%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SET609^3 : TPTP v6.1.0. Released v3.6.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n100.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 13:30:49 EDT 2014

% Result   : Timeout 300.01s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SET609^3 : TPTP v6.1.0. Released v3.6.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n100.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 10:18:31 CDT 2014
% % CPUTime  : 300.01 
% 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 0x1e5a638>, <kernel.DependentProduct object at 0x1e5a518>) 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 0x1e5a638>, <kernel.DependentProduct object at 0x1e5aab8>) 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 0x1e5aab8>, <kernel.DependentProduct object at 0x1e5ae60>) 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 0x1e5ae60>, <kernel.DependentProduct object at 0x1e5ad40>) 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 0x1e39cb0>, <kernel.DependentProduct object at 0x1e5ae60>) 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 0x1e5ae60>, <kernel.DependentProduct object at 0x1e5a098>) 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 0x20b1b48>, <kernel.DependentProduct object at 0x1e5ae60>) 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 0x20b1200>, <kernel.DependentProduct object at 0x1e5a200>) 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 0x1e5a200>, <kernel.DependentProduct object at 0x1e5ae60>) 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 0x1e5ae60>, <kernel.DependentProduct object at 0x1e5a170>) 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 0x1e5a170>, <kernel.DependentProduct object at 0x1e5af80>) 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 0x1e3c8c0>, <kernel.DependentProduct object at 0x1e5af80>) 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 0x1e3c560>, <kernel.DependentProduct object at 0x1e5af38>) 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 0x1e3c560>, <kernel.DependentProduct object at 0x1e5afc8>) 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)) ((setminus X) ((setminus Y) Z))) ((union ((setminus X) Y)) ((intersection X) Z)))) of role conjecture named thm
% Conjecture to prove = (forall (X:(fofType->Prop)) (Y:(fofType->Prop)) (Z:(fofType->Prop)), (((eq (fofType->Prop)) ((setminus X) ((setminus Y) Z))) ((union ((setminus X) Y)) ((intersection X) Z)))):Prop
% Parameter fofType_DUMMY:fofType.
% We need to prove ['(forall (X:(fofType->Prop)) (Y:(fofType->Prop)) (Z:(fofType->Prop)), (((eq (fofType->Prop)) ((setminus X) ((setminus Y) Z))) ((union ((setminus X) Y)) ((intersection X) Z))))']
% 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)) ((setminus X) ((setminus Y) Z))) ((union ((setminus X) Y)) ((intersection X) Z))))
% Found eq_ref00:=(eq_ref0 ((setminus X) ((setminus Y) Z))):(((eq (fofType->Prop)) ((setminus X) ((setminus Y) Z))) ((setminus X) ((setminus Y) Z)))
% Found (eq_ref0 ((setminus X) ((setminus Y) Z))) as proof of (((eq (fofType->Prop)) ((setminus X) ((setminus Y) Z))) b)
% Found ((eq_ref (fofType->Prop)) ((setminus X) ((setminus Y) Z))) as proof of (((eq (fofType->Prop)) ((setminus X) ((setminus Y) Z))) b)
% Found ((eq_ref (fofType->Prop)) ((setminus X) ((setminus Y) Z))) as proof of (((eq (fofType->Prop)) ((setminus X) ((setminus Y) Z))) b)
% Found ((eq_ref (fofType->Prop)) ((setminus X) ((setminus Y) Z))) as proof of (((eq (fofType->Prop)) ((setminus X) ((setminus Y) Z))) 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 ((setminus X) Y)) ((intersection X) Z)))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) ((union ((setminus X) Y)) ((intersection X) Z)))
% Found x2:(P ((setminus X) ((setminus Y) Z)))
% Found (fun (x2:(P ((setminus X) ((setminus Y) Z))))=> x2) as proof of (P ((setminus X) ((setminus Y) Z)))
% Found (fun (x2:(P ((setminus X) ((setminus Y) Z))))=> x2) as proof of (P0 ((setminus X) ((setminus Y) Z)))
% Found x2:(P ((setminus X) ((setminus Y) Z)))
% Found (fun (x2:(P ((setminus X) ((setminus Y) Z))))=> x2) as proof of (P ((setminus X) ((setminus Y) Z)))
% Found (fun (x2:(P ((setminus X) ((setminus Y) Z))))=> x2) as proof of (P0 ((setminus X) ((setminus Y) Z)))
% Found x2:(P ((setminus X) ((setminus Y) Z)))
% Found (fun (x2:(P ((setminus X) ((setminus Y) Z))))=> x2) as proof of (P ((setminus X) ((setminus Y) Z)))
% Found (fun (x2:(P ((setminus X) ((setminus Y) Z))))=> x2) as proof of (P0 ((setminus X) ((setminus Y) Z)))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) ((setminus X) ((setminus Y) Z)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((setminus X) ((setminus Y) Z)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((setminus X) ((setminus Y) Z)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((setminus X) ((setminus Y) Z)))
% Found eq_ref00:=(eq_ref0 ((union ((setminus X) Y)) ((intersection X) Z))):(((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (eq_ref0 ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found ((eq_ref (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found ((eq_ref (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found ((eq_ref (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found x2:(P ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (fun (x2:(P ((union ((setminus X) Y)) ((intersection X) Z))))=> x2) as proof of (P ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (fun (x2:(P ((union ((setminus X) Y)) ((intersection X) Z))))=> x2) as proof of (P0 ((union ((setminus X) Y)) ((intersection X) Z)))
% Found x2:(P ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (fun (x2:(P ((union ((setminus X) Y)) ((intersection X) Z))))=> x2) as proof of (P ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (fun (x2:(P ((union ((setminus X) Y)) ((intersection X) Z))))=> x2) as proof of (P0 ((union ((setminus X) Y)) ((intersection X) Z)))
