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' (All (Ex a,A,G),B,G) '<' Ex ('not' (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' (All (Ex a,A,G),B,G) '<' Ex ('not' (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' (All (Ex a,A,G),B,G) '<' Ex ('not' (All a,B,G)),A,G
let A, B be a_partition of Y; :: thesis: ( G is independent implies 'not' (All (Ex a,A,G),B,G) '<' Ex ('not' (All a,B,G)),A,G )
assume G is independent ; :: thesis: 'not' (All (Ex a,A,G),B,G) '<' Ex ('not' (All a,B,G)),A,G
then A1: Ex ('not' (Ex a,A,G)),B,G '<' 'not' (All (All a,B,G),A,G) by Th7;
A2: 'not' (All (Ex a,A,G),B,G) = Ex (All ('not' a),A,G),B,G by BVFUNC11:21;
Ex ('not' (Ex a,A,G)),B,G = Ex (All ('not' a),A,G),B,G by BVFUNC_2:21;
hence 'not' (All (Ex a,A,G),B,G) '<' Ex ('not' (All a,B,G)),A,G by A1, A2, BVFUNC_2:20; :: thesis: verum