reserve E,V for set, G,G1,G2 for _Graph, c,c1,c2 for Cardinal, n for Nat;
reserve f for VColoring of G;

theorem Th8:
  for F being PGraphMapping of G1,G st dom F_V = the_Vertices_of G1
  holds f*F_V is VColoring of G1
proof
  let F be PGraphMapping of G1,G;
  assume A1: dom F_V = the_Vertices_of G1;
  rng F_V c= the_Vertices_of G;
  then rng F_V c= dom f by PARTFUN1:def 2;
  then dom(f*(F_V)) = dom F_V by RELAT_1:27;
  hence thesis by A1, PARTFUN1:def 2;
end;
