%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SET601^3 : TPTP v6.1.0. Released v3.6.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n093.star.cs.uiowa.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory   : 32286.75MB
% OS       : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:30:48 EDT 2014

% Result   : Timeout 300.00s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SET601^3 : TPTP v6.1.0. Released v3.6.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n093.star.cs.uiowa.edu
% % Model    : x86_64 x86_64
% % CPU      : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory   : 32286.75MB
% % OS       : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 10:14:36 CDT 2014
% % CPUTime  : 300.00 
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% Failed to open /home/cristobal/cocATP/CASC/TPTP/Axioms/SET008^0.ax, trying next directory
% FOF formula (<kernel.Constant object at 0x1de9128>, <kernel.DependentProduct object at 0x1de99e0>) of role type named in_decl
% Using role type
% Declaring in:(fofType->((fofType->Prop)->Prop))
% FOF formula (((eq (fofType->((fofType->Prop)->Prop))) in) (fun (X:fofType) (M:(fofType->Prop))=> (M X))) of role definition named in
% A new definition: (((eq (fofType->((fofType->Prop)->Prop))) in) (fun (X:fofType) (M:(fofType->Prop))=> (M X)))
% Defined: in:=(fun (X:fofType) (M:(fofType->Prop))=> (M X))
% FOF formula (<kernel.Constant object at 0x1de9128>, <kernel.DependentProduct object at 0x1de9d40>) of role type named is_a_decl
% Using role type
% Declaring is_a:(fofType->((fofType->Prop)->Prop))
% FOF formula (((eq (fofType->((fofType->Prop)->Prop))) is_a) (fun (X:fofType) (M:(fofType->Prop))=> (M X))) of role definition named is_a
% A new definition: (((eq (fofType->((fofType->Prop)->Prop))) is_a) (fun (X:fofType) (M:(fofType->Prop))=> (M X)))
% Defined: is_a:=(fun (X:fofType) (M:(fofType->Prop))=> (M X))
% FOF formula (<kernel.Constant object at 0x1de9d40>, <kernel.DependentProduct object at 0x1de9488>) of role type named emptyset_decl
% Using role type
% Declaring emptyset:(fofType->Prop)
% FOF formula (((eq (fofType->Prop)) emptyset) (fun (X:fofType)=> False)) of role definition named emptyset
% A new definition: (((eq (fofType->Prop)) emptyset) (fun (X:fofType)=> False))
% Defined: emptyset:=(fun (X:fofType)=> False)
% FOF formula (<kernel.Constant object at 0x1de9488>, <kernel.DependentProduct object at 0x1de9290>) of role type named unord_pair_decl
% Using role type
% Declaring unord_pair:(fofType->(fofType->(fofType->Prop)))
% FOF formula (((eq (fofType->(fofType->(fofType->Prop)))) unord_pair) (fun (X:fofType) (Y:fofType) (U:fofType)=> ((or (((eq fofType) U) X)) (((eq fofType) U) Y)))) of role definition named unord_pair
% A new definition: (((eq (fofType->(fofType->(fofType->Prop)))) unord_pair) (fun (X:fofType) (Y:fofType) (U:fofType)=> ((or (((eq fofType) U) X)) (((eq fofType) U) Y))))
% Defined: unord_pair:=(fun (X:fofType) (Y:fofType) (U:fofType)=> ((or (((eq fofType) U) X)) (((eq fofType) U) Y)))
% FOF formula (<kernel.Constant object at 0x1de9290>, <kernel.DependentProduct object at 0x1de9758>) of role type named singleton_decl
% Using role type
% Declaring singleton:(fofType->(fofType->Prop))
% FOF formula (((eq (fofType->(fofType->Prop))) singleton) (fun (X:fofType) (U:fofType)=> (((eq fofType) U) X))) of role definition named singleton
% A new definition: (((eq (fofType->(fofType->Prop))) singleton) (fun (X:fofType) (U:fofType)=> (((eq fofType) U) X)))
% Defined: singleton:=(fun (X:fofType) (U:fofType)=> (((eq fofType) U) X))
% FOF formula (<kernel.Constant object at 0x1de9758>, <kernel.DependentProduct object at 0x1de9d40>) of role type named union_decl
% Using role type
% Declaring union:((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))
% FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) union) (fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((or (X U)) (Y U)))) of role definition named union
% A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) union) (fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((or (X U)) (Y U))))
% Defined: union:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((or (X U)) (Y U)))
% FOF formula (<kernel.Constant object at 0x1de9d40>, <kernel.DependentProduct object at 0x1de9cf8>) of role type named excl_union_decl
% Using role type
% Declaring excl_union:((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))
% FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) excl_union) (fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((or ((and (X U)) ((Y U)->False))) ((and ((X U)->False)) (Y U))))) of role definition named excl_union
% A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) excl_union) (fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((or ((and (X U)) ((Y U)->False))) ((and ((X U)->False)) (Y U)))))
% Defined: excl_union:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((or ((and (X U)) ((Y U)->False))) ((and ((X U)->False)) (Y U))))
