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 ('not' (Ex a,A,G)),B,G '<' 'not' (All (All 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 ('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_2:21;
hence All ('not' (Ex a,A,G)),B,G '<' 'not' (All (All a,B,G),A,G) by Th2; :: thesis: verum