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
Ex (All ('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
Ex (All ('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
Ex (All ('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 Ex (All ('not' a),A,G),B,G '<' 'not' (All (All a,B,G),A,G) )
assume A1: G is independent ; :: thesis: Ex (All ('not' a),A,G),B,G '<' 'not' (All (All a,B,G),A,G)
then ( Ex (All ('not' a),A,G),B,G '<' All (Ex ('not' a),B,G),A,G & All (Ex ('not' a),B,G),A,G '<' 'not' (All (All a,A,G),B,G) ) by Th4, PARTIT_2:19;
then Ex (All ('not' a),A,G),B,G '<' 'not' (All (All a,A,G),B,G) by BVFUNC_1:18;
hence Ex (All ('not' a),A,G),B,G '<' 'not' (All (All a,B,G),A,G) by A1, PARTIT_2:17; :: thesis: verum