
theorem Th94:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  for v being Vertex of G1 st F is directed weak_SG-embedding holds
    v.inDegree() c= (F_V/.v).inDegree() &
    v.outDegree() c= (F_V/.v).outDegree()
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2, v be Vertex of G1;
  assume A1: F is directed weak_SG-embedding;
  then A2: dom F_E = the_Edges_of G1 by GLIB_010:def 11;
  set f = F_E|(v.edgesIn());
  A3: dom f = dom F_E /\ v.edgesIn() by RELAT_1:61
    .= v.edgesIn() by A2, XBOOLE_1:28;
  A4: rng f = F_E.:(v.edgesIn()) by RELAT_1:115;
  dom F_V = the_Vertices_of G1 by A1, GLIB_010:def 11;
  then F_E.:(v.edgesIn()) c= (F_V/.v).edgesIn() by A1, Th87;
  then A5: card(F_E.:(v.edgesIn())) c= card (F_V/.v).edgesIn() by CARD_1:11;
  f is one-to-one by A1, FUNCT_1:52;
  hence v.inDegree() c= (F_V/.v).inDegree() by A3, A4, A5, CARD_1:70;
  set f = F_E|(v.edgesOut());
  A6: dom f = dom F_E /\ v.edgesOut() by RELAT_1:61
    .= v.edgesOut() by A2, XBOOLE_1:28;
  A7: rng f = F_E.:(v.edgesOut()) by RELAT_1:115;
  dom F_V = the_Vertices_of G1 by A1, GLIB_010:def 11;
  then F_E.:(v.edgesOut()) c= (F_V/.v).edgesOut() by A1, Th87;
  then A8: card(F_E.:(v.edgesOut())) c= card (F_V/.v).edgesOut() by CARD_1:11;
  f is one-to-one by A1, FUNCT_1:52;
  hence v.outDegree() c= (F_V/.v).outDegree() by A6, A7, A8, CARD_1:70;
end;
