let Y be non empty set ; :: thesis: for a being Function of 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 Function of 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_2:19

.= 'not' (All ((All (('not' a),P,G)),Q,G)) by BVFUNC_2:19

.= 'not' (All ((All (('not' a),Q,G)),P,G)) by A1, Th15

.= 'not' (All (('not' (Ex (a,Q,G))),P,G)) by BVFUNC_2:19

.= 'not' ('not' (Ex ((Ex (a,Q,G)),P,G))) by BVFUNC_2:19

.= Ex ((Ex (a,Q,G)),P,G) ; :: thesis: verum

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 Function of 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_2:19

.= 'not' (All ((All (('not' a),P,G)),Q,G)) by BVFUNC_2:19

.= 'not' (All ((All (('not' a),Q,G)),P,G)) by A1, Th15

.= 'not' (All (('not' (Ex (a,Q,G))),P,G)) by BVFUNC_2:19

.= 'not' ('not' (Ex ((Ex (a,Q,G)),P,G))) by BVFUNC_2:19

.= Ex ((Ex (a,Q,G)),P,G) ; :: thesis: verum