
theorem Th76:
for G being with_finite_clique# SimpleGraph
  holds clique# G = stability# Complement G
proof
let G be with_finite_clique# SimpleGraph;
 set CG = Complement G; set sCG = stability# Complement G, cG = clique# G;
  consider C being finite Clique of G such that
A1: order C = cG by Def15;
A2: Vertices G = Vertices CG by Lm4;
  reconsider A = union C as StableSet of CG by Th72;
A3: card A = cG by A1;
  now let T be finite StableSet of CG;
   G SubgraphInducedBy T is Clique of G by Th75;
      then order (G SubgraphInducedBy T) <= cG by Def15;
    hence card T <= cG by A2,Lm9;
  end;
 hence cG = sCG by A3,Def24;
end;
