
theorem Th7:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  holds dom F_E c= G1.edgesBetween(dom F_V) &
    rng F_E c= G2.edgesBetween(rng F_V)
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2;
  for e being object holds e in dom F_E implies e in G1.edgesBetween(dom F_V)
  proof
    let e be object;
    assume A1: e in dom F_E;
    then (the_Source_of G1).e in dom F_V & (the_Target_of G1).e in dom F_V
      by Th5;
    hence thesis by A1, GLIB_000:31;
  end;
  hence dom F_E c= G1.edgesBetween(dom F_V) by TARSKI:def 3;
  for e being object holds e in rng F_E implies e in G2.edgesBetween(rng F_V)
  proof
    let e be object;
    assume A2: e in rng F_E;
    then (the_Source_of G2).e in rng F_V & (the_Target_of G2).e in rng F_V
      by Th6;
    hence thesis by A2, GLIB_000:31;
  end;
  hence thesis by TARSKI:def 3;
end;
