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

theorem Th25:
  for G2, E for G1 being reverseEdgeDirections of G2, E
  st E c= the_Edges_of G2 & G2 is non-Dmulti &
    for e1,e2,v1,v2 being object st e1 Joins v1,v2,G2 & e2 Joins v1,v2,G2
    holds (e1 in E & e2 in E) or (not e1 in E & not e2 in E)
  holds G1 is non-Dmulti
proof
  let G2, E;
  let G1 be reverseEdgeDirections of G2, E;
  assume that
    A1: E c= the_Edges_of G2 and
    A2: G2 is non-Dmulti and
    A3: for e1,e2,v1,v2 being object st e1 Joins v1,v2,G2 & e2 Joins v1,v2,G2
      holds (e1 in E & e2 in E) or (not e1 in E & not e2 in E);
  for e1,e2,v1,v2 being object holds e1 DJoins
    v1,v2,G1 & e2 DJoins v1,v2,G1 implies e1 = e2
  proof
    let e1,e2,v1,v2 be object;
    assume A4: e1 DJoins v1,v2,G1 & e2 DJoins v1,v2,G1;
    then e1 Joins v1,v2,G1 & e2 Joins v1,v2,G1 by GLIB_000:16;
    then e1 Joins v1,v2,G2 & e2 Joins v1,v2,G2 by Th9;
    then per cases by A3;
    suppose e1 in E & e2 in E;
      then e1 DJoins v2,v1,G2 & e2 DJoins v2,v1,G2 by A1, A4, Th7;
      hence thesis by A2, GLIB_000:def 21;
    end;
    suppose (not e1 in E & not e2 in E);
      then e1 DJoins v1,v2,G2 & e2 DJoins v1,v2,G2 by A1, A4, Th8;
      hence thesis by A2, GLIB_000:def 21;
    end;
  end;
  hence thesis by GLIB_000:def 21;
end;
