
theorem Th128:
  for G1, G2 being _Graph, G being GraphUnion of G1, G2
  for v2 being Vertex of G2, v being Vertex of G
  st G1 tolerates G2 & the_Vertices_of G1 misses the_Vertices_of G2
  holds v = v2 implies G.reachableFrom(v) = G2.reachableFrom(v2)
proof
  let G1, G2 be _Graph, G be GraphUnion of G1, G2;
  let v2 be Vertex of G2, v be Vertex of G;
  assume that A1: G1 tolerates G2 and
    A2: the_Vertices_of G1 misses the_Vertices_of G2 & v = v2;
  G is Supergraph of G2 by A1, GLIB_014:26;
  then G2 is Subgraph of G by GLIB_006:57;
  then A3: G2.reachableFrom(v2) c= G.reachableFrom(v) by A2, GLIB_002:14;
  now
    let y be object;
    assume y in G.reachableFrom(v);
    then consider W being Walk of G such that
      A4: W is_Walk_from v,y by GLIB_002:def 5;
    now
      assume W is Walk of G1;
      then reconsider W1 = W as Walk of G1;
      W.first() = v2 by A2, A4, GLIB_001:def 23;
      then v2 in W.vertices() by GLIB_001:88;
      then v2 in W1.vertices() by GLIB_001:98;
      hence contradiction by A2, XBOOLE_0:3;
    end;
    then reconsider W2 = W as Walk of G2 by A1, A2, Th126;
    W2 is_Walk_from v2,y by A2, A4, GLIB_001:19;
    hence y in G2.reachableFrom(v2) by GLIB_002:def 5;
  end;
  then G.reachableFrom(v) c= G2.reachableFrom(v2) by TARSKI:def 3;
  hence thesis by A3, XBOOLE_0:def 10;
end;
