
theorem Th99:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  for v being Vertex of G1 st F is isomorphism
  holds v.degree() = (F_V/.v).degree()
proof
  let G1, G2 be _Graph, F0 be PGraphMapping of G1, G2;
  let v be Vertex of G1;
  assume A1: F0 is isomorphism;
  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 being PGraphMapping of G1, G3 such that
    A3: F = F0 & F is directed and
    (F0 is weak_SG-embedding implies F is weak_SG-embedding) &
    (F0 is strong_SG-embedding implies F is strong_SG-embedding) and
    A4: (F0 is isomorphism implies F is isomorphism) by A2;
  v.inDegree()=(F_V/.v).inDegree() & v.outDegree()=(F_V/.v).outDegree()
    by A1, A3, A4, Th98;
  then A5: (F_V/.v).degree() = v.degree();
  F0 is total & dom(F_V) = dom(F0_V) & v in the_Vertices_of G1 by A1, A3;
  then A6: v in dom F0_V & v in dom F_V by GLIB_010:def 11;
  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;
