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
  W.edges() c= G.edgesBetween(W.vertices())
proof
  now
    let e be object;
    assume e in W.edges();
    then consider
    v1,v2 being Vertex of G, n being odd Element of NAT such that
A1: n+2 <= len W and
A2: v1 = W.n and
    e = W.(n+1) and
A3: v2 = W.(n+2) and
A4: e Joins v1,v2,G by Lm47;
    n < len W by A1,Th1;
    then
A5: v1 in W.vertices() by A2,Lm45;
    v2 in W.vertices() by A1,A3,Lm45;
    hence e in G.edgesBetween(W.vertices()) by A4,A5,GLIB_000:32;
  end;
  hence thesis by TARSKI:def 3;
end;