% Found x01:(P (((setminus X) ((setminus Y) Z)) x))
% Found (fun (x01:(P (((setminus X) ((setminus Y) Z)) x)))=> x01) as proof of (P (((setminus X) ((setminus Y) Z)) x))
% Found (fun (x01:(P (((setminus X) ((setminus Y) Z)) x)))=> x01) as proof of (P0 (((setminus X) ((setminus Y) Z)) x))
% Found x01:(P (((setminus X) ((setminus Y) Z)) x))
% Found (fun (x01:(P (((setminus X) ((setminus Y) Z)) x)))=> x01) as proof of (P (((setminus X) ((setminus Y) Z)) x))
% Found (fun (x01:(P (((setminus X) ((setminus Y) Z)) x)))=> x01) as proof of (P0 (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found x01:(P (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P0 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found x01:(P (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P0 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found x:(P ((setminus X) ((setminus Y) Z)))
% Instantiate: b:=((setminus X) ((setminus Y) Z)):(fofType->Prop)
% Found x as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((union ((setminus X) Y)) ((intersection X) Z))):(((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (eq_ref0 ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found ((eq_ref (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found ((eq_ref (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found ((eq_ref (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found x:(P ((setminus X) ((setminus Y) Z)))
% Instantiate: f:=((setminus X) ((setminus Y) Z)):(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 ((setminus X) Y)) ((intersection X) Z)) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((union ((setminus X) Y)) ((intersection X) Z)) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((union ((setminus X) Y)) ((intersection X) Z)) x0))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((union ((setminus X) Y)) ((intersection X) Z)) x0))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (((union ((setminus X) Y)) ((intersection X) Z)) x)))
% Found x:(P ((setminus X) ((setminus Y) Z)))
% Instantiate: f:=((setminus X) ((setminus Y) Z)):(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 ((setminus X) Y)) ((intersection X) Z)) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((union ((setminus X) Y)) ((intersection X) Z)) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((union ((setminus X) Y)) ((intersection X) Z)) x0))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((union ((setminus X) Y)) ((intersection X) Z)) x0))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (((union ((setminus X) Y)) ((intersection X) Z)) x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((setminus X) ((setminus Y) Z))):(((eq (fofType->Prop)) ((setminus X) ((setminus Y) Z))) (fun (x:fofType)=> (((setminus X) ((setminus Y) Z)) x)))
% Found (eta_expansion_dep00 ((setminus X) ((setminus Y) Z))) as proof of (((eq (fofType->Prop)) ((setminus X) ((setminus Y) Z))) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) ((setminus X) ((setminus Y) Z))) as proof of (((eq (fofType->Prop)) ((setminus X) ((setminus Y) Z))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((setminus X) ((setminus Y) Z))) as proof of (((eq (fofType->Prop)) ((setminus X) ((setminus Y) Z))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((setminus X) ((setminus Y) Z))) as proof of (((eq (fofType->Prop)) ((setminus X) ((setminus Y) Z))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((setminus X) ((setminus Y) Z))) as proof of (((eq (fofType->Prop)) ((setminus X) ((setminus Y) Z))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) ((union ((setminus X) Y)) ((intersection X) Z)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union ((setminus X) Y)) ((intersection X) Z)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union ((setminus X) Y)) ((intersection X) Z)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union ((setminus X) Y)) ((intersection X) Z)))
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 ((union ((setminus X) Y)) ((intersection X) Z))):(((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (eq_ref0 ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found ((eq_ref (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found ((eq_ref (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found ((eq_ref (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found x:(P0 b)
% Instantiate: b:=(fun (U:fofType)=> ((and (X U)) ((((setminus Y) Z) U)->False))):(fofType->Prop)
% Found (fun (x:(P0 b))=> x) as proof of (P0 ((setminus X) ((setminus Y) Z)))
% Found (fun (P0:((fofType->Prop)->Prop)) (x:(P0 b))=> x) as proof of ((P0 b)->(P0 ((setminus X) ((setminus Y) Z))))
% Found (fun (P0:((fofType->Prop)->Prop)) (x:(P0 b))=> x) as proof of (P b)
% Found x3:(P ((setminus X) ((setminus Y) Z)))
% Found (fun (x3:(P ((setminus X) ((setminus Y) Z))))=> x3) as proof of (P ((setminus X) ((setminus Y) Z)))
% Found (fun (x3:(P ((setminus X) ((setminus Y) Z))))=> x3) as proof of (P0 ((setminus X) ((setminus Y) Z)))
% Found eta_expansion000:=(eta_expansion00 ((setminus X) ((setminus Y) Z))):(((eq (fofType->Prop)) ((setminus X) ((setminus Y) Z))) (fun (x:fofType)=> (((setminus X) ((setminus Y) Z)) x)))
% Found (eta_expansion00 ((setminus X) ((setminus Y) Z))) as proof of (((eq (fofType->Prop)) ((setminus X) ((setminus Y) Z))) b)
% Found ((eta_expansion0 Prop) ((setminus X) ((setminus Y) Z))) as proof of (((eq (fofType->Prop)) ((setminus X) ((setminus Y) Z))) b)
% Found (((eta_expansion fofType) Prop) ((setminus X) ((setminus Y) Z))) as proof of (((eq (fofType->Prop)) ((setminus X) ((setminus Y) Z))) b)
