
theorem Th83:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  st F is isomorphism holds G1 is with_max_degree iff G2 is with_max_degree
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2;
  assume A1: F is isomorphism;
  hereby
    assume G1 is with_max_degree;
    then consider v being Vertex of G1 such that
      A2: v.degree() = G1.supDegree() and
      for w being Vertex of G1 holds w.degree() c= v.degree() by Th79;
    v.degree() = (F_V/.v).degree() by A1, GLIBPRE0:93;
    then (F_V/.v).degree() = G2.supDegree() by A1, A2, Th55;
    hence G2 is with_max_degree by Lm3;
  end;
  assume G2 is with_max_degree;
  then consider v being Vertex of G2 such that
    A3: v.degree() = G2.supDegree() and
    for w being Vertex of G2 holds w.degree() c= v.degree() by Th79;
  rng F_V = the_Vertices_of G2 by A1, GLIB_010:def 12;
  then consider v0 being object such that
    A4: v0 in dom F_V & F_V.v0 = v by FUNCT_1:def 3;
  reconsider v0 as Vertex of G1 by A4;
  F_V/.v0 = v by A4, PARTFUN1:def 6;
  then v.degree() = v0.degree() by A1, GLIBPRE0:93;
  then v0.degree() = G1.supDegree() by A1, A3, Th55;
  hence thesis by Lm3;
end;
