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

theorem Th83:
  for F being PGraphMapping of G1,G st dom F_E = the_Edges_of G1
  holds g*(F_E) is EColoring of G1
proof
  let F be PGraphMapping of G1,G;
  assume A1: dom F_E = the_Edges_of G1;
  rng F_E c= the_Edges_of G;
  then rng F_E c= dom g by PARTFUN1:def 2;
  then dom(g*(F_E)) = dom F_E by RELAT_1:27;
  hence thesis by A1, PARTFUN1:def 2;
end;
