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
All (('not' (All (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
All (('not' (All (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
All (('not' (All (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 All (('not' (All (a,A,G))),B,G) '<' 'not' (All ((All (a,B,G)),A,G)) )
assume G is independent ; :: thesis: All (('not' (All (a,A,G))),B,G) '<' 'not' (All ((All (a,B,G)),A,G))
then All ((All (a,B,G)),A,G) = All ((All (a,A,G)),B,G) by PARTIT_2:15;
then ( All (('not' (All (a,A,G))),B,G) = 'not' (Ex ((All (a,A,G)),B,G)) & All ((All (a,B,G)),A,G) '<' Ex ((All (a,A,G)),B,G) ) by Th8, BVFUNC_2:19;
hence All (('not' (All (a,A,G))),B,G) '<' 'not' (All ((All (a,B,G)),A,G)) by PARTIT_2:11; :: thesis: verum