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 Th150:
  f is proper implies ex f9 being VColoring of G
    st f9 is proper & rng f9 misses rng g & card rng f = card rng f9
proof
  assume A1: f is proper;
  set h = <: rng f --> rng g, id rng f :>, f9 = h*f;
  A2: dom h = rng f by Lm8;
  then reconsider f9 as VColoring of G by Th1;
  take f9;
  thus f9 is proper by A1, A2, Th12;
  A3: rng f9 = rng h by A2, RELAT_1:28;
  hence rng f9 misses rng g by Lm11;
  thus card rng f = card [: rng f, {rng g} :] by CARD_1:69
    .= card [: {rng g}, rng f :] by CARD_2:4
    .= card rng f9 by A3, Lm10;
end;
