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;
reserve t for TColoring of G;

theorem Th145:
  for F being PGraphMapping of G1,G st F is total
  holds [ t_V*(F_V) , t_E*(F_E) ] is TColoring of G1
proof
  let F be PGraphMapping of G1,G;
  assume A1: F is total;
  then A2: t_V*(F_V) is VColoring of G1 by Th9;
  t_E*(F_E) is EColoring of G1 by A1, Th84;
  hence thesis by A2, Def9;
end;
