
theorem Th33: :: CompleteCli:
for n being Nat holds clique# CompleteRelStr n = n
proof
 let n be Nat;
 set R = CompleteRelStr n;
A1: card card n = card n;
    [#]R = n by Def7;
   then
A2: ex C being finite Clique of R st card C = n by A1;
   for T being finite Clique of R holds card T <= n proof
   let T be finite Clique of R;
   card n = n;
   then
 A3: card the carrier of R = n by Def7;
 A4: card T <= clique# R by DILWORTH:def 4;
     clique# R <= n by A3,Th11;
   hence thesis by A4,XXREAL_0:2;
 end;
 hence clique# R = n by A2,DILWORTH:def 4;
end;