% FOF formula (<kernel.Constant object at 0x1de9cf8>, <kernel.DependentProduct object at 0x1de9758>) of role type named intersection_decl
% Using role type
% Declaring intersection:((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))
% FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) intersection) (fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((and (X U)) (Y U)))) of role definition named intersection
% A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) intersection) (fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((and (X U)) (Y U))))
% Defined: intersection:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((and (X U)) (Y U)))
% FOF formula (<kernel.Constant object at 0x1de9758>, <kernel.DependentProduct object at 0x1de9f80>) of role type named setminus_decl
% Using role type
% Declaring setminus:((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))
% FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) setminus) (fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((and (X U)) ((Y U)->False)))) of role definition named setminus
% A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) setminus) (fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((and (X U)) ((Y U)->False))))
% Defined: setminus:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((and (X U)) ((Y U)->False)))
% FOF formula (<kernel.Constant object at 0x1e46cb0>, <kernel.DependentProduct object at 0x1e465a8>) of role type named complement_decl
% Using role type
% Declaring complement:((fofType->Prop)->(fofType->Prop))
% FOF formula (((eq ((fofType->Prop)->(fofType->Prop))) complement) (fun (X:(fofType->Prop)) (U:fofType)=> ((X U)->False))) of role definition named complement
% A new definition: (((eq ((fofType->Prop)->(fofType->Prop))) complement) (fun (X:(fofType->Prop)) (U:fofType)=> ((X U)->False)))
% Defined: complement:=(fun (X:(fofType->Prop)) (U:fofType)=> ((X U)->False))
% FOF formula (<kernel.Constant object at 0x1e467a0>, <kernel.DependentProduct object at 0x1e46e60>) of role type named disjoint_decl
% Using role type
% Declaring disjoint:((fofType->Prop)->((fofType->Prop)->Prop))
% FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->Prop))) disjoint) (fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (((eq (fofType->Prop)) ((intersection X) Y)) emptyset))) of role definition named disjoint
% A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->Prop))) disjoint) (fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (((eq (fofType->Prop)) ((intersection X) Y)) emptyset)))
% Defined: disjoint:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (((eq (fofType->Prop)) ((intersection X) Y)) emptyset))
% FOF formula (<kernel.Constant object at 0x1e46e60>, <kernel.DependentProduct object at 0x1de94d0>) of role type named subset_decl
% Using role type
% Declaring subset:((fofType->Prop)->((fofType->Prop)->Prop))
% FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->Prop))) subset) (fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (forall (U:fofType), ((X U)->(Y U))))) of role definition named subset
% A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->Prop))) subset) (fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (forall (U:fofType), ((X U)->(Y U)))))
% Defined: subset:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (forall (U:fofType), ((X U)->(Y U))))
% FOF formula (<kernel.Constant object at 0x1e465a8>, <kernel.DependentProduct object at 0x1de9170>) of role type named meets_decl
% Using role type
% Declaring meets:((fofType->Prop)->((fofType->Prop)->Prop))
% FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->Prop))) meets) (fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))))) of role definition named meets
% A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->Prop))) meets) (fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U))))))
% Defined: meets:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))))
% FOF formula (<kernel.Constant object at 0x1e465a8>, <kernel.DependentProduct object at 0x1de9290>) of role type named misses_decl
% Using role type
% Declaring misses:((fofType->Prop)->((fofType->Prop)->Prop))
% FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->Prop))) misses) (fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U))))->False))) of role definition named misses
% A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->Prop))) misses) (fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U))))->False)))
% Defined: misses:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U))))->False))
% FOF formula (forall (X:(fofType->Prop)) (Y:(fofType->Prop)) (Z:(fofType->Prop)), (((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))) of role conjecture named thm
% Conjecture to prove = (forall (X:(fofType->Prop)) (Y:(fofType->Prop)) (Z:(fofType->Prop)), (((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))):Prop
% Parameter fofType_DUMMY:fofType.
