
theorem Th17:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  st dom F_E = the_Edges_of G1 holds F_V is PVertexMapping of G1, G2
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2;
  assume A1: dom F_E = the_Edges_of G1;
  now
    let v,w be Vertex of G1;
    assume v in dom F_V & w in dom F_V & v,w are_adjacent;
    then consider e being object such that
      A2: e Joins v,w,G1 by CHORD:def 3;
    take e;
    thus e in dom F_E by A1, A2, GLIB_000:def 13;
    thus e Joins v,w,G1 by A2;
  end;
  hence thesis by Th16;
end;
