
theorem Th66:
for G being SimpleGraph, S being Clique of G, L being set
 st L c= Vertices S holds G SubgraphInducedBy L is Clique of G
proof
 let G be SimpleGraph, S be Clique of G, L be set such that
A1: L c= Vertices S;
  set g = G SubgraphInducedBy L;
 now let x, y be set such that
   A2: x in union g and
   A3: y in union g;
   A4: x in L by A2,Lm7;
   A5: y in L by A3,Lm7;
   A6: {x,y} in S by A4,A5,A1,Th53;
     thus {x,y} in g by A4,A5,A6,Lm10;
   end;
  then g = CompleteSGraph Vertices g by Th32;
 hence g is Clique of G;
end;
