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 ((Ex (a,A,G)),B,G) '<' Ex ((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 All ((Ex (a,A,G)),B,G) '<' Ex ((Ex (a,B,G)),A,G)
let G be Subset of (PARTITIONS Y); :: thesis: for A, B being a_partition of Y holds All ((Ex (a,A,G)),B,G) '<' Ex ((Ex (a,B,G)),A,G)
let A, B be a_partition of Y; :: thesis: All ((Ex (a,A,G)),B,G) '<' Ex ((Ex (a,B,G)),A,G)
A1: Ex (a,B,G) = B_SUP (a,(CompF (B,G))) by BVFUNC_2:def 10;
let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not (All ((Ex (a,A,G)),B,G)) . z = TRUE or (Ex ((Ex (a,B,G)),A,G)) . z = TRUE )
assume A2: (All ((Ex (a,A,G)),B,G)) . z = TRUE ; :: thesis: (Ex ((Ex (a,B,G)),A,G)) . z = TRUE

A3: now :: thesis: for x being Element of Y st x in EqClass (z,(CompF (B,G))) holds
(Ex (a,A,G)) . x = TRUE

assume ex x being Element of Y st
( x in EqClass (z,(CompF (B,G))) & not (Ex (a,A,G)) . x = TRUE ) ; :: thesis: contradiction
then (B_INF ((Ex (a,A,G)),(CompF (B,G)))) . z = FALSE by BVFUNC_1:def 16;
hence contradiction by A2, BVFUNC_2:def 9; :: thesis: verum

end;

A4: z in EqClass (z,(CompF (B,G))) by EQREL_1:def 6;

now :: thesis: ex x being Element of Y st
( x in EqClass (z,(CompF (A,G))) & a . x = TRUE )

assume for x being Element of Y holds
( not x in EqClass (z,(CompF (A,G))) or not a . x = TRUE ) ; :: thesis: contradiction
then (B_SUP (a,(CompF (A,G)))) . z = FALSE by BVFUNC_1:def 17;
then (Ex (a,A,G)) . z = FALSE by BVFUNC_2:def 10;
hence contradiction by A4, A3; :: thesis: verum

end;

then consider x1 being Element of Y such that
A5: x1 in EqClass (z,(CompF (A,G))) and
A6: a . x1 = TRUE ;
x1 in EqClass (x1,(CompF (B,G))) by EQREL_1:def 6;
then (Ex (a,B,G)) . x1 = TRUE by A1, A6, BVFUNC_1:def 17;
then (B_SUP ((Ex (a,B,G)),(CompF (A,G)))) . z = TRUE by A5, BVFUNC_1:def 17;
hence (Ex ((Ex (a,B,G)),A,G)) . z = TRUE by BVFUNC_2:def 10; :: thesis: verum