% Found (((eta_expansion fofType) Prop) ((setminus X) ((setminus Y) Z))) as proof of (((eq (fofType->Prop)) ((setminus X) ((setminus Y) Z))) b)
% Found (((eta_expansion fofType) Prop) ((setminus X) ((setminus Y) Z))) as proof of (((eq (fofType->Prop)) ((setminus X) ((setminus Y) Z))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) ((union ((setminus X) Y)) ((intersection X) Z)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union ((setminus X) Y)) ((intersection X) Z)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union ((setminus X) Y)) ((intersection X) Z)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union ((setminus X) Y)) ((intersection X) Z)))
% Found x:(P ((union ((setminus X) Y)) ((intersection X) Z)))
% Instantiate: b:=((union ((setminus X) Y)) ((intersection X) Z)):(fofType->Prop)
% Found x as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((setminus X) ((setminus Y) Z))):(((eq (fofType->Prop)) ((setminus X) ((setminus Y) Z))) ((setminus X) ((setminus Y) Z)))
% Found (eq_ref0 ((setminus X) ((setminus Y) Z))) as proof of (((eq (fofType->Prop)) ((setminus X) ((setminus Y) Z))) b)
% Found ((eq_ref (fofType->Prop)) ((setminus X) ((setminus Y) Z))) as proof of (((eq (fofType->Prop)) ((setminus X) ((setminus Y) Z))) b)
% Found ((eq_ref (fofType->Prop)) ((setminus X) ((setminus Y) Z))) as proof of (((eq (fofType->Prop)) ((setminus X) ((setminus Y) Z))) b)
% Found ((eq_ref (fofType->Prop)) ((setminus X) ((setminus Y) Z))) as proof of (((eq (fofType->Prop)) ((setminus X) ((setminus Y) Z))) b)
% Found eta_expansion000:=(eta_expansion00 ((setminus X) ((setminus Y) Z))):(((eq (fofType->Prop)) ((setminus X) ((setminus Y) Z))) (fun (x:fofType)=> (((setminus X) ((setminus Y) Z)) x)))
% Found (eta_expansion00 ((setminus X) ((setminus Y) Z))) as proof of (((eq (fofType->Prop)) ((setminus X) ((setminus Y) Z))) b0)
% Found ((eta_expansion0 Prop) ((setminus X) ((setminus Y) Z))) as proof of (((eq (fofType->Prop)) ((setminus X) ((setminus Y) Z))) b0)
% Found (((eta_expansion fofType) Prop) ((setminus X) ((setminus Y) Z))) as proof of (((eq (fofType->Prop)) ((setminus X) ((setminus Y) Z))) b0)
% Found (((eta_expansion fofType) Prop) ((setminus X) ((setminus Y) Z))) as proof of (((eq (fofType->Prop)) ((setminus X) ((setminus Y) Z))) b0)
% Found (((eta_expansion fofType) Prop) ((setminus X) ((setminus Y) Z))) as proof of (((eq (fofType->Prop)) ((setminus X) ((setminus Y) Z))) 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) ((union ((setminus X) Y)) ((intersection X) Z)))
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) ((union ((setminus X) Y)) ((intersection X) Z)))
% Found x:(P ((union ((setminus X) Y)) ((intersection X) Z)))
% Instantiate: f:=((union ((setminus X) Y)) ((intersection X) Z)):(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)) (((setminus X) ((setminus Y) Z)) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((setminus X) ((setminus Y) Z)) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((setminus X) ((setminus Y) Z)) x0))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((setminus X) ((setminus Y) Z)) x0))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (((setminus X) ((setminus Y) Z)) x)))
% Found x:(P ((union ((setminus X) Y)) ((intersection X) Z)))
% Instantiate: f:=((union ((setminus X) Y)) ((intersection X) Z)):(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)) (((setminus X) ((setminus Y) Z)) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((setminus X) ((setminus Y) Z)) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((setminus X) ((setminus Y) Z)) x0))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((setminus X) ((setminus Y) Z)) x0))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (((setminus X) ((setminus Y) Z)) x)))
% Found x0:(P (((setminus X) ((setminus Y) Z)) x))
% Instantiate: b:=(((setminus X) ((setminus Y) Z)) x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found x0:(P (((setminus X) ((setminus Y) Z)) x))
% Instantiate: b:=(((setminus X) ((setminus Y) Z)) x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) 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) ((setminus X) ((setminus Y) Z)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((setminus X) ((setminus Y) Z)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((setminus X) ((setminus Y) Z)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((setminus X) ((setminus Y) Z)))
% Found eq_ref00:=(eq_ref0 ((union ((setminus X) Y)) ((intersection X) Z))):(((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (eq_ref0 ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found ((eq_ref (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found ((eq_ref (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found ((eq_ref (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) ((setminus X) ((setminus Y) Z)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((setminus X) ((setminus Y) Z)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((setminus X) ((setminus Y) Z)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((setminus X) ((setminus Y) Z)))
% Found eq_ref00:=(eq_ref0 ((union ((setminus X) Y)) ((intersection X) Z))):(((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (eq_ref0 ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found ((eq_ref (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found ((eq_ref (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found ((eq_ref (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) ((setminus X) ((setminus Y) Z)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((setminus X) ((setminus Y) Z)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((setminus X) ((setminus Y) Z)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((setminus X) ((setminus Y) Z)))
