
theorem Th54:
for G being with_finite_clique# SimpleGraph
  st clique# G = 0 holds Vertices G = {}
proof
 let G be with_finite_clique# SimpleGraph;
 assume
A1: clique# G = 0;
  assume Vertices G <> {};
     then consider v being object such that
   A2: v in Vertices G by XBOOLE_0:def 1;
     consider D being finite Clique of G  such that
   A3: Vertices D = {v} by A2,Th52;
     order D <= 0 by A1,Def15;
    hence contradiction by A3;
end;
