let Y be non empty set ; for a being Function of Y,BOOLEAN
for G being Subset of (PARTITIONS Y)
for A, B being a_partition of Y holds All ((All (a,A,G)),B,G) '<' Ex ((Ex (a,B,G)),A,G)
let a be Function of Y,BOOLEAN; for G being Subset of (PARTITIONS Y)
for A, B being a_partition of Y holds All ((All (a,A,G)),B,G) '<' Ex ((Ex (a,B,G)),A,G)
let G be Subset of (PARTITIONS Y); for A, B being a_partition of Y holds All ((All (a,A,G)),B,G) '<' Ex ((Ex (a,B,G)),A,G)
let A, B be a_partition of Y; All ((All (a,A,G)),B,G) '<' Ex ((Ex (a,B,G)),A,G)
A1:
All ((All (a,A,G)),B,G) '<' All (a,A,G)
by BVFUNC_2:11;
All (a,A,G) '<' Ex ((Ex (a,B,G)),A,G)
by Th8, BVFUNC_2:12;
hence
All ((All (a,A,G)),B,G) '<' Ex ((Ex (a,B,G)),A,G)
by A1, BVFUNC_1:15; verum