
theorem Th23: :: cliRstaCR:
for R being with_finite_clique# symmetric RelStr
  holds clique# R = stability# ComplRelStr R
proof
 let R be with_finite_clique# symmetric RelStr;
    set k = stability# ComplRelStr R;
   consider A being finite StableSet of ComplRelStr R such that
A1: card(A) = k by DILWORTH:def 6;
   A is Clique of R by Th22;
   then
A2: ex C being finite Clique of R st card C = k by A1;
   now let T be finite Clique of R;
     T is StableSet of ComplRelStr R by Th19;
    hence card T <= k by DILWORTH:def 6;
   end;
 hence clique# R = stability# ComplRelStr R by A2,DILWORTH:def 4;
end;
