
theorem Th8:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  st F is semi-continuous & dom F_V=the_Vertices_of G1 & G2.loops() c= rng F_E
  holds G2 is loopfull implies G1 is loopfull
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2;
  assume that
    A1: F is semi-continuous and
    A2: dom F_V = the_Vertices_of G1 & G2.loops() c= rng F_E and
    A3: G2 is loopfull;
  now
    let v be Vertex of G1;
    F_V.v in rng F_V by A2, FUNCT_1:3;
    then consider e being object such that
      A4: e Joins F_V.v,F_V.v,G2 by A3;
    e in G2.loops() by A4, GLIB_009:def 2;
    then consider e0 being object such that
      A5: e0 in dom F_E & F_E.e0 = e by A2, FUNCT_1:def 3;
    take e0;
    thus e0 Joins v,v,G1 by A1, A2, A4, A5, GLIB_010:def 15;
  end;
  hence G1 is loopfull;
end;
