let Y be non empty set ; :: thesis: for a being Element of Funcs (Y,BOOLEAN)
for G being Subset of (PARTITIONS Y)
for A, B being a_partition of Y st G is independent holds
'not' (All ((Ex (a,A,G)),B,G)) '<' All ((Ex (('not' a),B,G)),A,G)
let a be Element of Funcs (Y,BOOLEAN); :: thesis: for G being Subset of (PARTITIONS Y)
for A, B being a_partition of Y st G is independent holds
'not' (All ((Ex (a,A,G)),B,G)) '<' All ((Ex (('not' 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
'not' (All ((Ex (a,A,G)),B,G)) '<' All ((Ex (('not' a),B,G)),A,G)
let A, B be a_partition of Y; :: thesis: ( G is independent implies 'not' (All ((Ex (a,A,G)),B,G)) '<' All ((Ex (('not' a),B,G)),A,G) )
assume G is independent ; :: thesis: 'not' (All ((Ex (a,A,G)),B,G)) '<' All ((Ex (('not' a),B,G)),A,G)
then 'not' (All ((Ex (a,A,G)),B,G)) '<' All (('not' (All (a,B,G))),A,G) by Th43;
hence 'not' (All ((Ex (a,A,G)),B,G)) '<' All ((Ex (('not' a),B,G)),A,G) by BVFUNC_2:18; :: thesis: verum