1;
  set j = (i-'1)+len W2;
  set M = the_Vertices_of G \/ the_Edges_of G;
  W2 is_odd_substring_of W, i by A4;
  then W2 is_substring_of W, i by Th12;
  then A5: 1 <= j & j <= len W & i <= j by Th11;
  len mid(W,i,j) = 3 by A1, A4, GLIB_001:14;
  then A8: mid(W,i,j).1 = W.(1+i-'1) & mid(W,i,j).2 = W.(2+i-'1) &
    mid(W,i,j).3 = W.(3+i-'1) by A4, A5, FINSEQ_6:118;
  1+i-'1 = 1+i-1 & 2+i-'1 = 2+i-1 & 3+i-'1 = 3+i-1 by NAT_D:37;
  then A9: W2.1 = W.i & W2.2 = W.(i+1) & W2.3 = W.(i+2) by A8, A4;
  W2 = <*u,e,v*> by A1, GLIB_001:def 5;
  then A10: W.i = u & W.(i+1) = e & W.(i+2) = v by A9;
  j = i+3-'1 by A2, A4, NAT_D:38
    .= i+3-1 by NAT_D:37
    .= i+2;
  then A11: 1 <= i+2 & i+2 <= len W by A5;
  i+0 <= i+1 & i+1 <= i+2 by XREAL_1:6;
  then A12: 1 <= i+1 & i+1 <= len W by A4, A11, XXREAL_0:2;
  reconsider k=i+1 as even Element of NAT;
  reconsider l=i+2 as odd Element of NAT;
  thus e in W.edges() by A12, A10, GLIB_001:99;
  i in NAT by ORDINAL1:def 12;
  hence u in W.vertices() by A4, A10, GLIB_001:87;
  thus v in W.vertices() by A11, A10, GLIB_001:87;
end;
