let Y be non empty set ; :: thesis: for a being Function of Y,BOOLEAN
for G being Subset of (PARTITIONS Y)
for A, B being a_partition of Y holds All (('not' (Ex (a,A,G))),B,G) '<' 'not' (Ex ((All (a,B,G)),A,G))
let a be Function of Y,BOOLEAN; :: thesis: for G being Subset of (PARTITIONS Y)
for A, B being a_partition of Y holds All (('not' (Ex (a,A,G))),B,G) '<' 'not' (Ex ((All (a,B,G)),A,G))
let G be Subset of (PARTITIONS Y); :: thesis: for A, B being a_partition of Y holds All (('not' (Ex (a,A,G))),B,G) '<' 'not' (Ex ((All (a,B,G)),A,G))
let A, B be a_partition of Y; :: thesis: All (('not' (Ex (a,A,G))),B,G) '<' 'not' (Ex ((All (a,B,G)),A,G))
let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not (All (('not' (Ex (a,A,G))),B,G)) . z = TRUE or ('not' (Ex ((All (a,B,G)),A,G))) . z = TRUE )
A1: ( All (('not' (Ex (a,A,G))),B,G) = B_INF (('not' (Ex (a,A,G))),(CompF (B,G))) & z in EqClass (z,(CompF (B,G))) ) by BVFUNC_2:def 9, EQREL_1:def 6;
assume (All (('not' (Ex (a,A,G))),B,G)) . z = TRUE ; :: thesis: ('not' (Ex ((All (a,B,G)),A,G))) . z = TRUE
then ('not' (Ex (a,A,G))) . z = TRUE by A1, BVFUNC_1:def 16;
then A2: ( Ex (a,A,G) = B_SUP (a,(CompF (A,G))) & 'not' ((Ex (a,A,G)) . z) = TRUE ) by BVFUNC_2:def 10, MARGREL1:def 19;
A3: All (a,B,G) = B_INF (a,(CompF (B,G))) by BVFUNC_2:def 9;
for x being Element of Y st x in EqClass (z,(CompF (A,G))) holds
(All (a,B,G)) . x <> TRUE

proof

let x be Element of Y; :: thesis: ( x in EqClass (z,(CompF (A,G))) implies (All (a,B,G)) . x <> TRUE )
assume x in EqClass (z,(CompF (A,G))) ; :: thesis: (All (a,B,G)) . x <> TRUE
then A4: a . x <> TRUE by A2, BVFUNC_1:def 17;
x in EqClass (x,(CompF (B,G))) by EQREL_1:def 6;
hence (All (a,B,G)) . x <> TRUE by A3, A4, BVFUNC_1:def 16; :: thesis: verum

end;

then ( Ex ((All (a,B,G)),A,G) = B_SUP ((All (a,B,G)),(CompF (A,G))) & (B_SUP ((All (a,B,G)),(CompF (A,G)))) . z = FALSE ) by BVFUNC_1:def 17, BVFUNC_2:def 10;
then 'not' ((Ex ((All (a,B,G)),A,G)) . z) = TRUE ;
hence ('not' (Ex ((All (a,B,G)),A,G))) . z = TRUE by MARGREL1:def 19; :: thesis: verum