
theorem Th6:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  for e being object st e in rng F_E
  holds (the_Source_of G2).e in rng F_V & (the_Target_of G2).e in rng F_V
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2;
  let e be object;
  assume e in rng F_E;
  then consider e0 being object such that
    A1: e0 in dom F_E & F_E.e0 = e by FUNCT_1:def 3;
  A2: (the_Source_of G1).e0 in dom F_V & (the_Target_of G1).e0 in dom F_V
    by A1, Th5;
  e0 Joins (the_Source_of G1).e0, (the_Target_of G1).e0, G1
    by A1, GLIB_000:def 13;
  then F_E.e0 Joins F_V.((the_Source_of G1).e0), F_V.((the_Target_of G1).e0),
    G2 by A1, A2, Th4;
  then per cases by A1, GLIB_000:def 13;
  suppose (the_Source_of G2).e = F_V.((the_Source_of G1).e0) &
      (the_Target_of G2).e = F_V.((the_Target_of G1).e0);
    hence thesis by A2, FUNCT_1:def 3;
  end;
  suppose (the_Target_of G2).e = F_V.((the_Source_of G1).e0) &
      (the_Source_of G2).e = F_V.((the_Target_of G1).e0);
    hence thesis by A2, FUNCT_1:def 3;
  end;
end;
