
theorem Th25:
  for C being FormalContext for CP being strict FormalConcept of C
  holds (the Extent of CP = {} implies CP = Concept-with-all-Attributes(C)) & (
  the Intent of CP = {} implies CP = Concept-with-all-Objects(C))
proof
  let C be FormalContext;
  let CP be strict FormalConcept of C;
A1: (AttributeDerivation(C)).(the Intent of CP) = the Extent of CP by Def9;
A2: (ObjectDerivation(C)).(the Extent of CP) = the Intent of CP by Def9;
A3: now
    assume
A4: the Intent of CP = {};
    then the Extent of CP = the carrier of C by A1,Th18;
    then CP is universal;
    hence CP = Concept-with-all-Objects(C) by A2,A1,A4,Def12;
  end;
  now
    assume
A5: the Extent of CP = {};
    then the Intent of CP = the carrier' of C by A2,Th17;
    then CP is co-universal;
    hence CP = Concept-with-all-Attributes(C) by A2,A1,A5,Def13;
  end;
  hence thesis by A3;
end;
