
theorem Th18:
  for G1, G2 being _Graph, F being directed PGraphMapping of G1, G2
  st F_V is one-to-one holds F is semi-Dcontinuous
proof
  let G1, G2 be _Graph, F be directed PGraphMapping of G1, G2;
  assume A1: F_V is one-to-one;
  let e,v,w be object;
  assume that
    A2: e in dom F_E & v in dom F_V & w in dom F_V and
    A3: F_E.e DJoins F_V.v, F_V.w, G2;
  set v1 = (the_Source_of G1).e, v2 = (the_Target_of G1).e;
  A4: e DJoins v1,v2,G1 by A2, GLIB_000:def 14;
  A5: v1 in dom F_V & v2 in dom F_V by A2, Th5;
  then F_E.e DJoins F_V.v1,F_V.v2,G2 by A2, A4, Def14;
  then F_V.v1 = F_V.v & F_V.v2 = F_V.w by A3, GLIB_000:125;
  then v1 = v & v2 = w by A1, A2, A5, FUNCT_1:def 4;
  hence e DJoins v,w,G1 by A4;
end;
