
theorem Th13:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  st F is weak_SG-embedding holds
    (G2 is vertex-finite implies G1 is vertex-finite) &
    (G2 is edge-finite implies G1 is edge-finite)
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2;
  assume A1: F is weak_SG-embedding;
  hereby
    assume A2: G2 is vertex-finite;
    dom F_V = the_Vertices_of G1 & F_V is one-to-one by A1, GLIB_010:def 11;
    then the_Vertices_of G1, F_V.:the_Vertices_of G1 are_equipotent
      by CARD_1:33;
    hence G1 is vertex-finite by A2, CARD_1:38;
  end;
  assume A3: G2 is edge-finite;
  dom F_E = the_Edges_of G1 & F_E is one-to-one by A1, GLIB_010:def 11;
  then the_Edges_of G1, F_E.:the_Edges_of G1 are_equipotent
    by CARD_1:33;
  hence G1 is edge-finite by A3, CARD_1:38;
end;
