
theorem Th13:
  for C being FormalContext for o being Object of C for a being
  Attribute of C holds o is-connected-with a iff (gamma(C)).o [= (delta(C)).a
proof
  let C be FormalContext;
  let o be Object of C;
  let a be Attribute of C;
  set aa = {a};
  set o9 = {o};
  set oo = (AttributeDerivation(C)).((ObjectDerivation(C)).{o});
  consider O being Subset of the carrier of C, A being Subset of the carrier'
  of C such that
A1: (gamma(C)).o = ConceptStr(#O,A#) and
A2: O = (AttributeDerivation(C)).((ObjectDerivation(C)).{o}) and
  A = (ObjectDerivation(C)).{o} by Def4;
  hereby
    assume o is-connected-with a;
    then a in {a9 where a9 is Attribute of C : o is-connected-with a9};
    then a in (ObjectDerivation(C)).({o}) by CONLAT_1:1;
    then
A3: {a} c= (ObjectDerivation(C)).({o}) by ZFMISC_1:31;
    consider O being Subset of the carrier of C, A being Subset of the
    carrier' of C such that
A4: (gamma(C)).o = ConceptStr(#O,A#) and
A5: O = (AttributeDerivation(C)).((ObjectDerivation(C)).{o}) and
    A = (ObjectDerivation(C)).{o} by Def4;
    consider O9 being Subset of the carrier of C, A9 being Subset of the
    carrier' of C such that
A6: (delta(C)).a = ConceptStr(#O9,A9#) and
A7: O9 = (AttributeDerivation(C)).{a} and
    A9 = (ObjectDerivation(C)).((AttributeDerivation(C)).{a}) by Def5;
A8: the Extent of ((delta(C)).a)@ = O9 by A6,CONLAT_1:def 21;
    the Extent of ((gamma(C)).o)@ = O by A4,CONLAT_1:def 21;
    then
    the Extent of ((gamma(C)).o)@ c= the Extent of ((delta(C)).a)@ by A3,A5,A7
,A8,CONLAT_1:4;
    then ((gamma(C)).o)@ is-SubConcept-of ((delta(C)).a)@ by CONLAT_1:def 16;
    hence ((gamma(C)).o) [= ((delta(C)).a) by CONLAT_1:43;
  end;
  consider O9 being Subset of the carrier of C, A9 being Subset of the
  carrier' of C such that
A9: (delta(C)).a = ConceptStr(#O9,A9#) and
A10: O9 = (AttributeDerivation(C)).{a} and
  A9 = (ObjectDerivation(C)).((AttributeDerivation(C)).{a}) by Def5;
  assume ((gamma(C)).o) [= ((delta(C)).a);
  then ((gamma(C)).o)@ is-SubConcept-of ((delta(C)).a)@ by CONLAT_1:43;
  then
A11: the Extent of ((gamma(C)).o)@ c= the Extent of ((delta(C)).a)@ by
CONLAT_1:def 16;
  the Extent of ((delta(C)).a)@ = O9 by A9,CONLAT_1:def 21;
  then O c= O9 by A11,A1,CONLAT_1:def 21;
  then aa c= (ObjectDerivation(C)).oo by A2,A10,CONLAT_1:11;
  then {a} c= (ObjectDerivation(C)).o9 by CONLAT_1:7;
  then a in (ObjectDerivation(C)).({o}) by ZFMISC_1:31;
  then a in {a9 where a9 is Attribute of C : o is-connected-with a9} by
CONLAT_1:1;
  then ex b being Attribute of C st b = a & o is-connected-with b;
  hence thesis;
end;
