let Y be non empty set ; :: thesis: 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 ((All (('not' a),A,G)),B,G) '<' 'not' (All ((All (a,B,G)),A,G))
let a be Function of Y,BOOLEAN; :: thesis: for G being Subset of (PARTITIONS Y)
for A, B being a_partition of Y st G is independent holds
Ex ((All (('not' a),A,G)),B,G) '<' 'not' (All ((All (a,B,G)),A,G))
let G be Subset of (PARTITIONS Y); :: thesis: for A, B being a_partition of Y st G is independent holds
Ex ((All (('not' a),A,G)),B,G) '<' 'not' (All ((All (a,B,G)),A,G))
let A, B be a_partition of Y; :: thesis: ( G is independent implies Ex ((All (('not' a),A,G)),B,G) '<' 'not' (All ((All (a,B,G)),A,G)) )
assume A1: G is independent ; :: thesis: Ex ((All (('not' a),A,G)),B,G) '<' 'not' (All ((All (a,B,G)),A,G))
then ( Ex ((All (('not' a),A,G)),B,G) '<' All ((Ex (('not' a),B,G)),A,G) & All ((Ex (('not' a),B,G)),A,G) '<' 'not' (All ((All (a,A,G)),B,G)) ) by Th29, PARTIT_2:17;
then Ex ((All (('not' a),A,G)),B,G) '<' 'not' (All ((All (a,A,G)),B,G)) by BVFUNC_1:15;
hence Ex ((All (('not' a),A,G)),B,G) '<' 'not' (All ((All (a,B,G)),A,G)) by A1, PARTIT_2:15; :: thesis: verum