let Y be non empty set ; 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; 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); 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; All (('not' (Ex (a,A,G))),B,G) '<' 'not' (Ex ((All (a,B,G)),A,G))
let z be Element of Y; BVFUNC_1:def 12 ( 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
; ('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;
( x in EqClass (z,(CompF (A,G))) implies (All (a,B,G)) . x <> TRUE )
assume
x in EqClass (
z,
(CompF (A,G)))
;
(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;
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; verum