
theorem Th49:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  st F is weak_SG-embedding
  holds (G2 is _trivial implies G1 is _trivial) &
    (G2 is non-multi implies G1 is non-multi) &
    (G2 is simple implies G1 is simple) &
    (G2 is _finite implies G1 is _finite)
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2;
  assume A1: F is weak_SG-embedding;
  hereby
    assume G2 is _trivial;
    then card the_Vertices_of G2 = 1 by GLIB_000:def 19;
    then A2: G2.order() = 1 by GLIB_000:def 24;
    G1.order() = 1
    proof
      assume G1.order() <> 1;
      then G1.order() in 1 by A1, A2, Th45, CARD_1:3;
      then G1.order() = 0 by CARD_1:49, TARSKI:def 1;
      then card the_Vertices_of G1 = 0 by GLIB_000:def 24;
      hence thesis; :: by contradiction
    end;
    then card the_Vertices_of G1 = 1 by GLIB_000:def 24;
    hence G1 is _trivial by GLIB_000:def 19;
  end;
  thus A3: G2 is non-multi implies G1 is non-multi
  proof
    assume A4: G2 is non-multi;
    for e1,e2,v1,v2 being object
      holds e1 Joins v1,v2,G1 & e2 Joins v1,v2,G1 implies e1 = e2
    proof
      let e1,e2,v1,v2 be object;
      assume A5: e1 Joins v1,v2,G1 & e2 Joins v1,v2,G1;
      then e1 in the_Edges_of G1 & e2 in the_Edges_of G1 by GLIB_000:def 13;
      then A6: e1 in dom F_E & e2 in dom F_E by A1, Def11;
      v1 in the_Vertices_of G1 & v2 in the_Vertices_of G1 by A5, GLIB_000:13;
      then v1 in dom F_V & v2 in dom F_V by A1, Def11;
      then F_E.e1 Joins F_V.v1,F_V.v2,G2 & F_E.e2 Joins F_V.v1,F_V.v2,G2
        by A5, A6, Th4;
      hence e1 = e2 by A1, A4, A6, GLIB_000:def 20, FUNCT_1:def 4;
    end;
    hence G1 is non-multi by GLIB_000:def 20;
  end;
  hereby
    assume G2 is simple;
    then G1 is loopless non-multi by A3, A1, Th35;
    hence G1 is simple;
  end;
  assume G2 is _finite;
  then card the_Vertices_of G2 is finite & card the_Edges_of G2 is finite;
  then A7: G2.order() is finite & G2.size() is finite
    by GLIB_000:def 24, GLIB_000:def 25;
  G1.order() c= G2.order() & G1.size() c= G2.size() by A1, Th45;
  then card the_Vertices_of G1 is finite & card the_Edges_of G1 is finite
    by A7, GLIB_000:def 24, GLIB_000:def 25;
  then the_Vertices_of G1 is finite & the_Edges_of G1 is finite;
  hence thesis by GLIB_000:def 17;
end;
