
theorem Th35:
  for C being FormalContext for CP1,CP2,CP3 being strict
FormalConcept of C holds (B-join(C)).(CP1,(B-join(C)).(CP2,CP3)) = (B-join(C)).
  ((B-join(C)).(CP1,CP2),CP3)
proof
  let C be FormalContext;
  let CP1,CP2,CP3 be strict FormalConcept of C;
  (B-join(C)).(CP1,CP2) in rng((B-join(C))) by Lm3;
  then reconsider CP = (B-join(C)).(CP1,CP2) as strict FormalConcept of C by
Th31;
A1: (ex O being Subset of the carrier of C, A being Subset of the carrier'
of C st (B-join(C)).(CP1,CP2) = ConceptStr(#O,A#) & O = ( AttributeDerivation(C
)).(( ObjectDerivation(C)). ((the Extent of CP1) \/ (the Extent of CP2))) & A =
(the Intent of CP1) /\ (the Intent of CP2) )& ex O9 being Subset of the carrier
  of C, A9 being Subset of the carrier' of C st (B-join(C)).(CP,CP3) =
ConceptStr(# O9,A9#) & O9 = ( AttributeDerivation(C)).((ObjectDerivation(C)). (
  (the Extent of CP) \/ (the Extent of CP3))) & A9 = (the Intent of CP) /\ (the
  Intent of CP3) by Def18;
  (B-join(C)).(CP2,CP3) in rng((B-join(C))) by Lm3;
  then reconsider CP9 = (B-join(C)).(CP2,CP3) as strict FormalConcept of C by
Th31;
A2: (ex O being Subset of the carrier of C, A being Subset of the carrier'
of C st (B-join(C)).(CP2,CP3) = ConceptStr(#O,A#) & O = ( AttributeDerivation(C
)).(( ObjectDerivation(C)). ((the Extent of CP2) \/ (the Extent of CP3))) & A =
(the Intent of CP2) /\ (the Intent of CP3) )& ex O9 being Subset of the carrier
  of C, A9 being Subset of the carrier' of C st (B-join(C)).(CP1,CP9) =
ConceptStr (#O9,A9#) & O9 = ( AttributeDerivation(C)).((ObjectDerivation(C)). (
(the Extent of CP1) \/ (the Extent of CP9))) & A9 = (the Intent of CP1) /\ (the
  Intent of CP9) by Def18;
  (AttributeDerivation(C)).((ObjectDerivation(C)). ((the Extent of CP1) \/
((AttributeDerivation(C)).((ObjectDerivation(C)). ((the Extent of CP2) \/ (the
  Extent of CP3)))))) = (AttributeDerivation(C)). (((ObjectDerivation(C)).(the
  Extent of CP1)) /\ ((ObjectDerivation(C)). ((AttributeDerivation(C)).((
ObjectDerivation(C)). ((the Extent of CP2) \/ (the Extent of CP3)))))) by Th15
    .= (AttributeDerivation(C)). (((ObjectDerivation(C)).(the Extent of CP1)
) /\ ((ObjectDerivation(C)). ((the Extent of CP2) \/ (the Extent of CP3)))) by
Th7
    .= (AttributeDerivation(C)). (((ObjectDerivation(C)).(the Extent of CP1)
  ) /\ (((ObjectDerivation(C)).(the Extent of CP2)) /\ ((ObjectDerivation(C)).(
  the Extent of CP3)))) by Th15
    .= (AttributeDerivation(C)). ((((ObjectDerivation(C)).(the Extent of CP1
)) /\ ((ObjectDerivation(C)).(the Extent of CP2))) /\ ((ObjectDerivation(C)).(
  the Extent of CP3))) by XBOOLE_1:16
    .= (AttributeDerivation(C)). (((ObjectDerivation(C)). ((the Extent of
  CP1) \/ (the Extent of CP2)) /\ ((ObjectDerivation(C)).(the Extent of CP3))))
  by Th15
    .= (AttributeDerivation(C)). (((ObjectDerivation(C)). ((
  AttributeDerivation(C)).((ObjectDerivation(C)). ((the Extent of CP1) \/ (the
  Extent of CP2)))) /\ ((ObjectDerivation(C)).(the Extent of CP3)))) by Th7
    .= (AttributeDerivation(C)).((ObjectDerivation(C)). (((
  AttributeDerivation(C)).((ObjectDerivation(C)). ((the Extent of CP1) \/ (the
  Extent of CP2)))) \/ (the Extent of CP3))) by Th15;
  hence thesis by A1,A2,XBOOLE_1:16;
end;
