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
'not' (Ex (Ex a,A,G),B,G) '<' 'not' (Ex (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
'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_2:18;
All a,B,G '<' Ex a,B,G by BVFUNC11:8;
then Ex (All a,B,G),A,G '<' Ex (Ex a,A,G),B,G by A1, PARTIT_2:15;
hence 'not' (Ex (Ex a,A,G),B,G) '<' 'not' (Ex (All a,B,G),A,G) by PARTIT_2:11; :: thesis: verum