
theorem Th34:
  for C being FormalContext for CP1,CP2,CP3 being strict
FormalConcept of C holds (B-meet(C)).(CP1,(B-meet(C)).(CP2,CP3)) = (B-meet(C)).
  ((B-meet(C)).(CP1,CP2),CP3)
proof
  let C be FormalContext;
  let CP1,CP2,CP3 be strict FormalConcept of C;
  (B-meet(C)).(CP1,CP2) in rng((B-meet(C))) by Lm2;
  then reconsider CP = (B-meet(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-meet(C)).(CP1,CP2) = ConceptStr(#O,A#) & O = (the Extent of CP1) /\
(the Extent of CP2) & A = (ObjectDerivation(C)).(( AttributeDerivation(C)). ((
  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-meet(C)).(CP,CP3) =
ConceptStr (#O9,A9#) & O9 = (the Extent of CP) /\ (the Extent of CP3) & A9 = (
  ObjectDerivation(C)).(( AttributeDerivation(C)). ((the Intent of CP) \/ (the
  Intent of CP3))) by Def17;
  (B-meet(C)).(CP2,CP3) in rng((B-meet(C))) by Lm2;
  then reconsider CP9 = (B-meet(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-meet(C)).(CP2,CP3) = ConceptStr(#O,A#) & O = (the Extent of CP2) /\
(the Extent of CP3) & A = (ObjectDerivation(C)).(( AttributeDerivation(C)). ((
  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-meet(C)).(CP1,CP9) =
ConceptStr (#O9,A9#) & O9 = (the Extent of CP1) /\ (the Extent of CP9) & A9 = (
  ObjectDerivation(C)).(( AttributeDerivation(C)). ((the Intent of CP1) \/ (the
  Intent of CP9))) by Def17;
  (ObjectDerivation(C)).((AttributeDerivation(C)). ((the Intent of CP1) \/
((ObjectDerivation(C)).((AttributeDerivation(C)). ((the Intent of CP2) \/ (the
  Intent of CP3)))))) = (ObjectDerivation(C)). (((AttributeDerivation(C)).(the
  Intent of CP1)) /\ ((AttributeDerivation(C)). ((ObjectDerivation(C)).((
  AttributeDerivation(C)). ((the Intent of CP2) \/ (the Intent of CP3)))))) by
Th16
    .= (ObjectDerivation(C)). (((AttributeDerivation(C)).(the Intent of CP1)
) /\ ((AttributeDerivation(C)). ((the Intent of CP2) \/ (the Intent of CP3))))
  by Th8
    .= (ObjectDerivation(C)). (((AttributeDerivation(C)).(the Intent of CP1)
) /\ (((AttributeDerivation(C)).(the Intent of CP2)) /\ ((AttributeDerivation(C
  )).(the Intent of CP3)))) by Th16
    .= (ObjectDerivation(C)). ((((AttributeDerivation(C)).(the Intent of CP1
)) /\ ((AttributeDerivation(C)).(the Intent of CP2))) /\ ((AttributeDerivation(
  C)).(the Intent of CP3))) by XBOOLE_1:16
    .= (ObjectDerivation(C)). (((AttributeDerivation(C)). ((the Intent of
CP1) \/ (the Intent of CP2)) /\ ((AttributeDerivation(C)).(the Intent of CP3)))
  ) by Th16
    .= (ObjectDerivation(C)). (((AttributeDerivation(C)). ((ObjectDerivation
(C)).((AttributeDerivation(C)). ((the Intent of CP1) \/ (the Intent of CP2))))
  /\ ((AttributeDerivation(C)).(the Intent of CP3)))) by Th8
    .= (ObjectDerivation(C)).((AttributeDerivation(C)). (((ObjectDerivation(
  C)).((AttributeDerivation(C)). ((the Intent of CP1) \/ (the Intent of CP2))))
  \/ (the Intent of CP3))) by Th16;
  hence thesis by A1,A2,XBOOLE_1:16;
end;
