
theorem Th22:
  for G1, G2 being _Graph, F being semi-continuous PGraphMapping of G1, G2
  st F_V is PVertexMapping of G1, G2 &
    for v,w being Vertex of G1 st v in dom F_V & w in dom F_V &
      F_V/.v, F_V/.w are_adjacent
    ex e9 being object st e9 in rng F_E & e9 Joins F_V.v,F_V.w,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 that
    A1: F_V is PVertexMapping of G1, G2 and
    A2: for v,w being Vertex of G1 st v in dom F_V & w in dom F_V &
        F_V/.v, F_V/.w are_adjacent
      ex e9 being object st e9 in rng F_E & e9 Joins F_V.v,F_V.w,G2;
  reconsider f = F_V as PVertexMapping of G1, G2 by A1;
  now
    let v,w,e9 be object;
    assume A3: v in dom f & w in dom f & e9 Joins f.v,f.w,G2;
    then e9 Joins f/.v,f.w,G2 by PARTFUN1:def 6;
    then e9 Joins f/.v,f/.w,G2 by A3, PARTFUN1:def 6;
    then consider e8 being object such that
      A4: e8 in rng F_E & e8 Joins f.v,f.w,G2 by A2, A3, CHORD:def 3;
    consider e being object such that
      A5: e in dom F_E & F_E.e = e8 by A4, FUNCT_1:def 3;
    take e;
    thus e Joins v,w,G1 by A3, A4, A5, GLIB_010:def 15;
  end;
  hence thesis by Th2;
end;
