%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEU828^1 : TPTP v6.1.0. Released v3.7.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n092.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 13:33:15 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  : SEU828^1 : TPTP v6.1.0. Released v3.7.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n092.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 11:35:36 CDT 2014
% % CPUTime  : 300.01 
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x19ac488>, <kernel.DependentProduct object at 0x19ac320>) of role type named seteq_type
% Using role type
% Declaring seteq:(((fofType->Prop)->Prop)->(((fofType->Prop)->Prop)->Prop))
% FOF formula (((eq (((fofType->Prop)->Prop)->(((fofType->Prop)->Prop)->Prop))) seteq) (fun (X:((fofType->Prop)->Prop)) (Y:((fofType->Prop)->Prop))=> (forall (U:(fofType->Prop)), ((iff (X U)) (Y U))))) of role definition named seteq
% A new definition: (((eq (((fofType->Prop)->Prop)->(((fofType->Prop)->Prop)->Prop))) seteq) (fun (X:((fofType->Prop)->Prop)) (Y:((fofType->Prop)->Prop))=> (forall (U:(fofType->Prop)), ((iff (X U)) (Y U)))))
% Defined: seteq:=(fun (X:((fofType->Prop)->Prop)) (Y:((fofType->Prop)->Prop))=> (forall (U:(fofType->Prop)), ((iff (X U)) (Y U))))
% FOF formula (<kernel.Constant object at 0x19ac488>, <kernel.DependentProduct object at 0x19ac098>) of role type named subseteq_type
% Using role type
% Declaring subseteq:((fofType->Prop)->((fofType->Prop)->Prop))
% FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->Prop))) subseteq) (fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (forall (U:fofType), ((X U)->(Y U))))) of role definition named subseteq
% A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->Prop))) subseteq) (fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (forall (U:fofType), ((X U)->(Y U)))))
% Defined: subseteq:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (forall (U:fofType), ((X U)->(Y U))))
% FOF formula (<kernel.Constant object at 0x19ac098>, <kernel.DependentProduct object at 0x19ac8c0>) of role type named powerset_type
% Using role type
% Declaring powerset:((fofType->Prop)->((fofType->Prop)->Prop))
% FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->Prop))) powerset) (fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> ((subseteq Y) X))) of role definition named poserset
% A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->Prop))) powerset) (fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> ((subseteq Y) X)))
% Defined: powerset:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> ((subseteq Y) X))
% FOF formula (<kernel.Constant object at 0x19ac290>, <kernel.DependentProduct object at 0x1c23cb0>) of role type named emptyset_type
% 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 ((seteq (powerset emptyset)) (fun (X:(fofType->Prop))=> (((eq (fofType->Prop)) X) emptyset))) of role conjecture named conj
% Conjecture to prove = ((seteq (powerset emptyset)) (fun (X:(fofType->Prop))=> (((eq (fofType->Prop)) X) emptyset))):Prop
% Parameter fofType_DUMMY:fofType.
% We need to prove ['((seteq (powerset emptyset)) (fun (X:(fofType->Prop))=> (((eq (fofType->Prop)) X) emptyset)))']
% Parameter fofType:Type.
% Definition seteq:=(fun (X:((fofType->Prop)->Prop)) (Y:((fofType->Prop)->Prop))=> (forall (U:(fofType->Prop)), ((iff (X U)) (Y U)))):(((fofType->Prop)->Prop)->(((fofType->Prop)->Prop)->Prop)).
% Definition subseteq:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (forall (U:fofType), ((X U)->(Y U)))):((fofType->Prop)->((fofType->Prop)->Prop)).
% Definition powerset:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> ((subseteq Y) X)):((fofType->Prop)->((fofType->Prop)->Prop)).
% Definition emptyset:=(fun (X:fofType)=> False):(fofType->Prop).
