
theorem Th18:
  for C being FormalContext for O being Subset of the carrier of C
  holds (ObjectDerivation(C)).O = (AttributeDerivation(C.:)).O
proof
  let C be FormalContext;
  let O be Subset of the carrier of C;
  reconsider O9 = O as Subset of(the carrier' of C.:);
  set A1 = {a where a is Attribute of C : for o being Object of C st o in O
  holds o is-connected-with a };
  set A2 = {a where a is Object of C.: : for o being Attribute of C.: st o in
  O9 holds a is-connected-with o };
A1: A1 c= A2
  proof
    let u be object;
    assume u in A1;
    then consider a being Attribute of C such that
A2: a = u and
A3: for o being Object of C st o in O holds o is-connected-with a;
    reconsider u9 = u as Object of C.: by A2;
    for o9 being Attribute of C.: st o9 in O9 holds u9 is-connected-with o9
    proof
      let o9 be Attribute of C.:;
      reconsider o = o9 as Object of C;
      assume o9 in O9;
      then o is-connected-with a by A3;
      then [o,a] in the Information of C by CONLAT_1:def 2;
      then [a,o] in the Information of C.: by RELAT_1:def 7;
      hence thesis by A2,CONLAT_1:def 2;
    end;
    hence thesis;
  end;
A4: A2 c= A1
  proof
    let u be object;
    assume u in A2;
    then consider a being Object of C.: such that
A5: a = u and
A6: for o being Attribute of C.: st o in O9 holds a is-connected-with o;
    reconsider u9 = u as Attribute of C by A5;
    for o being Object of C st o in O holds o is-connected-with u9
    proof
      let o9 be Object of C;
      reconsider o = o9 as Attribute of C.:;
      assume o9 in O;
      then a is-connected-with o by A6;
      then [a,o] in the Information of C.: by CONLAT_1:def 2;
      then [o9,u9] in the Information of C by A5,RELAT_1:def 7;
      hence thesis by CONLAT_1:def 2;
    end;
    hence thesis;
  end;
  (ObjectDerivation(C)).O = {a where a is Attribute of C : for o being
Object of C st o in O holds o is-connected-with a } & (AttributeDerivation(C.:)
  ).O9 = {a where a is Object of C.: : for o being Attribute of C.: st o in O9
  holds a is-connected-with o } by CONLAT_1:def 3,def 4;
  hence thesis by A1,A4;
end;
