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 holds 'not' (Ex ((Ex (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 holds 'not' (Ex ((Ex (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 holds 'not' (Ex ((Ex (a,A,G)),B,G)) '<' Ex (('not' (All (a,B,G))),A,G)
let A, B be a_partition of Y; :: thesis: 'not' (Ex ((Ex (a,A,G)),B,G)) '<' Ex (('not' (All (a,B,G))),A,G)
( All (('not' (Ex (a,A,G))),B,G) '<' 'not' (All ((All (a,B,G)),A,G)) & 'not' (Ex ((Ex (a,A,G)),B,G)) = All (('not' (Ex (a,A,G))),B,G) ) by Th28, BVFUNC_2:19;
hence 'not' (Ex ((Ex (a,A,G)),B,G)) '<' Ex (('not' (All (a,B,G))),A,G) by BVFUNC_2:18; :: thesis: verum