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 st G is independent holds
Ex ('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 st G is independent holds
Ex ('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 st G is independent holds
Ex ('not' (Ex a,A,G)),B,G '<' 'not' (All (All a,B,G),A,G)
let A, B be a_partition of Y; :: thesis: ( G is independent implies Ex ('not' (Ex a,A,G)),B,G '<' 'not' (All (All a,B,G),A,G) )
assume A1: G is independent ; :: thesis: Ex ('not' (Ex a,A,G)),B,G '<' 'not' (All (All a,B,G),A,G)
A2: Ex ('not' (Ex a,A,G)),B,G = 'not' (All (Ex a,A,G),B,G) by BVFUNC_2:20;
A3: All (All a,B,G),A,G = All (All a,A,G),B,G by A1, PARTIT_2:17;
All a,A,G '<' Ex a,A,G by BVFUNC11:8;
then All (All a,A,G),B,G '<' All (Ex a,A,G),B,G by PARTIT_2:13;
hence Ex ('not' (Ex a,A,G)),B,G '<' 'not' (All (All a,B,G),A,G) by A2, A3, PARTIT_2:11; :: thesis: verum