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 G being _Graph, W being Walk of G, e,x being set holds e
Joins W.last(),x,G & not x in W.vertices() implies card W.addEdge(e).vertices()
  = card W.vertices() + 1
proof
  let G be _Graph, W be Walk of G, e, x be set;
  assume that
A1: e Joins W.last(),x,G and
A2: not x in W.vertices();
  card W.addEdge(e).vertices() = card (W.vertices()\/{x}) by A1,Th93;
  hence thesis by A2,CARD_2:41;
end;
