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 Th152:
  for t1 being TColoring of G1, t2 being TColoring of G2
  st G1 == G2 & t1 = t2 & t1 is proper holds t2 is proper
proof
  let t1 be TColoring of G1, t2 be TColoring of G2;
  assume A1: G1 == G2 & t1 = t2 & t1 is proper;
  then A2: t2_V is proper by Th16;
  A3: t2_E is proper by A1, Th90;
  now
    let e,v,w be object;
    assume e Joins v,w,G2;
    then e Joins v,w,G1 by A1, GLIB_000:88;
    hence t2_V.v <> t2_E.e by A1, Th146;
  end;
  hence thesis by A2, A3, Th146;
end;
