
theorem
  for C being FormalContext for D being non empty Subset of
  ConceptLattice(C) holds the Extent of "/\"(D,C) = meet {the Extent of
  ConceptStr(#E,I#) where E is Subset of the carrier of C, I is Subset of the
  carrier' of C : ConceptStr(#E,I#) in D} & the Intent of "/\"(D,C) = (
ObjectDerivation(C)).((AttributeDerivation(C)). union {the Intent of ConceptStr
(#E,I#) where E is Subset of the carrier of C, I is Subset of the carrier' of C
  : ConceptStr(#E,I#) in D})
proof
  let C be FormalContext;
  let D be non empty Subset of ConceptLattice(C);
  set A = (ObjectDerivation(C)).((AttributeDerivation(C)). union {the Intent
of ConceptStr(#E,I#) where E is Subset of the carrier of C, I is Subset of the
  carrier' of C : ConceptStr(#E,I#) in D});
  set O9 = (AttributeDerivation(C)). union {the Intent of ConceptStr(#E,I#)
  where E is Subset of the carrier of C, I is Subset of the carrier' of C :
  ConceptStr(#E,I#) in D};
  set y = the Element of D;
  {the Intent of ConceptStr(#E,I#) where E is Subset of the carrier of C,
  I is Subset of the carrier' of C : ConceptStr(#E,I#) in D} c= bool (the
  carrier' of C)
  proof
    let x be object;
    assume x in {the Intent of ConceptStr(#E,I#) where E is Subset of the
    carrier of C, I is Subset of the carrier' of C : ConceptStr(#E,I#) in D};
    then ex E being Subset of the carrier of C, I being Subset of the carrier'
    of C st x = the Intent of ConceptStr(#E,I#) & ConceptStr(#E,I#) in D;
    hence thesis;
  end;
  then reconsider
  AA = {the Intent of ConceptStr(#E,I#) where E is Subset of the
  carrier of C, I is Subset of the carrier' of C : ConceptStr(#E,I#) in D} as
  Subset-Family of (the carrier' of C);
  A c= the carrier' of C
  proof
    set u = union {the Intent of ConceptStr(#E,I#) where E is Subset of the
    carrier of C, I is Subset of the carrier' of C : ConceptStr(#E,I#) in D};
    u c= the carrier' of C
    proof
      let x be object;
      assume x in u;
      then consider Y being set such that
A1:   x in Y and
A2:   Y in {the Intent of ConceptStr(#E,I#) where E is Subset of the
carrier of C, I is Subset of the carrier' of C : ConceptStr(#E,I#) in D} by
TARSKI:def 4;
      ex E being Subset of the carrier of C, I being Subset of the
carrier' of C st Y = the Intent of ConceptStr(#E,I#) & ConceptStr(#E,I#) in D
      by A2;
      hence thesis by A1;
    end;
    then reconsider u as Subset of the carrier' of C;
    let x be object;
A3: (ObjectDerivation(C)).((AttributeDerivation(C)).u) is Element of bool
    (the carrier' of C);
    assume x in A;
    hence thesis by A3;
  end;
  then reconsider a = A as Subset of the carrier' of C;
A4: ConceptLattice(C) = LattStr(#B-carrier(C),B-join(C),B-meet(C)#) by
CONLAT_1:def 20;
A5: for x being object
holds x in D implies x is strict FormalConcept of C & ex
E being Subset of the carrier of C, I being Subset of the carrier' of C st x =
  ConceptStr(#E,I#)
  proof
    let x be object;
    assume x in D;
    then x is strict FormalConcept of C by A4,CONLAT_1:31;
    hence thesis;
  end;
  then ex E9 being Subset of the carrier of C, I9 being Subset of the carrier'
  of C st y = ConceptStr(#E9,I9#);
  then
  the Intent of y in {the Intent of ConceptStr(#E,I#) where E is Subset of
  the carrier of C, I is Subset of the carrier' of C : ConceptStr(#E,I#) in D};
  then reconsider AA as non empty Subset-Family of (the carrier' of C);
A6: {the Extent of ConceptStr(#E,I#) where E is Subset of the carrier of C,
  I is Subset of the carrier' of C : ConceptStr(#E,I#) in D} c= {(
AttributeDerivation(C)).A9 where A9 is Subset of the carrier' of C : A9 in {the
