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 (All ('not' a),A,G),B,G '<' 'not' (Ex (Ex 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 (All ('not' a),A,G),B,G '<' 'not' (Ex (Ex 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 (All ('not' a),A,G),B,G '<' 'not' (Ex (Ex a,B,G),A,G)
let A, B be a_partition of Y; :: thesis: ( G is independent implies All (All ('not' a),A,G),B,G '<' 'not' (Ex (Ex a,B,G),A,G) )
A1: ( 'not' (Ex (Ex a,A,G),B,G) = All ('not' (Ex a,A,G)),B,G & All ('not' (Ex a,A,G)),B,G = All (All ('not' a),A,G),B,G ) by BVFUNC_2:21;
assume G is independent ; :: thesis: All (All ('not' a),A,G),B,G '<' 'not' (Ex (Ex a,B,G),A,G)
hence All (All ('not' a),A,G),B,G '<' 'not' (Ex (Ex a,B,G),A,G) by A1, PARTIT_2:18; :: thesis: verum