
theorem Th23:
  for G1, G2 being _Graph, F being semi-continuous PGraphMapping of G1, G2
  st dom F_E = the_Edges_of G1 & rng F_E = the_Edges_of G2
  holds F_V is continuous PVertexMapping of G1, G2
proof
  let G1, G2 be _Graph, F be semi-continuous PGraphMapping of G1, G2;
  assume A1: dom F_E = the_Edges_of G1 & rng F_E = the_Edges_of G2;
  then A2: F_V is PVertexMapping of G1, G2 by Th17;
  now
    let v,w be Vertex of G1;
    assume A3: v in dom F_V & w in dom F_V & F_V/.v, F_V/.w are_adjacent;
    then consider e being object such that
      A4: e Joins F_V/.v,F_V/.w,G2 by CHORD:def 3;
    take e;
    thus e in rng F_E by A1, A4, GLIB_000:def 13;
    e Joins F_V.v,F_V/.w,G2 by A3, A4, PARTFUN1:def 6;
    hence e Joins F_V.v,F_V.w,G2 by A3, PARTFUN1:def 6;
  end;
  hence thesis by A2, Th22;
end;
