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 st G is independent holds
'not' (Ex ((Ex (a,A,G)),B,G)) '<' 'not' (Ex ((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 st G is independent holds
'not' (Ex ((Ex (a,A,G)),B,G)) '<' 'not' (Ex ((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
'not' (Ex ((Ex (a,A,G)),B,G)) '<' 'not' (Ex ((All (a,B,G)),A,G))
let A, B be a_partition of Y; :: thesis: ( G is independent implies 'not' (Ex ((Ex (a,A,G)),B,G)) '<' 'not' (Ex ((All (a,B,G)),A,G)) )
assume G is independent ; :: thesis: 'not' (Ex ((Ex (a,A,G)),B,G)) '<' 'not' (Ex ((All (a,B,G)),A,G))
then A1: Ex ((Ex (a,A,G)),B,G) = Ex ((Ex (a,B,G)),A,G) by PARTIT_2:16;
All (a,B,G) '<' Ex (a,B,G) by Th8;
then Ex ((All (a,B,G)),A,G) '<' Ex ((Ex (a,A,G)),B,G) by A1, PARTIT_2:13;
hence 'not' (Ex ((Ex (a,A,G)),B,G)) '<' 'not' (Ex ((All (a,B,G)),A,G)) by PARTIT_2:11; :: thesis: verum