% Trying to prove ((seteq (powerset emptyset)) (fun (X:(fofType->Prop))=> (((eq (fofType->Prop)) X) emptyset)))
% Found x0:=(x (fun (x0:(fofType->Prop))=> (x0 U0))):((U U0)->(emptyset U0))
% Found (x (fun (x0:(fofType->Prop))=> (x0 U0))) as proof of ((U U0)->(emptyset U0))
% Found (fun (U0:fofType)=> (x (fun (x0:(fofType->Prop))=> (x0 U0)))) as proof of ((U U0)->(emptyset U0))
% Found (fun (x:(((eq (fofType->Prop)) U) emptyset)) (U0:fofType)=> (x (fun (x0:(fofType->Prop))=> (x0 U0)))) as proof of ((powerset emptyset) U)
% Found (fun (x:(((eq (fofType->Prop)) U) emptyset)) (U0:fofType)=> (x (fun (x0:(fofType->Prop))=> (x0 U0)))) as proof of ((((eq (fofType->Prop)) U) emptyset)->((powerset emptyset) U))
% Found eta_expansion000:=(eta_expansion00 U):(((eq (fofType->Prop)) U) (fun (x:fofType)=> (U x)))
% Found (eta_expansion00 U) as proof of (((eq (fofType->Prop)) U) b)
% Found ((eta_expansion0 Prop) U) as proof of (((eq (fofType->Prop)) U) b)
% Found (((eta_expansion fofType) Prop) U) as proof of (((eq (fofType->Prop)) U) b)
% Found (((eta_expansion fofType) Prop) U) as proof of (((eq (fofType->Prop)) U) b)
% Found (((eta_expansion fofType) Prop) U) as proof of (((eq (fofType->Prop)) U) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found x00:(P U)
% Found (fun (x00:(P U))=> x00) as proof of (P U)
% Found (fun (x00:(P U))=> x00) as proof of (P0 U)
% Found x00:(P U)
% Found (fun (x00:(P U))=> x00) as proof of (P U)
% Found (fun (x00:(P U))=> x00) as proof of (P0 U)
% Found x00:(P U)
% Found (fun (x00:(P U))=> x00) as proof of (P U)
% Found (fun (x00:(P U))=> x00) as proof of (P0 U)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found eq_ref00:=(eq_ref0 emptyset):(((eq (fofType->Prop)) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found x0:=(x (fun (x0:(fofType->Prop))=> (x0 U0))):((U U0)->(emptyset U0))
% Found (x (fun (x0:(fofType->Prop))=> (x0 U0))) as proof of ((U U0)->(emptyset U0))
% Found (fun (U0:fofType)=> (x (fun (x0:(fofType->Prop))=> (x0 U0)))) as proof of ((U U0)->(emptyset U0))
% Found (fun (x:(((eq (fofType->Prop)) U) emptyset)) (U0:fofType)=> (x (fun (x0:(fofType->Prop))=> (x0 U0)))) as proof of ((powerset emptyset) U)
% Found (fun (x:(((eq (fofType->Prop)) U) emptyset)) (U0:fofType)=> (x (fun (x0:(fofType->Prop))=> (x0 U0)))) as proof of ((((eq (fofType->Prop)) U) emptyset)->((powerset emptyset) U))
% Found x00:(P emptyset)
% Found (fun (x00:(P emptyset))=> x00) as proof of (P emptyset)
% Found (fun (x00:(P emptyset))=> x00) as proof of (P0 emptyset)
% Found x00:(P emptyset)
% Found (fun (x00:(P emptyset))=> x00) as proof of (P emptyset)
% Found (fun (x00:(P emptyset))=> x00) as proof of (P0 emptyset)
% Found x10:(P (U x0))
% Found (fun (x10:(P (U x0)))=> x10) as proof of (P (U x0))
% Found (fun (x10:(P (U x0)))=> x10) as proof of (P0 (U x0))
% Found x10:(P (U x0))
% Found (fun (x10:(P (U x0)))=> x10) as proof of (P (U x0))
% Found (fun (x10:(P (U x0)))=> x10) as proof of (P0 (U x0))
% Found eq_ref00:=(eq_ref0 ((((eq (fofType->Prop)) U) emptyset)->((powerset emptyset) U))):(((eq Prop) ((((eq (fofType->Prop)) U) emptyset)->((powerset emptyset) U))) ((((eq (fofType->Prop)) U) emptyset)->((powerset emptyset) U)))
% Found (eq_ref0 ((((eq (fofType->Prop)) U) emptyset)->((powerset emptyset) U))) as proof of (((eq Prop) ((((eq (fofType->Prop)) U) emptyset)->((powerset emptyset) U))) b)
% Found ((eq_ref Prop) ((((eq (fofType->Prop)) U) emptyset)->((powerset emptyset) U))) as proof of (((eq Prop) ((((eq (fofType->Prop)) U) emptyset)->((powerset emptyset) U))) b)
% Found ((eq_ref Prop) ((((eq (fofType->Prop)) U) emptyset)->((powerset emptyset) U))) as proof of (((eq Prop) ((((eq (fofType->Prop)) U) emptyset)->((powerset emptyset) U))) b)
% Found ((eq_ref Prop) ((((eq (fofType->Prop)) U) emptyset)->((powerset emptyset) U))) as proof of (((eq Prop) ((((eq (fofType->Prop)) U) emptyset)->((powerset emptyset) U))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found x10:(P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P0 (emptyset x0))
