
theorem
  for G1 being _Graph, G2 being G1-Disomorphic _Graph
  for G3 being G2-Disomorphic _Graph, F being DIsomorphism of G1, G2
  st G1 == G3 holds F" is DIsomorphism of G2, G3
proof
  let G1 be _Graph, G2 be G1-Disomorphic _Graph;
  let G3 be G2-Disomorphic _Graph, F being DIsomorphism of G1, G2;
  assume A1: G1 == G3;
  then reconsider F2 = F" as PGraphMapping of G2, G3 by Th9;
  A2: F2 is total
  proof
    thus dom F2_V = rng F_V by FUNCT_1:33
      .= the_Vertices_of G2 by Def12;
    thus dom F2_E = rng F_E by FUNCT_1:33
      .= the_Edges_of G2 by Def12;
  end;
  A3: F2 is onto
  proof
    thus rng F2_V = dom F_V by FUNCT_1:33
      .= the_Vertices_of G1 by Def11
      .= the_Vertices_of G3 by A1, GLIB_000:def 34;
    thus rng F2_E = dom F_E by FUNCT_1:33
      .= the_Edges_of G1 by Def11
      .= the_Edges_of G3 by A1, GLIB_000:def 34;
  end;
  A4: F2 is one-to-one;
  now
    let e,v,w be object;
    assume e in dom F2_E & v in dom F2_V & w in dom F2_V;
    then A5: e in dom(F"_E) & v in dom(F"_V) & w in dom(F"_V);
    assume e DJoins v,w,G2;
    then (F"_E).e DJoins (F"_V).v,(F"_V).w,G1 by A5, Def14;
    hence (F"_E).e DJoins (F"_V).v,(F"_V).w,G3 by A1, GLIB_000:88;
  end;
  then F2 is directed;
  hence thesis by A2, A3, A4;
end;
