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 Th66:
  for n being odd Element of NAT st n <= len W holds 2 * ((n+1)
  div 2) - 1 = n & 1 <= (n+1) div 2 & (n+1) div 2 <= len W.vertexSeq()
proof
  let n be odd Element of NAT;
  assume
A1: n <= len W;
  set m = (n+1) div 2;
  2 divides n+1 by PEPIN:22;
  then
A2: 2 * m = n+1 by NAT_D:3;
  hence 2 * m - 1 = n;
A3: now
    assume m < 1;
    then m < 0 + 1;
    then m = 0 by NAT_1:13;
    hence contradiction by A2;
  end;
  then reconsider maa1 = m-1 as Element of NAT by INT_1:5;
  thus 1 <= m by A3;
  now
    assume len W.vertexSeq() < m;
    then len W.vertexSeq() < maa1 + 1;
    then len W.vertexSeq() <= maa1 by NAT_1:13;
    then 2 * len W.vertexSeq() <= 2 * maa1 by NAT_1:4;
    then len W + 1 <= (2 * m) - (2 * 1) by Def14;
    then len W + 1 + 2 <= n + 1 - 2 + 2 by A2,XREAL_1:7;
    then len W + 1 + 2 < n + 1 + 1 by NAT_1:13;
    then len W + 3 - 3 < n + 2 - 2 by XREAL_1:14;
    hence contradiction by A1;
  end;
  hence thesis;
end;
