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 ('not' (Ex a,A,G)),B,G '<' 'not' (Ex (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 ('not' (Ex a,A,G)),B,G '<' 'not' (Ex (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 ('not' (Ex a,A,G)),B,G '<' 'not' (Ex (All a,B,G),A,G)
let A, B be a_partition of Y; :: thesis: ( G is independent implies Ex ('not' (Ex a,A,G)),B,G '<' 'not' (Ex (All a,B,G),A,G) )
assume G is independent ; :: thesis: Ex ('not' (Ex a,A,G)),B,G '<' 'not' (Ex (All a,B,G),A,G)
then A1: 'not' (All (Ex a,A,G),B,G) '<' 'not' (Ex (All a,B,G),A,G) by PARTIT_2:11, PARTIT_2:19;
'not' (All (Ex a,A,G),B,G) = Ex (All ('not' a),A,G),B,G by BVFUNC11:21;
hence Ex ('not' (Ex a,A,G)),B,G '<' 'not' (Ex (All a,B,G),A,G) by A1, BVFUNC_2:21; :: thesis: verum