
theorem Th6:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  st rng F_V = the_Vertices_of G2 & G1.loops() c= dom F_E
  holds G1 is loopfull implies G2 is loopfull
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2;
  assume A1: rng F_V = the_Vertices_of G2 & G1.loops() c= dom F_E;
  assume A2: G1 is loopfull;
  now
    let v be Vertex of G2;
    consider v0 being object such that
      A3: v0 in dom F_V & F_V.v0 = v by A1, FUNCT_1:def 3;
    consider e0 being object such that
      A4: e0 Joins v0,v0,G1 by A2, A3;
    reconsider e = F_E.e0 as object;
    take e;
    e0 in G1.loops() by A4, GLIB_009:def 2;
    hence e Joins v,v,G2 by A1, A3, A4, GLIB_010:4;
  end;
  hence G2 is loopfull;
end;