% Found eq_ref00:=(eq_ref0 ((union ((setminus X) Y)) ((intersection X) Z))):(((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (eq_ref0 ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found ((eq_ref (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found ((eq_ref (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found ((eq_ref (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) 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 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) 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 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found x2:(P1 ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (fun (x2:(P1 ((union ((setminus X) Y)) ((intersection X) Z))))=> x2) as proof of (P1 ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (fun (x2:(P1 ((union ((setminus X) Y)) ((intersection X) Z))))=> x2) as proof of (P2 ((union ((setminus X) Y)) ((intersection X) Z)))
% Found x2:(P1 ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (fun (x2:(P1 ((union ((setminus X) Y)) ((intersection X) Z))))=> x2) as proof of (P1 ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (fun (x2:(P1 ((union ((setminus X) Y)) ((intersection X) Z))))=> x2) as proof of (P2 ((union ((setminus X) Y)) ((intersection X) Z)))
% Found x2:(P1 ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (fun (x2:(P1 ((union ((setminus X) Y)) ((intersection X) Z))))=> x2) as proof of (P1 ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (fun (x2:(P1 ((union ((setminus X) Y)) ((intersection X) Z))))=> x2) as proof of (P2 ((union ((setminus X) Y)) ((intersection X) Z)))
% Found x2:(P1 ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (fun (x2:(P1 ((union ((setminus X) Y)) ((intersection X) Z))))=> x2) as proof of (P1 ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (fun (x2:(P1 ((union ((setminus X) Y)) ((intersection X) Z))))=> x2) as proof of (P2 ((union ((setminus X) Y)) ((intersection X) Z)))
% Found x3:(P ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (fun (x3:(P ((union ((setminus X) Y)) ((intersection X) Z))))=> x3) as proof of (P ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (fun (x3:(P ((union ((setminus X) Y)) ((intersection X) Z))))=> x3) as proof of (P0 ((union ((setminus X) Y)) ((intersection X) Z)))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) ((setminus X) ((setminus Y) Z)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((setminus X) ((setminus Y) Z)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((setminus X) ((setminus Y) Z)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((setminus X) ((setminus Y) Z)))
% Found eq_ref00:=(eq_ref0 ((union ((setminus X) Y)) ((intersection X) Z))):(((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (eq_ref0 ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found ((eq_ref (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found ((eq_ref (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found ((eq_ref (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found x01:(P1 (((setminus X) ((setminus Y) Z)) x))
% Found (fun (x01:(P1 (((setminus X) ((setminus Y) Z)) x)))=> x01) as proof of (P1 (((setminus X) ((setminus Y) Z)) x))
% Found (fun (x01:(P1 (((setminus X) ((setminus Y) Z)) x)))=> x01) as proof of (P2 (((setminus X) ((setminus Y) Z)) x))
% Found x01:(P1 (((setminus X) ((setminus Y) Z)) x))
% Found (fun (x01:(P1 (((setminus X) ((setminus Y) Z)) x)))=> x01) as proof of (P1 (((setminus X) ((setminus Y) Z)) x))
% Found (fun (x01:(P1 (((setminus X) ((setminus Y) Z)) x)))=> x01) as proof of (P2 (((setminus X) ((setminus Y) Z)) x))
% Found x01:(P1 (((setminus X) ((setminus Y) Z)) x))
% Found (fun (x01:(P1 (((setminus X) ((setminus Y) Z)) x)))=> x01) as proof of (P1 (((setminus X) ((setminus Y) Z)) x))
% Found (fun (x01:(P1 (((setminus X) ((setminus Y) Z)) x)))=> x01) as proof of (P2 (((setminus X) ((setminus Y) Z)) x))
% Found x01:(P1 (((setminus X) ((setminus Y) Z)) x))
% Found (fun (x01:(P1 (((setminus X) ((setminus Y) Z)) x)))=> x01) as proof of (P1 (((setminus X) ((setminus Y) Z)) x))
% Found (fun (x01:(P1 (((setminus X) ((setminus Y) Z)) x)))=> x01) as proof of (P2 (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 b0):(((eq (fofType->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found eq_ref00:=(eq_ref0 ((union ((setminus X) Y)) ((intersection X) Z))):(((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (eq_ref0 ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b0)
% Found ((eq_ref (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b0)
% Found ((eq_ref (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b0)
% Found ((eq_ref (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b0)
% Found x02:(P (((setminus X) ((setminus Y) Z)) x))
% Found (fun (x02:(P (((setminus X) ((setminus Y) Z)) x)))=> x02) as proof of (P (((setminus X) ((setminus Y) Z)) x))
% Found (fun (x02:(P (((setminus X) ((setminus Y) Z)) x)))=> x02) as proof of (P0 (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found x02:(P (((setminus X) ((setminus Y) Z)) x))
% Found (fun (x02:(P (((setminus X) ((setminus Y) Z)) x)))=> x02) as proof of (P (((setminus X) ((setminus Y) Z)) x))
% Found (fun (x02:(P (((setminus X) ((setminus Y) Z)) x)))=> x02) as proof of (P0 (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found x0:(P0 b)
% Instantiate: b:=((and (X x)) ((((setminus Y) Z) x)->False)):Prop
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 (((setminus X) ((setminus Y) Z)) x))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 (((setminus X) ((setminus Y) Z)) x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found x0:(P0 b)
% Instantiate: b:=((and (X x)) ((((setminus Y) Z) x)->False)):Prop
