
theorem Th17:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  st F_V is one-to-one holds F is semi-continuous
proof
  let G1, G2 be _Graph, F be 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 Joins F_V.v, F_V.w, G2;
  set v1 = (the_Source_of G1).e, v2 = (the_Target_of G1).e;
  A4: e Joins v1,v2,G1 by A2, GLIB_000:def 13;
  A5: v1 in dom F_V & v2 in dom F_V by A2, Th5;
  then F_E.e Joins F_V.v1,F_V.v2,G2 by A2, A4, Th4;
  then per cases by A3, GLIB_000:15;
  suppose F_V.v1 = F_V.v & F_V.v2 = F_V.w;
    then v1 = v & v2 = w by A1, A2, A5, FUNCT_1:def 4;
    hence e Joins v,w,G1 by A4;
  end;
  suppose F_V.v1 = F_V.w & F_V.v2 = F_V.v;
    then v1 = w & v2 = v by A1, A2, A5, FUNCT_1:def 4;
    hence e Joins v,w,G1 by A4, GLIB_000:14;
  end;
end;
