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 Th148:
  t is proper iff for e1,e2,v,w1,w2 being object
    st e1 Joins v,w1,G & e2 Joins v,w2,G
    holds t_V.v <> t_V.w1 & t_V.v <> t_E.e1 &
      (e1 <> e2 implies t_E.e1 <> t_E.e2)
proof
  thus t is proper implies for e1,e2,v,w1,w2 being object
    st e1 Joins v,w1,G & e2 Joins v,w2,G
    holds t_V.v <> t_V.w1 & t_V.v <> t_E.e1 &
      (e1 <> e2 implies t_E.e1 <> t_E.e2)
    by Th10, Th86, Th146;
  assume A1: for e1,e2,v,w1,w2 being object
    st e1 Joins v,w1,G & e2 Joins v,w2,G
    holds t_V.v <> t_V.w1 & t_V.v <> t_E.e1 &
      (e1 <> e2 implies t_E.e1 <> t_E.e2);
  for e,v,w being object st e Joins v,w,G holds t_V.v <> t_V.w by A1;
  then A2: t_V is proper by Th10;
  for e1,e2,v,w1,w2 be object st e1 Joins v,w1,G & e2 Joins v,w2,G
      & t_E.e1 = t_E.e2 holds e1 = e2 by A1;
  then A3: t_E is proper by Th86;
  for e,v,w being object st e Joins v,w,G holds t_V.v <> t_E.e by A1;
  hence thesis by A2, A3, Th146;
end;
