reserve G,G1,G2 for _Graph;
reserve W,W1,W2 for Walk of G;
reserve e,x,y,z for set;
reserve v for Vertex of G;
reserve n,m for Element of NAT;

theorem Th10:
  for n being odd Element of NAT st 1 < n & n <= len W holds W.(n-
  1) in W.vertexAt(n).edgesInOut()
proof
  let n be odd Element of NAT;
  assume that
A1: 1 < n and
A2: n <= len W;
  reconsider naa1 = n-1 as even Element of NAT by A1,INT_1:5;
  1+1 <= n by A1,NAT_1:13;
  then
A3: 1+1-1 <= n-1 by XREAL_1:13;
  n - 1 <= len W - 0 by A2,XREAL_1:13;
  then naa1 in dom W by A3,FINSEQ_3:25;
  then consider n5 being odd Element of NAT such that
A4: n5 = naa1-1 and
A5: naa1-1 in dom W and
  naa1+1 in dom W and
A6: W.naa1 Joins W.(n5), W.(naa1+1),G by Lm2;
  n5 <= len W by A4,A5,FINSEQ_3:25;
  then W.(n5) = W.vertexAt(n5) by Def8;
  then W.(n-1) Joins W.vertexAt(n5), W.vertexAt(n), G by A2,A6,Def8;
  hence thesis by GLIB_000:14,62;
end;
