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_2:20;
then A1: All (Ex ('not' a),A,G),B,G = 'not' (Ex (All a,A,G),B,G) by BVFUNC_2:21;
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, BVFUNC11:11, PARTIT_2:11; :: thesis: verum