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
  G is _trivial implies G.eChromaticNum() = G.size()
proof
  assume A1: G is _trivial;
  now
    let c;
    assume G is c-ecolorable;
    then consider g being proper EColoring of G such that
      A2: card rng g c= c;
    set v = the Vertex of G;
    g | v.edgesInOut() = g | the_Edges_of G by A1, GLIB_000:148
      .= g;
    then g is one-to-one by Def5;
    then card rng g = card dom g by CARD_1:70
      .= G.size() by PARTFUN1:def 2;
    hence G.size() c= c by A2;
  end;
  hence thesis by Th100, Th122;
end;
