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;

theorem Th123:
  for G being loopless _Graph holds G.supDegree() c= G.eChromaticNum()
proof
  let G be loopless _Graph;
  G is G.eChromaticNum()-ecolorable by Lm14;
  then consider g being proper EColoring of G such that
    A1: card rng g c= G.eChromaticNum();
  set D = the set of all v.degree() where v is Vertex of G;
  now
    let x be object;
    assume x in G.supDegree();
    then x in union D by GLIB_013:def 6;
    then consider d being set such that
      A2: x in d & d in D by TARSKI:def 4;
    consider v being Vertex of G such that
      A3: d = v.degree() by A2;
    card rng(g|v.edgesInOut()) c= card rng g by RELAT_1:70, CARD_1:11;
    then A4: card rng(g|v.edgesInOut()) c= G.eChromaticNum() by A1, XBOOLE_1:1;
    g|v.edgesInOut() is one-to-one by Def5;
    then card rng(g|v.edgesInOut()) = card dom(g|v.edgesInOut()) by CARD_1:70
      .= card(dom g /\ v.edgesInOut()) by RELAT_1:61
      .= card(the_Edges_of G /\ v.edgesInOut()) by PARTFUN1:def 2
      .= card v.edgesInOut() by XBOOLE_1:28
      .= d by A3, GLIB_000:19;
    hence x in G.eChromaticNum() by A2, A4;
  end;
  hence thesis by TARSKI:def 3;
end;
