
theorem Th12:
  for C being FormalContext holds rng(gamma(C)) is supremum-dense
  & rng(delta(C)) is infimum-dense
proof
  let C be FormalContext;
  set G = rng(gamma(C));
  thus G is supremum-dense
  proof
    let a be Element of ConceptLattice(C);
A1: {ConceptStr(#O,A#) where O is Subset of the carrier of C, A is Subset
of the carrier' of C : ex o being Object of C st o in the Extent of a@ & O = (
AttributeDerivation(C)).((ObjectDerivation(C)).{o}) & A = (ObjectDerivation(C))
    .{o}} c= G
    proof
      let x be object;
      assume x in {ConceptStr(#O,A#) where O is Subset of the carrier of C, A
is Subset of the carrier' of C : ex o being Object of C st o in the Extent of a
      @ & O = (AttributeDerivation(C)).((ObjectDerivation(C)).{o}) & A = (
      ObjectDerivation(C)).{o}};
      then consider O being Subset of the carrier of C, A being Subset of the
      carrier' of C such that
A2:   x = ConceptStr(#O,A#) and
A3:   ex o being Object of C st o in the Extent of a@ & O = (
AttributeDerivation(C)).((ObjectDerivation(C)).{o}) & A = (ObjectDerivation(C))
      .{o};
      consider o being Object of C such that
      o in the Extent of a@ and
      O = (AttributeDerivation(C)).((ObjectDerivation(C)).{o}) and
A4:   A = (ObjectDerivation(C)).{o} by A3;
      consider y being Element of ConceptLattice(C) such that
A5:   (gamma(C)).o = y;
      dom(gamma(C)) = the carrier of C & ex O9 being Subset of the carrier
of C, A9 being Subset of the carrier' of C st y = ConceptStr(#O9,A9#) & O9 = (
AttributeDerivation(C)) .((ObjectDerivation(C)).{o}) & A9 = (ObjectDerivation(
      C)).{o} by A5,Def4,FUNCT_2:def 1;
      hence thesis by A2,A3,A4,A5,FUNCT_1:def 3;
    end;
    "\/"({ConceptStr(#O,A#) where O is Subset of the carrier of C, A is
Subset of the carrier' of C : ex o being Object of C st o in the Extent of a@ &
    O = ( AttributeDerivation(C)).((ObjectDerivation(C)).{o}) & A = (
ObjectDerivation(C)) .{o}}, ConceptLattice(C)) = a@ & a = a@ by Th9,
CONLAT_1:def 21;
    hence thesis by A1;
  end;
  let b be Element of ConceptLattice(C);
  set G = rng(delta(C));
A6: {ConceptStr(#O,A#) where O is Subset of the carrier of C, A is Subset of
the carrier' of C : ex a being Attribute of C st a in the Intent of b@ & O = (
AttributeDerivation(C)).{a} & A = (ObjectDerivation(C)).((AttributeDerivation(C
  )).{a})} c= G
  proof
    let x be object;
    assume x in {ConceptStr(#O,A#) where O is Subset of the carrier of C, A
is Subset of the carrier' of C : ex a being Attribute of C st a in the Intent
    of b@ & O = (AttributeDerivation(C)).{a} & A = (ObjectDerivation(C)).((
    AttributeDerivation(C)).{a})};
    then consider O being Subset of the carrier of C, A being Subset of the
    carrier' of C such that
A7: x = ConceptStr(#O,A#) and
A8: ex a being Attribute of C st a in the Intent of b@ & O = (
AttributeDerivation(C)).{a} & A = (ObjectDerivation(C)).((AttributeDerivation(C
    )).{a});
    consider a being Attribute of C such that
    a in the Intent of b@ and
A9: O = (AttributeDerivation(C)).{a} and
    A = (ObjectDerivation(C)).((AttributeDerivation(C)).{a}) by A8;
    consider y being Element of ConceptLattice(C) such that
A10: (delta(C)).a = y;
    dom(delta(C)) = the carrier' of C & ex O9 being Subset of the carrier
of C, A9 being Subset of the carrier' of C st y = ConceptStr(#O9,A9#) & O9 = (
    AttributeDerivation(C)) .{a} & A9 = (ObjectDerivation(C)).((
    AttributeDerivation(C)).{a}) by A10,Def5,FUNCT_2:def 1;
    hence thesis by A7,A8,A9,A10,FUNCT_1:def 3;
  end;
  "/\"({ConceptStr(#O,A#) where O is Subset of the carrier of C, A is
Subset of the carrier' of C : ex a being Attribute of C st a in the Intent of b
  @ & O = ( AttributeDerivation(C)).{a} & A = (ObjectDerivation(C)).((
  AttributeDerivation(C )).{a})}, ConceptLattice(C)) = b@ & b = b@ by Th10,
CONLAT_1:def 21;
  hence thesis by A6;
end;
