reserve E,V for set, G,G1,G2 for _Graph, c,c1,c2 for Cardinal, n for Nat;
reserve f for VColoring of G;

theorem Th26:
  for F being PGraphMapping of G1,G, f9 being VColoring of G1
  st F is total & f9 = f*F_V & f is proper holds f9 is proper
proof
  let F be PGraphMapping of G1, G, f9 be VColoring of G1;
  assume A1: F is total & f9 = f*(F_V) & f is proper;
  now
    let e,v,w be object;
    assume A2: e Joins v,w,G1;
    then e in the_Edges_of G1 & v in the_Vertices_of G1 &
      w in the_Vertices_of G1 by GLIB_000:def 13, GLIB_000:13;
    then A3: e in dom F_E & v in dom F_V & w in dom F_V by A1, GLIB_010:def 11;
    then F_E.e Joins F_V.v,F_V.w,G by A2, GLIB_010:4;
    then f.(F_V.v) <> f.(F_V.w) by A1, Th10;
    then (f*F_V).v <> f.(F_V.w) by A3, FUNCT_1:13;
    hence f9.v <> f9.w by A1, A3, FUNCT_1:13;
  end;
  hence thesis by Th10;
end;
