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 Th149:
  g is proper implies ex g9 being proper EColoring of G
    st rng f misses rng g9 & card rng g = card rng g9
proof
  assume A1: g is proper;
  set h = <: rng g --> rng f, id rng g :>, g9 = h*g;
  A2: dom h = rng g by Lm8;
  then reconsider g9 as EColoring of G by Th77;
  reconsider g9 as proper EColoring of G by A1, Th87;
  take g9;
  A3: rng g9 = rng h by A2, RELAT_1:28;
  hence rng f misses rng g9 by Lm11;
  thus card rng g = card [: rng g, {rng f} :] by CARD_1:69
    .= card [: {rng f}, rng g :] by CARD_2:4
    .= card rng g9 by A3, Lm10;
end;
