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 Th160:
  for F being PGraphMapping of G1,G, t9 being TColoring of G1
  st F is weak_SG-embedding & t9 = [ t_V*(F_V) , t_E*(F_E) ] & t is proper
  holds t9 is proper
proof
  let F be PGraphMapping of G1,G, t9 be TColoring of G1;
  assume A1: F is weak_SG-embedding & t9 = [t_V*(F_V),t_E*(F_E)] & t is proper;
  then A2: t9_V is proper by Th26;
  A3: t9_E is proper by A1, Th98;
  now
    let e,v,w be object;
    assume A4: e Joins v,w,G1;
    then e in the_Edges_of G1 by GLIB_000:def 13;
    then A5: e in dom F_E by A1, GLIB_010:def 11;
    ((the_Source_of G1).e = v & (the_Target_of G1).e = w) or
      ((the_Source_of G1).e = w & (the_Target_of G1).e = v)
      by A4, GLIB_000:def 13;
    then A6: v in dom F_V & w in dom F_V by A5, GLIB_010:5;
    then F_E.e Joins F_V.v,F_V.w,G by A4, A5, GLIB_010:4;
    then A7: t_V.(F_V.v) <> t_E.(F_E.e) by A1, Th146;
    t_V.(F_V.v) = t9_V.v & t_E.(F_E.e) = t9_E.e by A1, A5, A6, FUNCT_1:13;
    hence t9_V.v <> t9_E.e by A7;
  end;
  hence thesis by A2, A3, Th146;
end;
