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 holds 'not' (Ex ((Ex (a,A,G)),B,G)) '<' Ex (('not' (Ex (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 holds 'not' (Ex ((Ex (a,A,G)),B,G)) '<' Ex (('not' (Ex (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' (Ex (a,B,G))),A,G)
let A, B be a_partition of Y; :: thesis: 'not' (Ex ((Ex (a,A,G)),B,G)) '<' Ex (('not' (Ex (a,B,G))),A,G)
( 'not' (Ex ((Ex (a,A,G)),B,G)) '<' 'not' (All ((Ex (a,B,G)),A,G)) & Ex (('not' (Ex (a,B,G))),A,G) = Ex ((All (('not' a),B,G)),A,G) ) by Th36, BVFUNC_2:19;
hence 'not' (Ex ((Ex (a,A,G)),B,G)) '<' Ex (('not' (Ex (a,B,G))),A,G) by Th21; :: thesis: verum