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
Ex (('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 st G is independent holds
Ex (('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 st G is independent holds
Ex (('not' (Ex (a,A,G))),B,G) '<' 'not' (Ex ((All (a,B,G)),A,G))
let A, B be a_partition of Y; ( G is independent implies Ex (('not' (Ex (a,A,G))),B,G) '<' 'not' (Ex ((All (a,B,G)),A,G)) )
assume
G is independent
; Ex (('not' (Ex (a,A,G))),B,G) '<' 'not' (Ex ((All (a,B,G)),A,G))
then A1:
'not' (All ((Ex (a,A,G)),B,G)) '<' 'not' (Ex ((All (a,B,G)),A,G))
by PARTIT_2:11, PARTIT_2:17;
'not' (All ((Ex (a,A,G)),B,G)) = Ex ((All (('not' a),A,G)),B,G)
by Th19;
hence
Ex (('not' (Ex (a,A,G))),B,G) '<' 'not' (Ex ((All (a,B,G)),A,G))
by A1, BVFUNC_2:19; verum