
theorem Th69:
for X being finite set holds chromatic# CompleteSGraph X = card X
proof
 let X be finite set;
  set n = card X;  set G = CompleteSGraph X;  set D = SmallestPartition X;
A1: card D = card X by TOPGEN_2:12;
A2: Vertices G = X by Lm1;
   reconsider D as a_partition of Vertices G by Lm1;
  A3: D is StableSet-wise proof
        let x be set;
        assume A4: x in D;
         then reconsider xx = x as Subset of Vertices G;
         xx is stable proof
           let x, y be set such that
         A5: x <> y and
         A6: x in xx and
         A7: y in xx;
               X is non empty by A4;
               then D = the set of all {a} where a is Element of X
                 by EQREL_1:37;
               then consider a being Element of X such that
         A8: xx = {a} by A4;
             a = x & y = a by A8,A6,A7,TARSKI:def 1;
          hence {x,y} nin G by A5;
         end;
        hence x is StableSet of G;
      end;
   for C being finite Coloring of G holds card X <= card C proof
    let C be finite Coloring of G;
    assume A9: card X > card C;
    then X is non empty;
    then consider p being set, x, y being object such that
   A10: p in C and
   A11: x in p and
   A12: y in p and
   A13: x <> y by A9,A2,Th7;
   A14: p is StableSet of G by A10,Def20;
       reconsider p as Subset of Vertices G by A10;
   A15: {x,y} nin G by A14,A11,A12,A13,Def19;
       p c= X by A2;
    hence contradiction by A11,A12,A15,Th34;
   end;
 hence chromatic# CompleteSGraph X = n by A1,A3,Def22;
end;