% We need to prove ['(forall (X:(fofType->Prop)) (Y:(fofType->Prop)) (Z:(fofType->Prop)), (((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))))']
% Parameter fofType:Type.
% Definition in:=(fun (X:fofType) (M:(fofType->Prop))=> (M X)):(fofType->((fofType->Prop)->Prop)).
% Definition is_a:=(fun (X:fofType) (M:(fofType->Prop))=> (M X)):(fofType->((fofType->Prop)->Prop)).
% Definition emptyset:=(fun (X:fofType)=> False):(fofType->Prop).
% Definition unord_pair:=(fun (X:fofType) (Y:fofType) (U:fofType)=> ((or (((eq fofType) U) X)) (((eq fofType) U) Y))):(fofType->(fofType->(fofType->Prop))).
% Definition singleton:=(fun (X:fofType) (U:fofType)=> (((eq fofType) U) X)):(fofType->(fofType->Prop)).
% Definition union:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((or (X U)) (Y U))):((fofType->Prop)->((fofType->Prop)->(fofType->Prop))).
% Definition excl_union:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((or ((and (X U)) ((Y U)->False))) ((and ((X U)->False)) (Y U)))):((fofType->Prop)->((fofType->Prop)->(fofType->Prop))).
% Definition intersection:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((and (X U)) (Y U))):((fofType->Prop)->((fofType->Prop)->(fofType->Prop))).
% Definition setminus:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((and (X U)) ((Y U)->False))):((fofType->Prop)->((fofType->Prop)->(fofType->Prop))).
% Definition complement:=(fun (X:(fofType->Prop)) (U:fofType)=> ((X U)->False)):((fofType->Prop)->(fofType->Prop)).
% Definition disjoint:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (((eq (fofType->Prop)) ((intersection X) Y)) emptyset)):((fofType->Prop)->((fofType->Prop)->Prop)).
% Definition subset:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (forall (U:fofType), ((X U)->(Y U)))):((fofType->Prop)->((fofType->Prop)->Prop)).
% Definition meets:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U))))):((fofType->Prop)->((fofType->Prop)->Prop)).
% Definition misses:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U))))->False)):((fofType->Prop)->((fofType->Prop)->Prop)).
