
theorem Th47:
for G being SimpleGraph
 st for x, y being set st x <> y & x in Vertices G & y in Vertices G
      holds {x,y} in Edges G
  holds G is clique
proof
 let G be SimpleGraph such that
A1: for x, y being set st x <> y & x in Vertices G & y in Vertices G
      holds {x,y} in Edges G;
   now
     let x, y be set such that
   A2: x in Vertices G and
   A3: y in Vertices G;
     per cases;
     suppose x <> y;
       then {x,y} in Edges G by A2,A3,A1;
       hence {x, y} in G;
     end;
     suppose x = y;
       then {x,y} = {x} by ENUMSET1:29;
      hence {x,y} in G by A2,Th24;
     end;
   end;
  then G = CompleteSGraph Vertices G by Th32;
 hence G is clique;
end;
