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
  for G being loopless _Graph
  holds G is complete implies G.vChromaticNum() = G.order()
proof
  let G be loopless _Graph;
  G is G.order()-vcolorable & G.order() c= G.order() by Th29;
  then A1: G.order() in VColSet(G);
  assume A2: G is complete;
  A3: G.vChromaticNum() c= G.order() by Th56;
  now
    let x be set;
    assume x in VColSet(G);
    then consider c being cardinal Subset of G.order() such that
      A4: x = c & G is c-vcolorable;
    consider f being VColoring of G such that
      A5: f is proper & card rng f c= c by A4;
    thus G.order() c= x
    proof
      assume not G.order() c= x;
      then c in G.order() by A4, ORDINAL1:16;
      then card rng f in G.order() by A5, ORDINAL1:12;
      then card rng f in card dom f by PARTFUN1:def 2;
      then card rng f <> card dom f;
      :: use the pigeon hole principle to find two vertices in the same color
      then f is non one-to-one by CARD_1:70;
      then consider v,w being object such that
        A6: v in dom f & w in dom f & f.v = f.w & v <> w by FUNCT_1:def 4;
      reconsider v,w as Vertex of G by A6;
      v,w are_adjacent by A2, A6, CHORD:def 6;
      hence contradiction by A5, A6;
    end;
  end;
  then G.order() c= G.vChromaticNum() by A1, SETFAM_1:5;
  hence thesis by A3, XBOOLE_0:def 10;
end;