% Trying to prove (forall (X:(fofType->Prop)) (Y:(fofType->Prop)) (Z:(fofType->Prop)), (((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))):(((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) (fun (x:fofType)=> (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))
% Found (eta_expansion_dep00 ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) as proof of (((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) as proof of (((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) as proof of (((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) as proof of (((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) as proof of (((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found x2:(P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found (fun (x2:(P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))))=> x2) as proof of (P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found (fun (x2:(P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))))=> x2) as proof of (P0 ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found x2:(P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found (fun (x2:(P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))))=> x2) as proof of (P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found (fun (x2:(P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))))=> x2) as proof of (P0 ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found x2:(P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found (fun (x2:(P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))))=> x2) as proof of (P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found (fun (x2:(P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))))=> x2) as proof of (P0 ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found eq_ref00:=(eq_ref0 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))):(((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (eq_ref0 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found ((eq_ref (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found ((eq_ref (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found ((eq_ref (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found eq_ref00:=(eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)):(((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)):(((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found x2:(P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (fun (x2:(P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))))=> x2) as proof of (P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (fun (x2:(P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))))=> x2) as proof of (P0 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found x2:(P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (fun (x2:(P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))))=> x2) as proof of (P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (fun (x2:(P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))))=> x2) as proof of (P0 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found x01:(P (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (fun (x01:(P (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))=> x01) as proof of (P (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (fun (x01:(P (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))=> x01) as proof of (P0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found x01:(P (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (fun (x01:(P (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))=> x01) as proof of (P (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (fun (x01:(P (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))=> x01) as proof of (P0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found x01:(P (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (fun (x01:(P (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))=> x01) as proof of (P (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (fun (x01:(P (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))=> x01) as proof of (P0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found x01:(P (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (fun (x01:(P (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))=> x01) as proof of (P (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (fun (x01:(P (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))=> x01) as proof of (P0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found x:(P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Instantiate: b:=((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))):(fofType->Prop)
% Found x as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))):(((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (eq_ref0 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found ((eq_ref (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found ((eq_ref (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found ((eq_ref (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found x:(P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Instantiate: f:=((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))):(fofType->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x0))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x0))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))
% Found x:(P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Instantiate: f:=((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))):(fofType->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x0))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x0))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))
% Found eq_ref00:=(eq_ref0 ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))):(((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found (eq_ref0 ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) as proof of (((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) b)
% Found ((eq_ref (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) as proof of (((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) b)
% Found ((eq_ref (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) as proof of (((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) b)
% Found ((eq_ref (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) as proof of (((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)):(((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)):(((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))):(((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (eq_ref0 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found ((eq_ref (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found ((eq_ref (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found ((eq_ref (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found x:(P0 b)
% Instantiate: b:=(fun (U:fofType)=> ((or (((intersection X) Y) U)) (((union ((intersection Y) Z)) ((intersection Z) X)) U))):(fofType->Prop)
% Found (fun (x:(P0 b))=> x) as proof of (P0 ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found (fun (P0:((fofType->Prop)->Prop)) (x:(P0 b))=> x) as proof of ((P0 b)->(P0 ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))))
% Found (fun (P0:((fofType->Prop)->Prop)) (x:(P0 b))=> x) as proof of (P b)
% Found x3:(P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found (fun (x3:(P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))))=> x3) as proof of (P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found (fun (x3:(P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))))=> x3) as proof of (P0 ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))):(((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) (fun (x:fofType)=> (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))
% Found (eta_expansion_dep00 ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) as proof of (((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) as proof of (((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) as proof of (((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) as proof of (((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) as proof of (((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found x:(P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Instantiate: b:=((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))):(fofType->Prop)
% Found x as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))):(((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found (eq_ref0 ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) as proof of (((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) b)
% Found ((eq_ref (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) as proof of (((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) b)
% Found ((eq_ref (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) as proof of (((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) b)
% Found ((eq_ref (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) as proof of (((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) b)
% Found eta_expansion000:=(eta_expansion00 ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))):(((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) (fun (x:fofType)=> (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))
% Found (eta_expansion00 ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) as proof of (((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) b0)
% Found ((eta_expansion0 Prop) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) as proof of (((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) b0)
% Found (((eta_expansion fofType) Prop) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) as proof of (((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) b0)
