
theorem Th29:
for G being SimpleGraph, x being set
 st x in G & x <> {}
  holds (ex y being set st x = {y} & y in Vertices G) or x in Edges G
proof
 let G be SimpleGraph, x be set such that
A1: x in G and
A2: x <> {};
   x in { {} } \/ singletons Vertices G \/ Edges G by A1,Th27;
   then x in {{}} \/ singletons Vertices G or x in Edges G by XBOOLE_0:def 3;
   then A3: x in {{}} or x in singletons Vertices G or x in Edges G
     by XBOOLE_0:def 3;
   per cases by A3,A2,TARSKI:def 1;
   suppose x in singletons Vertices G;
      then consider f being Subset of Vertices G such that
   A4: x = f and
   A5: f is 1-element;
      consider v being set such that
   A6: v in Vertices G and
   A7: f = {v} by A5,Th9;
     thus thesis by A7,A6,A4;
   end;
   suppose x in Edges G;
     hence thesis;
   end;
end;
