
theorem :: Gsingle:
for G being SimpleGraph, x being set st Vertices G = {x} holds G = { {}, {x} }
proof
 let G be SimpleGraph, a be set such that
A1: Vertices G = {a};
A2: now
  assume Edges G <> {};
   then consider e being object such that
 A3: e in Edges G by XBOOLE_0:def 1;
    consider x, y being set such that
 A4: x <> y and
 A5: x in Vertices G and
 A6: y in Vertices G and e = {x, y} by A3,Th11;
   x = a by A5,A1,TARSKI:def 1;
  hence contradiction by A4,A6,A1,TARSKI:def 1;
 end;
A7: singletons Vertices G = { {a} } by A1,Th6;
  thus G = { {} } \/ singletons Vertices G \/ Edges G by Th27
    .= { {}, {a} } by A7,A2,ENUMSET1:1;
end;
