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 (('not' (Ex (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 (('not' (Ex (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 (('not' (Ex (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 (('not' (Ex (a,A,G))),B,G) '<' 'not' (All ((All (a,B,G)),A,G)) )
All (a,A,G) '<' Ex (a,A,G) by Th8;
then A1: All ((All (a,A,G)),B,G) '<' All ((Ex (a,A,G)),B,G) by PARTIT_2:12;
assume G is independent ; :: thesis: Ex (('not' (Ex (a,A,G))),B,G) '<' 'not' (All ((All (a,B,G)),A,G))
then ( Ex (('not' (Ex (a,A,G))),B,G) = 'not' (All ((Ex (a,A,G)),B,G)) & All ((All (a,B,G)),A,G) = All ((All (a,A,G)),B,G) ) by BVFUNC_2:18, PARTIT_2:15;
hence Ex (('not' (Ex (a,A,G))),B,G) '<' 'not' (All ((All (a,B,G)),A,G)) by A1, PARTIT_2:11; :: thesis: verum