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 st G is independent holds
'not' (Ex ((Ex (a,A,G)),B,G)) = All (('not' (Ex (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 st G is independent holds
'not' (Ex ((Ex (a,A,G)),B,G)) = All (('not' (Ex (a,B,G))),A,G)
let G be Subset of (PARTITIONS Y); for A, B being a_partition of Y st G is independent holds
'not' (Ex ((Ex (a,A,G)),B,G)) = All (('not' (Ex (a,B,G))),A,G)
let A, B be a_partition of Y; ( G is independent implies 'not' (Ex ((Ex (a,A,G)),B,G)) = All (('not' (Ex (a,B,G))),A,G) )
assume
G is independent
; 'not' (Ex ((Ex (a,A,G)),B,G)) = All (('not' (Ex (a,B,G))),A,G)
then
'not' (Ex ((Ex (a,A,G)),B,G)) = 'not' (Ex ((Ex (a,B,G)),A,G))
by PARTIT_2:16;
hence
'not' (Ex ((Ex (a,A,G)),B,G)) = All (('not' (Ex (a,B,G))),A,G)
by BVFUNC_2:19; verum