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

theorem Th21:
  G is c-regular iff for C being Component of G holds C is c-regular
proof
  thus G is c-regular implies for C being Component of G holds C is c-regular
  proof
    assume A1: G is c-regular;
    let C be Component of G;
    let w be Vertex of C;
    the_Vertices_of C c= the_Vertices_of G;
    then reconsider v = w as Vertex of G by TARSKI:def 3;
    thus w.degree() = v.degree() by GLIBPRE0:44
      .= c by A1;
  end;
  assume A2: for C being Component of G holds C is c-regular;
  let v be Vertex of G;
  set C = the inducedSubgraph of G, G.reachableFrom(v);
  A3: the_Vertices_of C = G.reachableFrom(v) by GLIB_000:def 37;
  A4: C is c-regular by A2;
  v in G.reachableFrom(v) by GLIB_002:9;
  then reconsider w = v as Vertex of C by A3;
  thus v.degree() = w.degree() by GLIBPRE0:44
    .= c by A4;
end;