% Found (((eta_expansion fofType) Prop) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) as proof of (((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) b0)
% Found (((eta_expansion fofType) Prop) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) as proof of (((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (fofType->Prop)) b0) (fun (x:fofType)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (fofType->Prop)) b0) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found x:(P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Instantiate: f:=((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))):(fofType->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x0))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x0))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))
% Found x:(P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Instantiate: f:=((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))):(fofType->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x0))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x0))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))
% Found x0:(P (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Instantiate: b:=(((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found x0:(P (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Instantiate: b:=(((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found eq_ref00:=(eq_ref0 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))):(((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (eq_ref0 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found ((eq_ref (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found ((eq_ref (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found ((eq_ref (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))):(((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) (fun (x:fofType)=> (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))
% Found (eta_expansion_dep00 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found eta_expansion000:=(eta_expansion00 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))):(((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) (fun (x:fofType)=> (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))
% Found (eta_expansion00 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found ((eta_expansion0 Prop) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found (((eta_expansion fofType) Prop) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found (((eta_expansion fofType) Prop) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found (((eta_expansion fofType) Prop) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)):(((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)):(((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)):(((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)):(((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)):(((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)):(((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found x2:(P1 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (fun (x2:(P1 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))))=> x2) as proof of (P1 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (fun (x2:(P1 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))))=> x2) as proof of (P2 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found x2:(P1 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (fun (x2:(P1 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))))=> x2) as proof of (P1 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (fun (x2:(P1 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))))=> x2) as proof of (P2 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found x2:(P1 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (fun (x2:(P1 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))))=> x2) as proof of (P1 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (fun (x2:(P1 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))))=> x2) as proof of (P2 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found x2:(P1 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (fun (x2:(P1 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))))=> x2) as proof of (P1 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (fun (x2:(P1 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))))=> x2) as proof of (P2 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found x3:(P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (fun (x3:(P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))))=> x3) as proof of (P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (fun (x3:(P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))))=> x3) as proof of (P0 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))):(((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) (fun (x:fofType)=> (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))
% Found (eta_expansion_dep00 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found x01:(P1 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (fun (x01:(P1 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))=> x01) as proof of (P1 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (fun (x01:(P1 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))=> x01) as proof of (P2 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found x01:(P1 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (fun (x01:(P1 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))=> x01) as proof of (P1 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (fun (x01:(P1 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))=> x01) as proof of (P2 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found x01:(P1 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (fun (x01:(P1 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))=> x01) as proof of (P1 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (fun (x01:(P1 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))=> x01) as proof of (P2 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found x01:(P1 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (fun (x01:(P1 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))=> x01) as proof of (P1 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (fun (x01:(P1 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))=> x01) as proof of (P2 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found eq_ref00:=(eq_ref0 b0):(((eq (fofType->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (fofType->Prop)) b0) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found eq_ref00:=(eq_ref0 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))):(((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (eq_ref0 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b0)
% Found ((eq_ref (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b0)
% Found ((eq_ref (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b0)
% Found ((eq_ref (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b0)
% Found x02:(P (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (fun (x02:(P (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))=> x02) as proof of (P (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (fun (x02:(P (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))=> x02) as proof of (P0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)):(((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found x02:(P (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (fun (x02:(P (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))=> x02) as proof of (P (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (fun (x02:(P (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))=> x02) as proof of (P0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)):(((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found x0:(P0 b)
% Instantiate: b:=((or (((intersection X) Y) x)) (((union ((intersection Y) Z)) ((intersection Z) X)) x)):Prop
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found x0:(P0 b)
% Instantiate: b:=((or (((intersection X) Y) x)) (((union ((intersection Y) Z)) ((intersection Z) X)) x)):Prop
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))):(((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) (fun (x:fofType)=> (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))
% Found (eta_expansion_dep00 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))):(((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) (fun (x:fofType)=> (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))
% Found (eta_expansion_dep00 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found eq_ref00:=(eq_ref0 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))):(((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (eq_ref0 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b0)
% Found ((eq_ref (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b0)
% Found ((eq_ref (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b0)
% Found ((eq_ref (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (fofType->Prop)) b0) (fun (x:fofType)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found x0:(P (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Instantiate: b:=(((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)):(((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found x0:(P (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Instantiate: b:=(((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)):(((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)):(((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b0)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b0)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b0)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)):(((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b0)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b0)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b0)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))):(((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) (fun (x:fofType)=> (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))
