
theorem :: CompleteClico
for n being non zero Nat holds cliquecover# CompleteRelStr n = 1
proof
let n be non zero Nat;
   set R = CompleteRelStr n; set cR = the carrier of R;
   reconsider C = {cR} as a_partition of cR by EQREL_1:39;
A1: now let x be set;
     assume x in C;
       then x = [#]R by TARSKI:def 1;
       hence x is Clique of R;
   end;
A2: now
     take C;
     thus C is finite;
     thus C is Clique-partition of R by A1,DILWORTH:def 11;
     thus card C = 1 by CARD_1:30;
   end;
   now
     let C be finite Clique-partition of R;
       0+1 <= card C by NAT_1:13;
     hence 1 <= card C;
   end;
 hence cliquecover# CompleteRelStr n = 1 by A2,Def5;
end;
