
theorem Th3:
  for C being FormalContext for D being non empty Subset-Family of(
  the carrier' of C) holds (AttributeDerivation(C)).(union D) = meet({(
  AttributeDerivation(C)).A where A is Subset of the carrier' of C : A in D})
proof
  let C be FormalContext;
  let D be non empty Subset-Family of(the carrier' of C);
  reconsider D9=D as non empty Subset-Family of the carrier' of C;
  set OU = (AttributeDerivation(C)).(union D);
  set M = meet({(AttributeDerivation(C)).A where A is Subset of the carrier'
  of C : A in D});
  now
    per cases;
    case
A1:   {(AttributeDerivation(C)).A where A is Subset of the carrier' of
C:    A in D} <> {};
      thus OU = M
      proof
        thus OU c= M
        proof
          let x be object;
          assume x in OU;
          then
          x in {o9 where o9 is Object of C : for a being Attribute of C st
          a in union D9 holds o9 is-connected-with a} by CONLAT_1:def 4;
          then
A2:       ex x9 being Object of C st x9 = x & for a being Attribute of C st
          a in union D holds x9 is-connected-with a;
          then reconsider x as Object of C;
A3:       for A being Subset of the carrier' of C st A in D for a being
          Attribute of C st a in A holds x is-connected-with a
          proof
            let A be Subset of the carrier' of C;
            assume
A4:         A in D;
            let a be Attribute of C;
            assume a in A;
            then a in union D by A4,TARSKI:def 4;
            hence thesis by A2;
          end;
A5:       for A being Subset of the carrier' of C st A in D holds x in (
          AttributeDerivation(C)).A
          proof
            let A be Subset of the carrier' of C;
            assume A in D;
            then for a being Attribute of C st a in A holds x
            is-connected-with a by A3;
            then x in {o where o is Object of C : for a being Attribute of C
            st a in A holds o is-connected-with a};
            hence thesis by CONLAT_1:def 4;
          end;
          for Y being set holds Y in {(AttributeDerivation(C)).A where A
          is Subset of the carrier' of C : A in D} implies x in Y
          proof
            let Y be set;
            assume Y in {(AttributeDerivation(C)).A where A is Subset of the
            carrier' of C : A in D};
            then ex A being Subset of the carrier' of C st Y = (
            AttributeDerivation(C)).A & A in D;
            hence thesis by A5;
          end;
          hence thesis by A1,SETFAM_1:def 1;
        end;

set d = the Element of {(AttributeDerivation(C)).A where A is Subset of the
carrier' of C : A in D};
        let x be object;
        assume
A6:     x in M;
        then
A7:     x in d by A1,SETFAM_1:def 1;
        d in {(AttributeDerivation(C)).A where A is Subset of the
        carrier' of C : A in D} by A1;
        then ex X being Subset of the carrier' of C st d = (
        AttributeDerivation(C)).X & X in D;
        then reconsider x as Object of C by A7;
A8:     for A being Subset of the carrier' of C st A in D holds x in (
        AttributeDerivation(C)).A
        proof
          let A be Subset of the carrier' of C;
          assume A in D;
          then (AttributeDerivation(C)).A in {(AttributeDerivation(C)).A9
          where A9 is Subset of the carrier' of C : A9 in D};
          hence thesis by A6,SETFAM_1:def 1;
        end;
A9:     for A being Subset of the carrier' of C st A in D for a being
        Attribute of C st a in A holds x is-connected-with a
        proof
          let A be Subset of the carrier' of C;
          assume A in D;
          then x in (AttributeDerivation(C)).A by A8;
          then x in {o where o is Object of C : for a being Attribute of C st
          a in A holds o is-connected-with a} by CONLAT_1:def 4;
          then
A10:      ex x9 being Object of C st x9 = x & for a being Attribute of C
          st a in A holds x9 is-connected-with a;
          let a be Attribute of C;
          assume a in A;
          hence thesis by A10;
        end;
        for a being Attribute of C st a in union D holds x is-connected-with a
        proof
          let a be Attribute of C;
          assume a in union D;
          then ex Y being set st a in Y & Y in D by TARSKI:def 4;
          hence thesis by A9;
        end;
        then x in {o9 where o9 is Object of C : for a being Attribute of C st
        a in union D9 holds o9 is-connected-with a};
        hence thesis by CONLAT_1:def 4;
      end;
    end;
    case
A11:  {(AttributeDerivation(C)).A where A is Subset of the carrier'
      of C : A in D} = {};
      D = {}
      proof
        set x = the Element of D;
        assume D <> {};
        (AttributeDerivation(C)).x in {(AttributeDerivation(C)).A where A
        is Subset of the carrier' of C : A in D};
        hence thesis by A11;
      end;
      hence thesis;
    end;
  end;
  hence thesis;
end;
