
theorem Th74:
for G being SimpleGraph, C being StableSet of G
  holds (Complement G) SubgraphInducedBy C is Clique of Complement G
proof
 let G be SimpleGraph, C be StableSet of G; set CG = Complement G;
   set CGSC = CG SubgraphInducedBy C; set uCGSC = union CGSC;
  now let a, b be set such that
  A1: a <> b and
  A2: a in uCGSC and
  A3: b in uCGSC;
  A4: a in C & b in C by A2,A3,Lm7;
  A5: {a,b} nin Edges G by A4,A1,Def19;
      A6: {a,b} in CompleteSGraph Vertices G by A4,Th34;
      {a,b} in CG by A5,A6,XBOOLE_0:def 5;
     then {a, b} in CGSC by A4,Lm10;
    hence {a, b} in Edges CGSC by A1,Th12;
  end;
 hence CGSC is Clique of Complement G by Th47;
end;