% Found (eta_expansion_dep00 ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) as proof of (((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) as proof of (((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) as proof of (((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) as proof of (((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) as proof of (((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found x0:(P (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Instantiate: b:=(((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)):(((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found x0:(P (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Instantiate: b:=(((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)):(((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found x2:(P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found (fun (x2:(P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))))=> x2) as proof of (P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found (fun (x2:(P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))))=> x2) as proof of (P0 ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found x2:(P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found (fun (x2:(P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))))=> x2) as proof of (P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found (fun (x2:(P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))))=> x2) as proof of (P0 ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found x2:(P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found (fun (x2:(P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))))=> x2) as proof of (P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found (fun (x2:(P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))))=> x2) as proof of (P0 ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found eq_ref00:=(eq_ref0 (b x)):(((eq Prop) (b x)) (b x))
% Found (eq_ref0 (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found eq_ref00:=(eq_ref0 (b x)):(((eq Prop) (b x)) (b x))
% Found (eq_ref0 (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found x2:(P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (fun (x2:(P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))))=> x2) as proof of (P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (fun (x2:(P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))))=> x2) as proof of (P0 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found x2:(P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (fun (x2:(P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))))=> x2) as proof of (P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (fun (x2:(P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))))=> x2) as proof of (P0 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found x01:(P b)
% Found (fun (x01:(P b))=> x01) as proof of (P b)
% Found (fun (x01:(P b))=> x01) as proof of (P0 b)
% Found x01:(P b)
% Found (fun (x01:(P b))=> x01) as proof of (P b)
% Found (fun (x01:(P b))=> x01) as proof of (P0 b)
% Found x01:(P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P0 (b x))
% Found x01:(P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P0 (b x))
% Found x01:(P1 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (fun (x01:(P1 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))=> x01) as proof of (P1 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (fun (x01:(P1 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))=> x01) as proof of (P2 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found x01:(P1 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (fun (x01:(P1 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))=> x01) as proof of (P1 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (fun (x01:(P1 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))=> x01) as proof of (P2 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found x01:(P1 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (fun (x01:(P1 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))=> x01) as proof of (P1 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (fun (x01:(P1 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))=> x01) as proof of (P2 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found x01:(P1 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (fun (x01:(P1 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))=> x01) as proof of (P1 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (fun (x01:(P1 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))=> x01) as proof of (P2 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found x01:(P1 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (fun (x01:(P1 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))=> x01) as proof of (P1 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (fun (x01:(P1 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))=> x01) as proof of (P2 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found x01:(P1 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (fun (x01:(P1 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))=> x01) as proof of (P1 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (fun (x01:(P1 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))=> x01) as proof of (P2 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found x01:(P1 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (fun (x01:(P1 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))=> x01) as proof of (P1 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (fun (x01:(P1 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))=> x01) as proof of (P2 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found x01:(P1 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (fun (x01:(P1 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))=> x01) as proof of (P1 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (fun (x01:(P1 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))=> x01) as proof of (P2 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (fofType->Prop)) b0) (fun (x:fofType)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found eta_expansion000:=(eta_expansion00 ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))):(((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) (fun (x:fofType)=> (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))
% Found (eta_expansion00 ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) as proof of (((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) b0)
% Found ((eta_expansion0 Prop) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) as proof of (((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) b0)
% Found (((eta_expansion fofType) Prop) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) as proof of (((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) b0)
% Found (((eta_expansion fofType) Prop) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) as proof of (((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) b0)
% Found (((eta_expansion fofType) Prop) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) as proof of (((eq (fofType->Prop)) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))) b0)
% Found x02:(P (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (fun (x02:(P (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))=> x02) as proof of (P (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (fun (x02:(P (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))=> x02) as proof of (P0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found x02:(P (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (fun (x02:(P (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))=> x02) as proof of (P (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (fun (x02:(P (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))=> x02) as proof of (P0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found x02:(P (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (fun (x02:(P (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))=> x02) as proof of (P (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (fun (x02:(P (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))=> x02) as proof of (P0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found x02:(P (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (fun (x02:(P (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))=> x02) as proof of (P (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (fun (x02:(P (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))=> x02) as proof of (P0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found eq_ref00:=(eq_ref0 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))):(((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (eq_ref0 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found ((eq_ref (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found ((eq_ref (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found ((eq_ref (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found eta_expansion000:=(eta_expansion00 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))):(((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) (fun (x:fofType)=> (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))
