
theorem Th63: :: Height4:
for G being with_finite_clique# SimpleGraph st clique# G = 1
  holds Vertices G is StableSet of G
proof
 let R be with_finite_clique# SimpleGraph;
 assume A1: clique# R = 1;
 set cR = Vertices R;
A2: cR c= cR;
 now
   let a, b be set such that
 A3: a <> b & a in cR & b in cR;
 A4: {a,b} c= cR by A3,ZFMISC_1:32;
   assume {a, b} in R;
   then reconsider H = R SubgraphInducedBy {a,b} as finite Clique of R
          by Th56;
    Vertices H = {a,b} by A4,Lm9;
    then order H = 2 by A3,CARD_2:57;
   hence contradiction by A1,Def15;
 end;
 hence cR is StableSet of R by A2,Def19;
end;
