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 Th147:
  t_V is proper & t_E is proper & rng(t_V) misses rng(t_E) implies t is proper
proof
  assume A1: t_V is proper & t_E is proper & rng(t_V) misses rng(t_E);
  now
    let e,v,w be object;
    assume e Joins v,w,G;
    then e in the_Edges_of G & v in the_Vertices_of G by GLIB_000:def 13, 13;
    then e in dom(t_E) & v in dom(t_V) by PARTFUN1:def 2;
    then t_E.e in rng(t_E) & t_V.v in rng(t_V) by FUNCT_1:3;
    hence t_V.v <> t_E.e by A1, XBOOLE_0:3;
  end;
  hence thesis by A1, Th146;
end;