Intent of ConceptStr(#E,I#) where E is Subset of the carrier of C, I is Subset
  of the carrier' of C : ConceptStr(#E,I#) in D}}
  proof
    let x be object;
    assume x in {the Extent of ConceptStr(#E,I#) where E is Subset of the
    carrier of C, I is Subset of the carrier' of C : ConceptStr(#E,I#) in D};
    then consider E being Subset of the carrier of C, I being Subset of the
    carrier' of C such that
A7: x = the Extent of ConceptStr(#E,I#) and
A8: ConceptStr(#E,I#) in D;
    ConceptStr(#E,I#) is FormalConcept of C by A5,A8;
    then
A9: x = (AttributeDerivation(C)).(the Intent of ConceptStr(#E,I#)) by A7,
CONLAT_1:def 10;
    the Intent of ConceptStr(#E,I#) in {the Intent of ConceptStr(#EE,II#)
    where EE is Subset of the carrier of C, II is Subset of the carrier' of C :
    ConceptStr(#EE,II#) in D} by A8;
    hence thesis by A9;
  end;
  {(AttributeDerivation(C)).A9 where A9 is Subset of the carrier' of C :
A9 in {the Intent of ConceptStr(#E,I#) where E is Subset of the carrier of C, I
  is Subset of the carrier' of C : ConceptStr(#E,I#) in D}} c= {the Extent of
  ConceptStr(#E,I#) where E is Subset of the carrier of C, I is Subset of the
  carrier' of C : ConceptStr(#E,I#) in D}
  proof
    let x be object;
    assume x in {(AttributeDerivation(C)).A9 where A9 is Subset of the
carrier' of C : A9 in {the Intent of ConceptStr(#E,I#) where E is Subset of the
    carrier of C, I is Subset of the carrier' of C : ConceptStr(#E,I#) in D}};
    then consider A9 being Subset of the carrier' of C such that
A10: x = (AttributeDerivation(C)).A9 and
A11: A9 in {the Intent of ConceptStr(#E,I#) where E is Subset of the
    carrier of C, I is Subset of the carrier' of C : ConceptStr(#E,I#) in D};
    consider E being Subset of the carrier of C, I being Subset of the
    carrier' of C such that
A12: A9 = the Intent of ConceptStr(#E,I#) and
A13: ConceptStr(#E,I#) in D by A11;
    ConceptStr(#E,I#) is FormalConcept of C by A5,A13;
    then x = the Extent of ConceptStr(#E,I#) by A10,A12,CONLAT_1:def 10;
    hence thesis by A13;
  end;
  then
A14: {(AttributeDerivation(C)).A9 where A9 is Subset of the carrier' of C :
A9 in {the Intent of ConceptStr(#E,I#) where E is Subset of the carrier of C, I
  is Subset of the carrier' of C : ConceptStr(#E,I#) in D}} = {the Extent of
  ConceptStr(#E,I#) where E is Subset of the carrier of C, I is Subset of the
  carrier' of C : ConceptStr(#E,I#) in D} by A6;
A15: O9 = meet({(AttributeDerivation(C)).A9 where A9 is Subset of the
  carrier' of C : A9 in AA}) by Th3;
  O9 c= the carrier of C
  proof
    set y = the Element of D;

set Y = the Element of {the Extent of ConceptStr(#E,I#) where E is Subset of
the carrier of C, I is Subset of the carrier' of C : ConceptStr(#E,I #) in D};
    let x be object;
    ex E9 being Subset of the carrier of C, I9 being Subset of the
    carrier' of C st y = ConceptStr(#E9,I9#) by A5;
    then
A16: the Extent of y in {the Extent of ConceptStr(#E,I#) where E is Subset
of the carrier of C, I is Subset of the carrier' of C : ConceptStr(#E,I#) in D}
    ;
    then Y in {the Extent of ConceptStr(#E,I#) where E is Subset of the
    carrier of C, I is Subset of the carrier' of C : ConceptStr(#E,I#) in D};
    then
A17: ex E1 being Subset of the carrier of C, I1 being Subset of the
carrier' of C st Y = the Extent of ConceptStr(#E1,I1#) & ConceptStr(#E1, I1#)
    in D;
    assume x in O9;
    then x in Y by A15,A14,A16,SETFAM_1:def 1;
    hence thesis by A17;
  end;
  then reconsider o = O9 as Subset of the carrier of C;