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 (((setminus X) ((setminus Y) Z)) x))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 (((setminus X) ((setminus Y) Z)) x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found eta_expansion000:=(eta_expansion00 ((union ((setminus X) Y)) ((intersection X) Z))):(((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) (fun (x:fofType)=> (((union ((setminus X) Y)) ((intersection X) Z)) x)))
% Found (eta_expansion00 ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found ((eta_expansion0 Prop) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found (((eta_expansion fofType) Prop) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found (((eta_expansion fofType) Prop) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found (((eta_expansion fofType) Prop) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found eq_ref00:=(eq_ref0 ((union ((setminus X) Y)) ((intersection X) Z))):(((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (eq_ref0 ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found ((eq_ref (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found ((eq_ref (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found ((eq_ref (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found eta_expansion000:=(eta_expansion00 ((union ((setminus X) Y)) ((intersection X) Z))):(((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) (fun (x:fofType)=> (((union ((setminus X) Y)) ((intersection X) Z)) x)))
% Found (eta_expansion00 ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b0)
% Found ((eta_expansion0 Prop) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b0)
% Found (((eta_expansion fofType) Prop) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b0)
% Found (((eta_expansion fofType) Prop) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b0)
% Found (((eta_expansion fofType) Prop) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (fofType->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found x0:(P (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Instantiate: b:=(((union ((setminus X) Y)) ((intersection X) Z)) x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found x0:(P (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Instantiate: b:=(((union ((setminus X) Y)) ((intersection X) Z)) x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b0)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b0)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b0)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b0)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b0)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b0)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found eq_ref00:=(eq_ref0 ((setminus X) ((setminus Y) Z))):(((eq (fofType->Prop)) ((setminus X) ((setminus Y) Z))) ((setminus X) ((setminus Y) Z)))
% Found (eq_ref0 ((setminus X) ((setminus Y) Z))) as proof of (((eq (fofType->Prop)) ((setminus X) ((setminus Y) Z))) b)
% Found ((eq_ref (fofType->Prop)) ((setminus X) ((setminus Y) Z))) as proof of (((eq (fofType->Prop)) ((setminus X) ((setminus Y) Z))) b)
% Found ((eq_ref (fofType->Prop)) ((setminus X) ((setminus Y) Z))) as proof of (((eq (fofType->Prop)) ((setminus X) ((setminus Y) Z))) b)
% Found ((eq_ref (fofType->Prop)) ((setminus X) ((setminus Y) Z))) as proof of (((eq (fofType->Prop)) ((setminus X) ((setminus Y) Z))) 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 ((setminus X) Y)) ((intersection X) Z)))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) ((union ((setminus X) Y)) ((intersection X) Z)))
% Found x2:(P ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (fun (x2:(P ((union ((setminus X) Y)) ((intersection X) Z))))=> x2) as proof of (P ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (fun (x2:(P ((union ((setminus X) Y)) ((intersection X) Z))))=> x2) as proof of (P0 ((union ((setminus X) Y)) ((intersection X) Z)))
% Found x0:(P (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Instantiate: b:=(((union ((setminus X) Y)) ((intersection X) Z)) x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found x0:(P (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Instantiate: b:=(((union ((setminus X) Y)) ((intersection X) Z)) x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found x2:(P ((setminus X) ((setminus Y) Z)))
% Found (fun (x2:(P ((setminus X) ((setminus Y) Z))))=> x2) as proof of (P ((setminus X) ((setminus Y) Z)))
% Found (fun (x2:(P ((setminus X) ((setminus Y) Z))))=> x2) as proof of (P0 ((setminus X) ((setminus Y) Z)))
% Found x2:(P ((setminus X) ((setminus Y) Z)))
% Found (fun (x2:(P ((setminus X) ((setminus Y) Z))))=> x2) as proof of (P ((setminus X) ((setminus Y) Z)))
% Found (fun (x2:(P ((setminus X) ((setminus Y) Z))))=> x2) as proof of (P0 ((setminus X) ((setminus Y) Z)))
% Found x2:(P ((setminus X) ((setminus Y) Z)))
% Found (fun (x2:(P ((setminus X) ((setminus Y) Z))))=> x2) as proof of (P ((setminus X) ((setminus Y) Z)))
% Found (fun (x2:(P ((setminus X) ((setminus Y) Z))))=> x2) as proof of (P0 ((setminus X) ((setminus Y) Z)))
% 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 ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((union ((setminus X) Y)) ((intersection X) 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) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found x2:(P ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (fun (x2:(P ((union ((setminus X) Y)) ((intersection X) Z))))=> x2) as proof of (P ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (fun (x2:(P ((union ((setminus X) Y)) ((intersection X) Z))))=> x2) as proof of (P0 ((union ((setminus X) Y)) ((intersection X) Z)))
% Found x2:(P ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (fun (x2:(P ((union ((setminus X) Y)) ((intersection X) Z))))=> x2) as proof of (P ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (fun (x2:(P ((union ((setminus X) Y)) ((intersection X) Z))))=> x2) as proof of (P0 ((union ((setminus X) Y)) ((intersection X) Z)))
