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 'not' (Ex ((Ex (a,A,G)),B,G)) '<' 'not' (All ((Ex (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 'not' (Ex ((Ex (a,A,G)),B,G)) '<' 'not' (All ((Ex (a,B,G)),A,G))
let G be Subset of (PARTITIONS Y); :: thesis: for A, B being a_partition of Y holds 'not' (Ex ((Ex (a,A,G)),B,G)) '<' 'not' (All ((Ex (a,B,G)),A,G))
let A, B be a_partition of Y; :: thesis: 'not' (Ex ((Ex (a,A,G)),B,G)) '<' 'not' (All ((Ex (a,B,G)),A,G))
let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not ('not' (Ex ((Ex (a,A,G)),B,G))) . z = TRUE or ('not' (All ((Ex (a,B,G)),A,G))) . z = TRUE )
assume ('not' (Ex ((Ex (a,A,G)),B,G))) . z = TRUE ; :: thesis: ('not' (All ((Ex (a,B,G)),A,G))) . z = TRUE
then A1: 'not' ((Ex ((Ex (a,A,G)),B,G)) . z) = TRUE by MARGREL1:def 19;

A2: now :: thesis: for x being Element of Y holds
( not x in EqClass (z,(CompF (B,G))) or not (Ex (a,A,G)) . x = TRUE )

assume ex x being Element of Y st
( x in EqClass (z,(CompF (B,G))) & (Ex (a,A,G)) . x = TRUE ) ; :: thesis: contradiction
then (B_SUP ((Ex (a,A,G)),(CompF (B,G)))) . z = TRUE by BVFUNC_1:def 17;
then (Ex ((Ex (a,A,G)),B,G)) . z = TRUE by BVFUNC_2:def 10;
hence contradiction by A1; :: thesis: verum

end;

A3: Ex (a,A,G) = B_SUP (a,(CompF (A,G))) by BVFUNC_2:def 10;
A4: for x being Element of Y st x in EqClass (z,(CompF (B,G))) holds
for y being Element of Y st y in EqClass (x,(CompF (A,G))) holds
a . y <> TRUE

proof

let x be Element of Y; :: thesis: ( x in EqClass (z,(CompF (B,G))) implies for y being Element of Y st y in EqClass (x,(CompF (A,G))) holds
a . y <> TRUE )
assume x in EqClass (z,(CompF (B,G))) ; :: thesis: for y being Element of Y st y in EqClass (x,(CompF (A,G))) holds
a . y <> TRUE
then (Ex (a,A,G)) . x <> TRUE by A2;
hence for y being Element of Y st y in EqClass (x,(CompF (A,G))) holds
a . y <> TRUE by A3, BVFUNC_1:def 17; :: thesis: verum

end;

for x being Element of Y st x in EqClass (z,(CompF (B,G))) holds
a . x <> TRUE

proof

let x be Element of Y; :: thesis: ( x in EqClass (z,(CompF (B,G))) implies a . x <> TRUE )
A5: x in EqClass (x,(CompF (A,G))) by EQREL_1:def 6;
assume x in EqClass (z,(CompF (B,G))) ; :: thesis: a . x <> TRUE
hence a . x <> TRUE by A4, A5; :: thesis: verum

end;

then (B_SUP (a,(CompF (B,G)))) . z = FALSE by BVFUNC_1:def 17;
then ( z in EqClass (z,(CompF (A,G))) & (Ex (a,B,G)) . z = FALSE ) by BVFUNC_2:def 10, EQREL_1:def 6;
then (B_INF ((Ex (a,B,G)),(CompF (A,G)))) . z = FALSE by BVFUNC_1:def 16;
then (All ((Ex (a,B,G)),A,G)) . z = FALSE by BVFUNC_2:def 9;
then 'not' ((All ((Ex (a,B,G)),A,G)) . z) = TRUE ;
hence ('not' (All ((Ex (a,B,G)),A,G))) . z = TRUE by MARGREL1:def 19; :: thesis: verum