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 ((Ex (('not' a),A,G)),B,G) '<' 'not' (All ((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
All ((Ex (('not' a),A,G)),B,G) '<' 'not' (All ((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
All ((Ex (('not' a),A,G)),B,G) '<' 'not' (All ((All (a,B,G)),A,G))
let A, B be a_partition of Y; :: thesis: ( G is independent implies All ((Ex (('not' a),A,G)),B,G) '<' 'not' (All ((All (a,B,G)),A,G)) )
Ex (('not' a),A,G) = 'not' (All (a,A,G)) by BVFUNC_2:18;
then A1: All ((Ex (('not' a),A,G)),B,G) = 'not' (Ex ((All (a,A,G)),B,G)) by BVFUNC_2:19;
assume G is independent ; :: thesis: All ((Ex (('not' a),A,G)),B,G) '<' 'not' (All ((All (a,B,G)),A,G))
hence All ((Ex (('not' a),A,G)),B,G) '<' 'not' (All ((All (a,B,G)),A,G)) by A1, Th11, PARTIT_2:11; :: thesis: verum