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
Ex ('not' (Ex a,A,G)),B,G '<' All (Ex ('not' 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
Ex ('not' (Ex a,A,G)),B,G '<' All (Ex ('not' 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
Ex ('not' (Ex a,A,G)),B,G '<' All (Ex ('not' a),B,G),A,G
let A, B be a_partition of Y; :: thesis: ( G is independent implies Ex ('not' (Ex a,A,G)),B,G '<' All (Ex ('not' a),B,G),A,G )
A1: ( 'not' (All (Ex a,A,G),B,G) = Ex (All ('not' a),A,G),B,G & Ex ('not' (Ex a,A,G)),B,G = Ex (All ('not' a),A,G),B,G ) by BVFUNC11:21, BVFUNC_2:21;
assume G is independent ; :: thesis: Ex ('not' (Ex a,A,G)),B,G '<' All (Ex ('not' a),B,G),A,G
hence Ex ('not' (Ex a,A,G)),B,G '<' All (Ex ('not' a),B,G),A,G by A1, Th26; :: thesis: verum