% Found x2:(P ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (fun (x2:(P ((union ((setminus X) Y)) ((intersection X) Z))))=> x2) as proof of (P ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (fun (x2:(P ((union ((setminus X) Y)) ((intersection X) Z))))=> x2) as proof of (P0 ((union ((setminus X) Y)) ((intersection X) Z)))
% Found x2:(P ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (fun (x2:(P ((union ((setminus X) Y)) ((intersection X) Z))))=> x2) as proof of (P ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (fun (x2:(P ((union ((setminus X) Y)) ((intersection X) Z))))=> x2) as proof of (P0 ((union ((setminus X) Y)) ((intersection X) Z)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) 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 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P1 (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P1 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P1 (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P2 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found x01:(P1 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P1 (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P1 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P1 (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P2 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found x01:(P1 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P1 (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P1 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P1 (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P2 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found x01:(P1 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P1 (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P1 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P1 (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P2 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found x01:(P1 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P1 (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P1 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P1 (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P2 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found x01:(P1 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P1 (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P1 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P1 (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P2 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found x01:(P1 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P1 (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P1 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P1 (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P2 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found x01:(P1 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P1 (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P1 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P1 (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P2 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (fofType->Prop)) b0) (fun (x:fofType)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (fofType->Prop)) b0) ((union ((setminus X) Y)) ((intersection X) Z)))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (fofType->Prop)) b0) ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) ((union ((setminus X) Y)) ((intersection X) Z)))
% 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) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found x02:(P (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x02:(P (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x02) as proof of (P (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x02:(P (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x02) as proof of (P0 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found x02:(P (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x02:(P (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x02) as proof of (P (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x02:(P (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x02) as proof of (P0 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) ((setminus X) ((setminus Y) Z)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((setminus X) ((setminus Y) Z)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((setminus X) ((setminus Y) Z)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((setminus X) ((setminus Y) Z)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((union ((setminus X) Y)) ((intersection X) Z))):(((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) (fun (x:fofType)=> (((union ((setminus X) Y)) ((intersection X) Z)) x)))
% Found (eta_expansion_dep00 ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) ((setminus X) ((setminus Y) Z)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((setminus X) ((setminus Y) Z)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((setminus X) ((setminus Y) Z)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((setminus X) ((setminus Y) Z)))
% Found eta_expansion000:=(eta_expansion00 ((union ((setminus X) Y)) ((intersection X) Z))):(((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) (fun (x:fofType)=> (((union ((setminus X) Y)) ((intersection X) Z)) x)))
