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) '<' Ex ((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) '<' Ex ((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) '<' Ex ((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) '<' Ex ((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 Th21, BVFUNC_2:19;
assume G is independent ; :: thesis: Ex (('not' (Ex (a,A,G))),B,G) '<' Ex ((Ex (('not' a),B,G)),A,G)
hence Ex (('not' (Ex (a,A,G))),B,G) '<' Ex ((Ex (('not' a),B,G)),A,G) by A1, Th47; :: thesis: verum