let Y be non empty set ; :: thesis: for a being Element of Funcs 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 Element of Funcs 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 15 :: 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

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 19;
hence contradiction by A2, BVFUNC_2:def 9; :: thesis: verum

end;

A4: z in EqClass z,(CompF B,G) by EQREL_1:def 8;

now

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 20;
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 8;
then (Ex a,B,G) . x1 = TRUE by A1, A6, BVFUNC_1:def 20;
then (B_SUP (Ex a,B,G),(CompF A,G)) . z = TRUE by A5, BVFUNC_1:def 20;
hence (Ex (Ex a,B,G),A,G) . z = TRUE by BVFUNC_2:def 10; :: thesis: verum