
theorem Th1:
  for C being FormalContext holds Bottom (ConceptLattice(C)) =
  Concept-with-all-Attributes(C) & Top (ConceptLattice(C)) =
  Concept-with-all-Objects(C)
proof
  let C be FormalContext;
A1: (@Concept-with-all-Objects(C))@ = Concept-with-all-Objects(C) by
CONLAT_1:def 21;
A2: for a being Element of ConceptLattice(C) holds a [= @
  Concept-with-all-Objects(C)
  proof
    let a be Element of ConceptLattice(C);
    a@ is-SubConcept-of (@Concept-with-all-Objects(C))@ by A1,CONLAT_1:30;
    hence thesis by CONLAT_1:43;
  end;
A3: for a being Element of ConceptLattice(C) holds @Concept-with-all-Objects
(C) "\/" a = @Concept-with-all-Objects(C) & a "\/" @Concept-with-all-Objects(C)
  = @Concept-with-all-Objects(C) by A2,LATTICES:def 3;
A4: (@Concept-with-all-Attributes(C))@ = Concept-with-all-Attributes(C) by
CONLAT_1:def 21;
A5: for a being Element of ConceptLattice(C) holds @
  Concept-with-all-Attributes(C) [= a
  proof
    let a be Element of ConceptLattice(C);
    (@Concept-with-all-Attributes(C))@ is-SubConcept-of a@ by A4,CONLAT_1:30;
    hence thesis by CONLAT_1:43;
  end;
  for a being Element of ConceptLattice(C) holds @
  Concept-with-all-Attributes(C) "/\" a = @Concept-with-all-Attributes(C) & a
  "/\" @Concept-with-all-Attributes(C) = @Concept-with-all-Attributes(C)
     by A5,LATTICES:4;
  hence thesis by A3,LATTICES:def 16,def 17;
end;
