
theorem Th97:
  for G1 being _Graph, G2 being G1-isomorphic _Graph
  for G3 being G2-isomorphic _Graph, F being Isomorphism of G1, G2
  st (ex E being set st G3 is reverseEdgeDirections of G1, E)
  holds F" is Isomorphism of G2, G3
proof
  let G1 be _Graph, G2 be G1-isomorphic _Graph;
  let G3 be G2-isomorphic _Graph, F being Isomorphism of G1, G2;
  given E being set such that
    A1: G3 is reverseEdgeDirections of G1, E;
  G2 is reverseEdgeDirections of G2, {} by GLIB_009:43;
  then reconsider F2 = F" as PGraphMapping of G2, G3 by A1, Th10;
  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_007:4;
    thus rng F2_E = dom F_E by FUNCT_1:33
      .= the_Edges_of G1 by Def11
      .= the_Edges_of G3 by A1, GLIB_007:4;
  end;
  F2 is one-to-one;
  hence thesis by A2, A3;
end;
