let C be FormalContext; :: thesis: for CP1, CP2 being strict FormalConcept of C holds (B-meet C) . (CP1,((B-join C) . (CP1,CP2))) = CP1
let CP1, CP2 be strict FormalConcept of C; :: thesis: (B-meet C) . (CP1,((B-join C) . (CP1,CP2))) = CP1
A1: the Intent of CP1 /\ the Intent of CP2 c= the Intent of CP1 by XBOOLE_1:17;
(B-join C) . (CP1,CP2) in rng (B-join C) by Lm3;
then reconsider CP9 = (B-join C) . (CP1,CP2) as strict FormalConcept of C by Th31;
A2: ( ex O being Subset of the carrier of C ex 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 ex 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, Def18;
(ObjectDerivation C) . ((AttributeDerivation C) . ( the Intent of CP1 \/ ( the Intent of CP1 /\ the Intent of CP2))) = (ObjectDerivation C) . (((AttributeDerivation C) . the Intent of CP1) /\ ((AttributeDerivation C) . ( the Intent of CP1 /\ the Intent of CP2))) by Th16;
then A3: (ObjectDerivation C) . ((AttributeDerivation C) . ( the Intent of CP1 \/ ( the Intent of CP1 /\ the Intent of CP2))) = (ObjectDerivation C) . ((AttributeDerivation C) . the Intent of CP1) by A1, Th4, XBOOLE_1:28
.= (ObjectDerivation C) . the Extent of CP1 by Def9
.= the Intent of CP1 by Def9 ;
the Extent of CP1 /\ ((AttributeDerivation C) . ((ObjectDerivation C) . ( the Extent of CP1 \/ the Extent of CP2))) = the Extent of CP1 /\ ((AttributeDerivation C) . (((ObjectDerivation C) . the Extent of CP1) /\ ((ObjectDerivation C) . the Extent of CP2))) by Th15
.= the Extent of CP1 /\ ((AttributeDerivation C) . ( the Intent of CP1 /\ ((ObjectDerivation C) . the Extent of CP2))) by Def9
.= the Extent of CP1 /\ ((AttributeDerivation C) . ( the Intent of CP1 /\ the Intent of CP2)) by Def9
.= ((AttributeDerivation C) . the Intent of CP1) /\ ((AttributeDerivation C) . ( the Intent of CP1 /\ the Intent of CP2)) by Def9
.= (AttributeDerivation C) . the Intent of CP1 by A1, Th4, XBOOLE_1:28
.= the Extent of CP1 by Def9 ;
hence (B-meet C) . (CP1,((B-join C) . (CP1,CP2))) = CP1 by A2, A3; :: thesis: verum