% Found (eta_expansion00 ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found ((eta_expansion0 Prop) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found (((eta_expansion fofType) Prop) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found (((eta_expansion fofType) Prop) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found (((eta_expansion fofType) Prop) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found x02:(P (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x02:(P (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x02) as proof of (P (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x02:(P (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x02) as proof of (P0 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found x02:(P (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x02:(P (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x02) as proof of (P (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x02:(P (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x02) as proof of (P0 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found x0:(P0 b)
% Instantiate: b:=((or (((setminus X) Y) x)) (((intersection X) Z) x)):Prop
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 (((union ((setminus X) Y)) ((intersection X) Z)) x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found x0:(P0 b)
% Instantiate: b:=((or (((setminus X) Y) x)) (((intersection X) Z) x)):Prop
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 (((union ((setminus X) Y)) ((intersection X) Z)) x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found x0:(P (((setminus X) ((setminus Y) Z)) x))
% Instantiate: b:=((setminus X) ((setminus Y) Z)):(fofType->Prop)
% Found x0 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((union ((setminus X) Y)) ((intersection X) Z))):(((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) (fun (x:fofType)=> (((union ((setminus X) Y)) ((intersection X) Z)) x)))
% Found (eta_expansion_dep00 ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found x0:(P (((setminus X) ((setminus Y) Z)) x))
% Instantiate: b:=((setminus X) ((setminus Y) Z)):(fofType->Prop)
% Found x0 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((union ((setminus X) Y)) ((intersection X) Z))):(((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) (fun (x:fofType)=> (((union ((setminus X) Y)) ((intersection X) Z)) x)))
% Found (eta_expansion_dep00 ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) 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 eta_expansion_dep000:=(eta_expansion_dep00 ((union ((setminus X) Y)) ((intersection X) Z))):(((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) (fun (x:fofType)=> (((union ((setminus X) Y)) ((intersection X) Z)) x)))
% Found (eta_expansion_dep00 ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found x0:(P0 (((setminus X) ((setminus Y) Z)) x))
% Instantiate: b:=((setminus X) ((setminus Y) Z)):(fofType->Prop)
% Found (fun (x0:(P0 (((setminus X) ((setminus Y) Z)) x)))=> x0) as proof of (P0 (b x))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 (((setminus X) ((setminus Y) Z)) x)))=> x0) as proof of ((P0 (((setminus X) ((setminus Y) Z)) x))->(P0 (b x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 (((setminus X) ((setminus Y) Z)) x)))=> x0) as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((union ((setminus X) Y)) ((intersection X) Z))):(((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) (fun (x:fofType)=> (((union ((setminus X) Y)) ((intersection X) Z)) x)))
% Found (eta_expansion_dep00 ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) as proof of (((eq (fofType->Prop)) ((union ((setminus X) Y)) ((intersection X) Z))) b)
% Found x0:(P0 (((setminus X) ((setminus Y) Z)) x))
% Instantiate: b:=((setminus X) ((setminus Y) Z)):(fofType->Prop)
% Found (fun (x0:(P0 (((setminus X) ((setminus Y) Z)) x)))=> x0) as proof of (P0 (b x))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 (((setminus X) ((setminus Y) Z)) x)))=> x0) as proof of ((P0 (((setminus X) ((setminus Y) Z)) x))->(P0 (b x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 (((setminus X) ((setminus Y) Z)) x)))=> x0) as proof of (P b)
% Found x0:(P (((setminus X) ((setminus Y) Z)) x))
% Instantiate: b:=(((setminus X) ((setminus Y) Z)) x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found x0:(P (((setminus X) ((setminus Y) Z)) x))
% Instantiate: b:=(((setminus X) ((setminus Y) Z)) x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found x2:(P ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (fun (x2:(P ((union ((setminus X) Y)) ((intersection X) Z))))=> x2) as proof of (P ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (fun (x2:(P ((union ((setminus X) Y)) ((intersection X) Z))))=> x2) as proof of (P0 ((union ((setminus X) Y)) ((intersection X) Z)))
% Found x2:(P ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (fun (x2:(P ((union ((setminus X) Y)) ((intersection X) Z))))=> x2) as proof of (P ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (fun (x2:(P ((union ((setminus X) Y)) ((intersection X) Z))))=> x2) as proof of (P0 ((union ((setminus X) Y)) ((intersection X) Z)))
% Found x2:(P ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (fun (x2:(P ((union ((setminus X) Y)) ((intersection X) Z))))=> x2) as proof of (P ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (fun (x2:(P ((union ((setminus X) Y)) ((intersection X) Z))))=> x2) as proof of (P0 ((union ((setminus X) Y)) ((intersection X) Z)))
% Found x2:(P ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (fun (x2:(P ((union ((setminus X) Y)) ((intersection X) Z))))=> x2) as proof of (P ((union ((setminus X) Y)) ((intersection X) Z)))
% Found (fun (x2:(P ((union ((setminus X) Y)) ((intersection X) Z))))=> x2) as proof of (P0 ((union ((setminus X) Y)) ((intersection X) Z)))
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) 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 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) 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 (((setminus X) ((setminus Y) Z)) x))
