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 ('not' (All 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
All ('not' (All 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
All ('not' (All 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 All ('not' (All a,A,G)),B,G '<' Ex (Ex ('not' a),B,G),A,G )
assume G is independent ; :: thesis: All ('not' (All a,A,G)),B,G '<' Ex (Ex ('not' a),B,G),A,G
then A1: Ex (Ex ('not' a),B,G),A,G = Ex (Ex ('not' a),A,G),B,G by PARTIT_2:18;
'not' (All a,A,G) = Ex ('not' a),A,G by BVFUNC_2:20;
hence All ('not' (All a,A,G)),B,G '<' Ex (Ex ('not' a),B,G),A,G by A1, Th8; :: thesis: verum