
theorem Th64:
  for S being vertex-disjoint GraphUnionSet, G being GraphUnion of S
  for H being Element of S, v being Vertex of G, w being Vertex of H
  st v = w holds G.reachableFrom(v) = H.reachableFrom(w)
proof
  let S be vertex-disjoint GraphUnionSet, G being GraphUnion of S;
  let H be Element of S, v be Vertex of G, w being Vertex of H;
  assume A1: v = w;
  H is Subgraph of G by GLIB_014:21;
  then A2: H.reachableFrom(w) c= G.reachableFrom(v) by A1, GLIB_002:14;
  now
    let x be object;
    assume x in G.reachableFrom(v);
    then consider W being Walk of G such that
      A3: W is_Walk_from v,x by GLIB_002:def 5;
    consider H9 being Element of S such that
      A4: W is Walk of H9 by Th58;
    reconsider W9 = W as Walk of H9 by A4;
    A5: W9 is_Walk_from v,x by A3, GLIB_001:19;
    then v is Vertex of H9 & x is Vertex of H9 by GLIB_001:18;
    then the_Vertices_of H meets the_Vertices_of H9 by A1, XBOOLE_0:3;
    then reconsider W0 = W9 as Walk of H by Def18;
    W0 is_Walk_from w,x by A1, A5, GLIB_001:19;
    hence x in H.reachableFrom(w) by GLIB_002:def 5;
  end;
  then G.reachableFrom(v) c= H.reachableFrom(w) by TARSKI:def 3;
  hence thesis by A2, XBOOLE_0:def 10;
end;
