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 All ((Ex (('not' a),A,G)),B,G) '<' Ex ((Ex (('not' 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 All ((Ex (('not' a),A,G)),B,G) '<' Ex ((Ex (('not' a),B,G)),A,G)
let G be Subset of (PARTITIONS Y); :: thesis: for A, B being a_partition of Y holds All ((Ex (('not' a),A,G)),B,G) '<' Ex ((Ex (('not' a),B,G)),A,G)
let A, B be a_partition of Y; :: thesis: All ((Ex (('not' a),A,G)),B,G) '<' Ex ((Ex (('not' a),B,G)),A,G)
'not' (Ex ((All (a,A,G)),B,G)) = All ((Ex (('not' a),A,G)),B,G) by Th22;
hence All ((Ex (('not' a),A,G)),B,G) '<' Ex ((Ex (('not' a),B,G)),A,G) by Th15; :: thesis: verum