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 Th98:
  e in W.edges() iff ex n being odd Element of NAT st n < len W & W.(n+1) = e
proof
  hereby
    assume e in W.edges();
    then consider n1 being even Element of NAT such that
A1: 1 <= n1 and
A2: n1 <= len W and
A3: W.n1 = e by Lm46;
    reconsider n = n1-1 as odd Element of NAT by A1,INT_1:5;
    take n;
    n1-1 < len W - 0 by A2,XREAL_1:15;
    hence n < len W;
    thus W.(n+1) = e by A3;
  end;
  given n being odd Element of NAT such that
A4: n < len W and
A5: W.(n+1) = e;
A6: 1 <= n+1 by NAT_1:12;
  n+1 <= len W by A4,NAT_1:13;
  hence thesis by A5,A6,Lm46;
end;