% Found (fun (x01:(P (((setminus X) ((setminus Y) Z)) x)))=> x01) as proof of (P (((setminus X) ((setminus Y) Z)) x))
% Found (fun (x01:(P (((setminus X) ((setminus Y) Z)) x)))=> x01) as proof of (P0 (((setminus X) ((setminus Y) Z)) x))
% Found x01:(P (((setminus X) ((setminus Y) Z)) x))
% Found (fun (x01:(P (((setminus X) ((setminus Y) Z)) x)))=> x01) as proof of (P (((setminus X) ((setminus Y) Z)) x))
% Found (fun (x01:(P (((setminus X) ((setminus Y) Z)) x)))=> x01) as proof of (P0 (((setminus X) ((setminus Y) Z)) x))
% Found x01:(P (((setminus X) ((setminus Y) Z)) x))
% Found (fun (x01:(P (((setminus X) ((setminus Y) Z)) x)))=> x01) as proof of (P (((setminus X) ((setminus Y) Z)) x))
% Found (fun (x01:(P (((setminus X) ((setminus Y) Z)) x)))=> x01) as proof of (P0 (((setminus X) ((setminus Y) Z)) x))
% Found x01:(P (((setminus X) ((setminus Y) Z)) x))
% Found (fun (x01:(P (((setminus X) ((setminus Y) Z)) x)))=> x01) as proof of (P (((setminus X) ((setminus Y) Z)) x))
% Found (fun (x01:(P (((setminus X) ((setminus Y) Z)) x)))=> x01) as proof of (P0 (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((setminus X) ((setminus Y) 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 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 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) 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 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) 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 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) 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 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) 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 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b0)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) 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 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 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P0 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) 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 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P0 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found x01:(P (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P0 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found x01:(P (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P0 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found x01:(P (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P0 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found x02:(P (((setminus X) ((setminus Y) Z)) x))
% Found (fun (x02:(P (((setminus X) ((setminus Y) Z)) x)))=> x02) as proof of (P (((setminus X) ((setminus Y) Z)) x))
% Found (fun (x02:(P (((setminus X) ((setminus Y) Z)) x)))=> x02) as proof of (P0 (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found x02:(P (((setminus X) ((setminus Y) Z)) x))
% Found (fun (x02:(P (((setminus X) ((setminus Y) Z)) x)))=> x02) as proof of (P (((setminus X) ((setminus Y) Z)) x))
% Found (fun (x02:(P (((setminus X) ((setminus Y) Z)) x)))=> x02) as proof of (P0 (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found eq_ref00:=(eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)):(((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (eq_ref0 (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found ((eq_ref Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) as proof of (((eq Prop) (((union ((setminus X) Y)) ((intersection X) Z)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus X) ((setminus Y) Z)) x))
% 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 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b0)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b0)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b0)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) 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 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b0)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b0)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b0)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b0)
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b0)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b0)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b0)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found x01:(P (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P0 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found x01:(P (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P0 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found x01:(P (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P0 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found x01:(P (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P0 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found x01:(P (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P0 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found x01:(P (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found (fun (x01:(P (((union ((setminus X) Y)) ((intersection X) Z)) x)))=> x01) as proof of (P0 (((union ((setminus X) Y)) ((intersection X) Z)) x))
% Found eq_ref00:=(eq_ref0 (((setminus X) ((setminus Y) Z)) x)):(((eq Prop) (((setminus X) ((setminus Y) Z)) x)) (((setminus X) ((setminus Y) Z)) x))
% Found (eq_ref0 (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b0)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b0)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b0)
% Found ((eq_ref Prop) (((setminus X) ((setminus Y) Z)) x)) as proof of (((eq Prop) (((setminus X) ((setminus Y) Z)) x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Pr
% EOF
%------------------------------------------------------------------------------