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 All (('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 holds All (('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 holds All (('not' (Ex (a,A,G))),B,G) '<' 'not' (All ((All (a,B,G)),A,G))
let A, B be a_partition of Y; :: thesis: All (('not' (Ex (a,A,G))),B,G) '<' 'not' (All ((All (a,B,G)),A,G))
All (('not' (Ex (a,A,G))),B,G) = All ((All (('not' a),A,G)),B,G) by BVFUNC_2:19;
hence All (('not' (Ex (a,A,G))),B,G) '<' 'not' (All ((All (a,B,G)),A,G)) by Th27; :: thesis: verum