reserve G, G2 for _Graph, V, E for set,
  v for object;

theorem Th9:
  for G2, E for G1 being reverseEdgeDirections of G2, E, v1,e,v2 being object
  holds (e Joins v1,v2,G2 iff e Joins v1,v2,G1)
proof
  let G2, E;
  let G1 be reverseEdgeDirections of G2, E;
  per cases;
  suppose A1: E c= the_Edges_of G2;
    let v1,e,v2 be object;
    thus e Joins v1,v2,G2 implies e Joins v1,v2,G1 by A1, Lm3;
    reconsider G3 = G2 as reverseEdgeDirections of G1, E by Th3;
    assume A2: e Joins v1,v2,G1;
    E c= the_Edges_of G1 by A1, Def1;
    then e Joins v1,v2,G3 by A2, Lm3;
    hence thesis;
  end;
  suppose not (E c= the_Edges_of G2);
    then G1 == G2 by Def1;
    hence thesis by GLIB_000:88;
  end;
end;
