
theorem
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  for v being Vertex of G1 st F is weak_SG-embedding
  holds v.degree() c= (F_V/.v).degree()
proof
  let G1, G2 be _Graph, F0 be PGraphMapping of G1, G2, v be Vertex of G1;
  assume A1: F0 is weak_SG-embedding;
  then F0_E is one-to-one;
  then consider E being Subset of the_Edges_of G2 such that
    A2: for G3 being reverseEdgeDirections of G2, E
    ex F being PGraphMapping of G1, G3 st F = F0 & F is directed &
      (F0 is weak_SG-embedding implies F is weak_SG-embedding) &
      (F0 is strong_SG-embedding implies F is strong_SG-embedding) &
      (F0 is isomorphism implies F is isomorphism) by Th93;
  set G3 = the reverseEdgeDirections of G2, E;
  consider F be PGraphMapping of G1, G3 such that
    A3: F = F0 & F is directed and
    A4: F0 is weak_SG-embedding implies F is weak_SG-embedding and
    (F0 is strong_SG-embedding implies F is strong_SG-embedding) &
    (F0 is isomorphism implies F is isomorphism) by A2;
  A5: v.degree() c= (F_V/.v).degree() by A1, A3, A4, Lm5;
  dom F_V = the_Vertices_of G1 by A1, A4, GLIB_010:def 11;
  then A6: v in dom F_V & v in dom F0_V by A3;
  F0_V/.v = F0_V.v by A6, PARTFUN1:def 6
    .= F_V/.v by A3, A6, PARTFUN1:def 6;
  hence thesis by A5, Th60;
end;
