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 (All a,A,G),B,G '<' All (Ex 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 (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_2:17;
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_2:13; :: thesis: verum