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

theorem
  G is Dregular iff G.minInDegree() = G.supInDegree() &
    G.minOutDegree() = G.supOutDegree() & G.minInDegree() = G.minOutDegree()
proof
  hereby
    assume G is Dregular;
    then consider c being Cardinal such that
      A1: G is c-Dregular;
    thus G.minInDegree() = c by A1, Th44
      .= G.supInDegree() by A1, Th44;
    thus G.minOutDegree() = c by A1, Th44
      .= G.supOutDegree() by A1, Th44;
    thus G.minInDegree() = c by A1, Th44
      .= G.minOutDegree() by A1, Th44;
  end;
  assume G.minInDegree() = G.supInDegree() &
    G.minOutDegree() = G.supOutDegree() & G.minInDegree() = G.minOutDegree();
  then G is G.minInDegree()-Dregular by Th45;
  hence thesis;
end;
