theorem
  for G2 being _finite _Graph, v1,v2 being Vertex of G2, e being object
  for G1 being addEdge of G2,v1,e,v2 st not e in the_Edges_of G2
  holds G1.size() = G2.size() + 1
proof
  let G2 be _finite _Graph;
  let v1,v2 be Vertex of G2, e being object;
  let G1 be addEdge of G2,v1,e,v2;
  assume A1: not e in the_Edges_of G2;
  then A2: the_Edges_of G1 = the_Edges_of G2 \/ {e} by Def11;
  thus G1.size() = card the_Edges_of G1 by GLIB_000:def 25
    .= card the_Edges_of G2 + card {e} by A1, A2, CARD_2:40, ZFMISC_1:50
    .= G2.size() + card {e} by GLIB_000:def 25
    .= G2.size() + 1 by CARD_2:42;
end;
