let Y be non empty set ; :: thesis: for a being Element of Funcs Y,BOOLEAN
for G being Subset of (PARTITIONS Y)
for P, Q being a_partition of Y st G is independent holds
Ex (Ex a,P,G),Q,G = Ex (Ex a,Q,G),P,G
let a be Element of Funcs Y,BOOLEAN ; :: thesis: for G being Subset of (PARTITIONS Y)
for P, Q being a_partition of Y st G is independent holds
Ex (Ex a,P,G),Q,G = Ex (Ex a,Q,G),P,G
let G be Subset of (PARTITIONS Y); :: thesis: for P, Q being a_partition of Y st G is independent holds
Ex (Ex a,P,G),Q,G = Ex (Ex a,Q,G),P,G
let P, Q be a_partition of Y; :: thesis: ( G is independent implies Ex (Ex a,P,G),Q,G = Ex (Ex a,Q,G),P,G )
assume A1: G is independent ; :: thesis: Ex (Ex a,P,G),Q,G = Ex (Ex a,Q,G),P,G
thus Ex (Ex a,P,G),Q,G = 'not' ('not' (Ex (Ex a,P,G),Q,G))
.= 'not' (All ('not' (Ex a,P,G)),Q,G) by BVFUNC_2:21
.= 'not' (All (All ('not' a),P,G),Q,G) by BVFUNC_2:21
.= 'not' (All (All ('not' a),Q,G),P,G) by A1, Th17
.= 'not' (All ('not' (Ex a,Q,G)),P,G) by BVFUNC_2:21
.= 'not' ('not' (Ex (Ex a,Q,G),P,G)) by BVFUNC_2:21
.= Ex (Ex a,Q,G),P,G ; :: thesis: verum