% Found (eta_expansion00 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found ((eta_expansion0 Prop) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found (((eta_expansion fofType) Prop) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found (((eta_expansion fofType) Prop) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found (((eta_expansion fofType) Prop) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found eq_ref00:=(eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)):(((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found x0:(P0 b)
% Instantiate: b:=((and (((union X) Y) x)) (((intersection ((union Y) Z)) ((union Z) X)) x)):Prop
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)):(((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found x0:(P0 b)
% Instantiate: b:=((and (((union X) Y) x)) (((intersection ((union Y) Z)) ((union Z) X)) x)):Prop
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found x0:(P (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Instantiate: b:=((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))):(fofType->Prop)
% Found x0 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))):(((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) (fun (x:fofType)=> (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))
% Found (eta_expansion_dep00 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found x0:(P (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Instantiate: b:=((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))):(fofType->Prop)
% Found x0 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))):(((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) (fun (x:fofType)=> (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))
% Found (eta_expansion_dep00 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found x0:(P (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Instantiate: b:=(((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found x0:(P (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Instantiate: b:=(((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)):(((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)):(((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))):(((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) (fun (x:fofType)=> (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))
% Found (eta_expansion_dep00 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found x0:(P0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Instantiate: b:=((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))):(fofType->Prop)
% Found (fun (x0:(P0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))=> x0) as proof of (P0 (b x))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))=> x0) as proof of ((P0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))->(P0 (b x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))=> x0) as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))):(((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) (fun (x:fofType)=> (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))
% Found (eta_expansion_dep00 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) as proof of (((eq (fofType->Prop)) ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))) b)
% Found x0:(P0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Instantiate: b:=((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))):(fofType->Prop)
% Found (fun (x0:(P0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))=> x0) as proof of (P0 (b x))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))=> x0) as proof of ((P0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))->(P0 (b x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))=> x0) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)):(((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)):(((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found x2:(P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (fun (x2:(P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))))=> x2) as proof of (P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (fun (x2:(P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))))=> x2) as proof of (P0 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found x2:(P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (fun (x2:(P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))))=> x2) as proof of (P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (fun (x2:(P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))))=> x2) as proof of (P0 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found x2:(P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (fun (x2:(P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))))=> x2) as proof of (P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (fun (x2:(P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))))=> x2) as proof of (P0 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found x2:(P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (fun (x2:(P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))))=> x2) as proof of (P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found (fun (x2:(P ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X)))))=> x2) as proof of (P0 ((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))))
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found x01:(P (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (fun (x01:(P (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))=> x01) as proof of (P (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (fun (x01:(P (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))=> x01) as proof of (P0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found x01:(P (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (fun (x01:(P (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))=> x01) as proof of (P (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (fun (x01:(P (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))=> x01) as proof of (P0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found x01:(P (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (fun (x01:(P (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))=> x01) as proof of (P (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (fun (x01:(P (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))=> x01) as proof of (P0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found x01:(P (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (fun (x01:(P (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))=> x01) as proof of (P (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (fun (x01:(P (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)))=> x01) as proof of (P0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found x2:(P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found (fun (x2:(P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))))=> x2) as proof of (P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found (fun (x2:(P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))))=> x2) as proof of (P0 ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found x2:(P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found (fun (x2:(P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))))=> x2) as proof of (P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found (fun (x2:(P ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X)))))=> x2) as proof of (P0 ((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))))
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)):(((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (eq_ref0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found ((eq_ref Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) as proof of (((eq Prop) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 (b x)):(((eq Prop) (b x)) (b x))
% Found (eq_ref0 (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found eq_ref00:=(eq_ref0 (b x)):(((eq Prop) (b x)) (b x))
% Found (eq_ref0 (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found x01:(P b)
% Found (fun (x01:(P b))=> x01) as proof of (P b)
% Found (fun (x01:(P b))=> x01) as proof of (P0 b)
% Found x01:(P (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (fun (x01:(P (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))=> x01) as proof of (P (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found (fun (x01:(P (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)))=> x01) as proof of (P0 (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)):(((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)):(((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)):(((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)):(((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)):(((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)):(((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)):(((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)):(((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)):(((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)):(((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)):(((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)):(((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found eq_ref00:=(eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)):(((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x))
% Found (eq_ref0 (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found ((eq_ref Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) as proof of (((eq Prop) (((union ((intersection X) Y)) ((union ((intersection Y) Z)) ((intersection Z) X))) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((union X) Y)) ((intersection ((union Y) Z)) ((union Z) X))) x)
% EOF
%------------------------------------------------------------------------------