
theorem Th19:
  for G1, G2 being _Graph, F being directed PGraphMapping of G1, G2
  st for v,w being object st v in dom F_V & w in dom F_V &
      ex e being object st e DJoins v,w,G1
    holds ex e being object st e in dom F_E & e DJoins v,w,G1
  holds F_V is directed PVertexMapping of G1, G2
proof
  let G1, G2 be _Graph, F be directed PGraphMapping of G1, G2;
  assume A1: for v,w being object st v in dom F_V & w in dom F_V &
      ex e being object st e DJoins v,w,G1
    holds ex e being object st e in dom F_E & e DJoins v,w,G1;
  now
    let v,w,e be object;
    assume A2: v in dom F_V & w in dom F_V & e Joins v,w,G1;
    then per cases by GLIB_000:16;
    suppose e DJoins v,w,G1;
      then consider e0 being object such that
        A3: e0 in dom F_E & e0 DJoins v,w,G1 by A1, A2;
      reconsider e9 = F_E.e0 as object;
      take e9;
      e0 Joins v,w,G1 by A3, GLIB_000:16;
      hence e9 Joins F_V.v,F_V.w,G2 by A2, A3, GLIB_010:4;
    end;
    suppose e DJoins w,v,G1;
      then consider e0 being object such that
        A4: e0 in dom F_E & e0 DJoins w,v,G1 by A1, A2;
      reconsider e9 = F_E.e0 as object;
      take e9;
      e0 Joins v,w,G1 by A4, GLIB_000:16;
      hence e9 Joins F_V.v,F_V.w,G2 by A2, A4, GLIB_010:4;
    end;
  end;
  then A5: F_V is PVertexMapping of G1, G2 by Th1;
  now
    let v,w,e be object;
    assume A6: v in dom F_V & w in dom F_V & e DJoins v,w,G1;
    then consider e0 being object such that
      A7: e0 in dom F_E & e0 DJoins v,w,G1 by A1;
    reconsider e9 = F_E.e0 as object;
    take e9;
    thus e9 DJoins F_V.v,F_V.w,G2 by A6, A7, GLIB_010:def 14;
  end;
  hence thesis by A5, Def2;
end;
