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' (All ((All (a,A,G)),B,G)) '<' Ex ((Ex (('not' a),A,G)),B,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' (All ((All (a,A,G)),B,G)) '<' Ex ((Ex (('not' a),A,G)),B,G)
let G be Subset of (PARTITIONS Y); for A, B being a_partition of Y st G is independent holds
'not' (All ((All (a,A,G)),B,G)) '<' Ex ((Ex (('not' a),A,G)),B,G)
let A, B be a_partition of Y; ( G is independent implies 'not' (All ((All (a,A,G)),B,G)) '<' Ex ((Ex (('not' a),A,G)),B,G) )
assume A1:
G is independent
; 'not' (All ((All (a,A,G)),B,G)) '<' Ex ((Ex (('not' a),A,G)),B,G)
then
Ex (('not' (All (a,B,G))),A,G) '<' Ex ((Ex (('not' a),A,G)),B,G)
by Th14;
hence
'not' (All ((All (a,A,G)),B,G)) '<' Ex ((Ex (('not' a),A,G)),B,G)
by A1, Th15; verum