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
'not' (All (All a,A,G),B,G) '<' Ex (Ex ('not' a),A,G),B,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
'not' (All (All a,A,G),B,G) '<' Ex (Ex ('not' a),A,G),B,G
let G be Subset of (PARTITIONS Y); :: thesis: for A, B being a_partition of Y st G is independent holds
'not' (All (All a,A,G),B,G) '<' Ex (Ex ('not' a),A,G),B,G
let A, B be a_partition of Y; :: thesis: ( G is independent implies 'not' (All (All a,A,G),B,G) '<' Ex (Ex ('not' a),A,G),B,G )
assume A1: G is independent ; :: thesis: 'not' (All (All a,A,G),B,G) '<' Ex (Ex ('not' a),A,G),B,G
then Ex ('not' (All a,B,G)),A,G '<' Ex (Ex ('not' a),A,G),B,G by Th16;
hence 'not' (All (All a,A,G),B,G) '<' Ex (Ex ('not' a),A,G),B,G by A1, Th17; :: thesis: verum