
theorem Th50:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  st F is directed weak_SG-embedding holds
    (G2 is non-Dmulti implies G1 is non-Dmulti) &
    (G2 is Dsimple implies G1 is Dsimple)
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2;
  assume A1: F is directed weak_SG-embedding;
  thus A2: G2 is non-Dmulti implies G1 is non-Dmulti
  proof
    assume A3: G2 is non-Dmulti;
    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 in the_Edges_of G1 & e2 in the_Edges_of G1 by GLIB_000:def 14;
      then A5: e1 in dom F_E & e2 in dom F_E by A1, Def11;
      e1 Joins v1,v2,G1 & e2 Joins v1,v2,G1 by A4, GLIB_000:16;
      then v1 in the_Vertices_of G1 & v2 in the_Vertices_of G1
        by GLIB_000:13;
      then A6: v1 in dom F_V & v2 in dom F_V by A1, Def11;
      then A7: F_E.e1 DJoins F_V.v1,F_V.v2,G2 by A1, A4, A5;
      F_E.e2 DJoins F_V.v1,F_V.v2,G2 by A1, A4, A5, A6;
      hence e1 = e2 by A1, A3, A5, A7, FUNCT_1:def 4, GLIB_000:def 21;
    end;
    hence G1 is non-Dmulti by GLIB_000:def 21;
  end;
  assume G2 is Dsimple;
  then G1 is loopless non-Dmulti by A2, A1, Th35;
  hence G1 is Dsimple;
end;