  union {the Intent of ConceptStr(#E,I#) where E is Subset of the carrier
  of C, I is Subset of the carrier' of C : ConceptStr(#E,I#) in D} c= the
  carrier' of C
  proof
    let x be object;
    assume x in union {the Intent of ConceptStr(#E,I#) where E is Subset of
the carrier of C, I is Subset of the carrier' of C : ConceptStr(#E,I#) in D};
    then consider Y being set such that
A18: x in Y and
A19: Y in {the Intent of ConceptStr(#E,I#) where E is Subset of the
    carrier of C,I is Subset of the carrier' of C : ConceptStr(#E,I#) in D} by
TARSKI:def 4;
    ex E being Subset of the carrier of C, I being Subset of the carrier'
of C st Y = the Intent of ConceptStr(#E,I#) & ConceptStr(#E,I#) in D by A19;
    hence thesis by A18;
  end;
  then reconsider CP9 = ConceptStr(#o,a#) as strict FormalConcept of C by
CONLAT_1:21;
  reconsider CP = CP9 as Element of ConceptLattice(C) by A4,CONLAT_1:31;
A20: the Extent of CP@ = meet {the Extent of ConceptStr(#E,I#) where E is
Subset of the carrier of C, I is Subset of the carrier' of C : ConceptStr(#E,I
  #) in D} by A15,A14,CONLAT_1:def 21;
A21: for r being Element of ConceptLattice(C) st r is_less_than D holds r [= CP
  proof
    let r be Element of ConceptLattice(C);
    assume
A22: r is_less_than D;
A23: for q being Element of ConceptLattice(C) st q in D holds the Extent
    of r@ c= the Extent of q@
    proof
      let q be Element of ConceptLattice(C);
      assume q in D;
      then r [= q by A22;
      then r@ is-SubConcept-of q@ by CONLAT_1:43;
      hence thesis by CONLAT_1:def 16;
    end;
    the Extent of r@ c= meet {the Extent of ConceptStr(#E,I#) where E is
Subset of the carrier of C, I is Subset of the carrier' of C : ConceptStr(#E,I
    #) in D}
    proof
      set y = the Element of D;
      let x be object;
      assume
A24:  x in the Extent of r@;
A25:  for Y being set holds Y in {the Extent of ConceptStr(#E,I#) where E
is Subset of the carrier of C, I is Subset of the carrier' of C : ConceptStr(#E
        ,I#) in D} implies x in Y
      proof
        let Y be set;
        assume Y in {the Extent of ConceptStr(#E,I#) where E is Subset of
the carrier of C, I is Subset of the carrier' of C : ConceptStr(#E,I#) in D};
        then consider
        Ey being Subset of the carrier of C, Iy being Subset of the
        carrier' of C such that
A26:    Y = the Extent of ConceptStr(#Ey,Iy#) and
A27:    ConceptStr(#Ey,Iy#) in D;
        reconsider C1 = ConceptStr(#Ey,Iy#) as Element of ConceptLattice(C) by
A27;
        the Extent of r@ c= the Extent of C1@ by A23,A27;
        then x in the Extent of C1@ by A24;
        hence thesis by A26,CONLAT_1:def 21;
      end;
      ex E9 being Subset of the carrier of C, I9 being Subset of the
      carrier' of C st y = ConceptStr(#E9,I9#) by A5;
      then the Extent of y in {the Extent of ConceptStr(#E,I#) where E is
Subset of the carrier of C, I is Subset of the carrier' of C : ConceptStr(#E,I
      #) in D};
      hence thesis by A25,SETFAM_1:def 1;
    end;
    then r@ is-SubConcept-of CP@ by A20,CONLAT_1:def 16;
    hence thesis by CONLAT_1:43;
  end;
  CP is_less_than D
  proof
    let q be Element of ConceptLattice(C);
    assume q in D;
    then q@ in D by CONLAT_1:def 21;
    then the Extent of q@ in {the Extent of ConceptStr(#E,I#) where E is
Subset of the carrier of C, I is Subset of the carrier' of C : ConceptStr(#E,I
    #) in D};
    then the Extent of CP@ c= the Extent of q@ by A20,SETFAM_1:3;
    then CP@ is-SubConcept-of q@ by CONLAT_1:def 16;
    hence thesis by CONLAT_1:43;
  end;
  hence thesis by A15,A14,A21,LATTICE3:34;
end;
