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
 for e,x being object holds
  e Joins W.last(),x,G implies W.addEdge(e).edges() = W.edges() \/ {e}
proof let e,x be object;
  set WB = G.walkOf(W.last(),e,W.last().adj(e));
  assume e Joins W.last(),x,G;
  then e in W.last().edgesInOut() by GLIB_000:62;
  then
A1: e Joins W.last(), W.last().adj(e), G by GLIB_000:67;
  then
A2: WB.first() = W.last() by Th14;
  WB.edges() = {e} by A1,Th106;
  hence thesis by A2,Th100;
end;
