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
'not' (Ex ((All (a,A,G)),B,G)) '<' Ex (('not' (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
'not' (Ex ((All (a,A,G)),B,G)) '<' Ex (('not' (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
'not' (Ex ((All (a,A,G)),B,G)) '<' Ex (('not' (All (a,B,G))),A,G)
let A, B be a_partition of Y; :: thesis: ( G is independent implies 'not' (Ex ((All (a,A,G)),B,G)) '<' Ex (('not' (All (a,B,G))),A,G) )
assume G is independent ; :: thesis: 'not' (Ex ((All (a,A,G)),B,G)) '<' Ex (('not' (All (a,B,G))),A,G)
then A1: All (('not' (All (a,A,G))),B,G) '<' 'not' (All ((All (a,B,G)),A,G)) by Th26;
'not' (Ex ((All (a,A,G)),B,G)) = All (('not' (All (a,A,G))),B,G) by BVFUNC_2:19;
hence 'not' (Ex ((All (a,A,G)),B,G)) '<' Ex (('not' (All (a,B,G))),A,G) by A1, BVFUNC_2:18; :: thesis: verum