reserve c,c1,c2 for Cardinal, G,G1,G2 for _Graph, v for Vertex of G;

theorem Th40:
  for F being PGraphMapping of G1, G2 st F is isomorphism
  holds G1 is regular iff G2 is regular
proof
  let F be PGraphMapping of G1, G2;
  assume A1: F is isomorphism;
  hereby
    assume G1 is regular;
    then consider c being Cardinal such that
      A2: G1 is c-regular;
    G2 is c-regular by A1, A2, Th30;
    hence G2 is regular;
  end;
  assume G2 is regular;
  then consider c being Cardinal such that
    A3: G2 is c-regular;
  G1 is c-regular by A1, A3, Th30;
  hence G1 is regular;
end;
