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
All ('not' (All 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
All ('not' (All 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
All ('not' (All 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 All ('not' (All a,A,G)),B,G '<' 'not' (All (All a,B,G),A,G) )
assume G is independent ; :: thesis: All ('not' (All a,A,G)),B,G '<' 'not' (All (All a,B,G),A,G)
then All (All a,B,G),A,G = All (All a,A,G),B,G by PARTIT_2:17;
then ( All ('not' (All a,A,G)),B,G = 'not' (Ex (All a,A,G),B,G) & All (All a,B,G),A,G '<' Ex (All a,A,G),B,G ) by BVFUNC11:8, BVFUNC_2:21;
hence All ('not' (All a,A,G)),B,G '<' 'not' (All (All a,B,G),A,G) by PARTIT_2:11; :: thesis: verum