
theorem Th55:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  st F is isomorphism
  holds G1.supDegree() = G2.supDegree() & G1.minDegree() = G2.minDegree()
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2;
  assume A1: F is isomorphism;
  then rng F_V = the_Vertices_of G2 by GLIB_010:def 12;
  then A2: G1.supDegree() c= G2.supDegree() & G1.minDegree() c= G2.minDegree()
    by A1, Th51, Th52;
  reconsider F0 = F as one-to-one PGraphMapping of G1, G2 by A1;
  A3: F0" is isomorphism by A1, GLIB_010:75;
  then rng (F0"_V) = the_Vertices_of G1 by GLIB_010:def 12;
  then G2.supDegree() c= G1.supDegree() & G2.minDegree() c= G1.minDegree()
    by A3, Th51, Th52;
  hence thesis by A2, XBOOLE_0:def 10;
end;
