
theorem Th5:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  for e being object st e in dom F_E
  holds (the_Source_of G1).e in dom F_V & (the_Target_of G1).e in dom F_V
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2;
  let e be object;
  assume A1: e in dom F_E;
  consider f,g being Function such that
    A2: F = [f,g] and
    dom f c= the_Vertices_of G1 & rng f c= the_Vertices_of G2 &
    dom g c= the_Edges_of G1 & rng g c= the_Edges_of G2 and
    A3: for e being object holds e in dom g implies
      (the_Source_of G1).e in dom f & (the_Target_of G1).e in dom f and
    for e,v,w being object st e in dom g & v in dom f & w in dom f
      holds e Joins v,w,G1 implies g.e Joins f.v,f.w,G2 by Def8;
  thus thesis by A1, A2, A3;
end;
