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
All ((All (a,A,G)),B,G) '<' All ((Ex (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
All ((All (a,A,G)),B,G) '<' All ((Ex (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 ((All (a,A,G)),B,G) '<' All ((Ex (a,B,G)),A,G)
let A, B be a_partition of Y; :: thesis: ( G is independent implies All ((All (a,A,G)),B,G) '<' All ((Ex (a,B,G)),A,G) )
assume G is independent ; :: thesis: All ((All (a,A,G)),B,G) '<' All ((Ex (a,B,G)),A,G)
then A1: All ((All (a,A,G)),B,G) = All ((All (a,B,G)),A,G) by PARTIT_2:15;
All (a,B,G) '<' Ex (a,B,G) by Th8;
hence All ((All (a,A,G)),B,G) '<' All ((Ex (a,B,G)),A,G) by A1, PARTIT_2:12; :: thesis: verum