
theorem Th41:
  for G2 being _Graph, v1,v2 being Vertex of G2, e being object
  for G1 being addEdge of G2,v1,e,v2
  for w1, w2 being Vertex of G1
  st not e in the_Edges_of G2 & w1 = v1 & w2 = v2
  holds w2 in G1.reachableFrom(w1)
proof
  let G2 be _Graph, v1,v2 be Vertex of G2, e be object;
  let G1 be addEdge of G2,v1,e,v2;
  let w1, w2 be Vertex of G1;
  assume that
    A1: not e in the_Edges_of G2 and
    A2: w1 = v1 & w2 = v2;
  e DJoins v1,v2,G1 by A1, GLIB_006:105;
  then e Joins w1,v2,G1 by A2, GLIB_000:16;
  then G1.walkOf(w1,e,w2) is_Walk_from w1,w2 by A2, GLIB_001:15;
  hence thesis by GLIB_002:def 5;
end;