% Found x10:(P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P0 (emptyset x0))
% Found eq_ref00:=(eq_ref0 U):(((eq (fofType->Prop)) U) U)
% Found (eq_ref0 U) as proof of (((eq (fofType->Prop)) U) b)
% Found ((eq_ref (fofType->Prop)) U) as proof of (((eq (fofType->Prop)) U) b)
% Found ((eq_ref (fofType->Prop)) U) as proof of (((eq (fofType->Prop)) U) b)
% Found ((eq_ref (fofType->Prop)) U) as proof of (((eq (fofType->Prop)) U) 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) emptyset)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found x0:(P U)
% Instantiate: b:=U:(fofType->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 emptyset):(((eq (fofType->Prop)) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found x00:(P U)
% Found (fun (x00:(P U))=> x00) as proof of (P U)
% Found (fun (x00:(P U))=> x00) as proof of (P0 U)
% Found x0:(P U)
% Instantiate: f:=U:(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (emptyset x)))
% Found x0:(P U)
% Instantiate: f:=U:(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (emptyset x)))
% Found x00:(P U)
% Found (fun (x00:(P U))=> x00) as proof of (P U)
% Found (fun (x00:(P U))=> x00) as proof of (P0 U)
% Found x00:(P U)
% Found (fun (x00:(P U))=> x00) as proof of (P U)
% Found (fun (x00:(P U))=> x00) as proof of (P0 U)
% Found eta_expansion000:=(eta_expansion00 U):(((eq (fofType->Prop)) U) (fun (x:fofType)=> (U x)))
% Found (eta_expansion00 U) as proof of (((eq (fofType->Prop)) U) b)
% Found ((eta_expansion0 Prop) U) as proof of (((eq (fofType->Prop)) U) b)
% Found (((eta_expansion fofType) Prop) U) as proof of (((eq (fofType->Prop)) U) b)
% Found (((eta_expansion fofType) Prop) U) as proof of (((eq (fofType->Prop)) U) b)
% Found (((eta_expansion fofType) Prop) U) as proof of (((eq (fofType->Prop)) U) 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) emptyset)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found x00:(P U)
% Found (fun (x00:(P U))=> x00) as proof of (P U)
% Found (fun (x00:(P U))=> x00) as proof of (P0 U)
% Found eta_expansion000:=(eta_expansion00 U):(((eq (fofType->Prop)) U) (fun (x:fofType)=> (U x)))
% Found (eta_expansion00 U) as proof of (((eq (fofType->Prop)) U) b)
% Found ((eta_expansion0 Prop) U) as proof of (((eq (fofType->Prop)) U) b)
% Found (((eta_expansion fofType) Prop) U) as proof of (((eq (fofType->Prop)) U) b)
% Found (((eta_expansion fofType) Prop) U) as proof of (((eq (fofType->Prop)) U) b)
% Found (((eta_expansion fofType) Prop) U) as proof of (((eq (fofType->Prop)) U) 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) emptyset)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found eta_expansion000:=(eta_expansion00 emptyset):(((eq (fofType->Prop)) emptyset) (fun (x:fofType)=> (emptyset x)))
% Found (eta_expansion00 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eta_expansion0 Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found x0:(P0 b)
% Instantiate: b:=U:(fofType->Prop)
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 U)
% Found (fun (P0:((fofType->Prop)->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 U))
% Found (fun (P0:((fofType->Prop)->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found x0:(P emptyset)
% Instantiate: b:=emptyset:(fofType->Prop)
% Found x0 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 U):(((eq (fofType->Prop)) U) (fun (x:fofType)=> (U x)))
% Found (eta_expansion_dep00 U) as proof of (((eq (fofType->Prop)) U) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) U) as proof of (((eq (fofType->Prop)) U) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) U) as proof of (((eq (fofType->Prop)) U) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) U) as proof of (((eq (fofType->Prop)) U) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) U) as proof of (((eq (fofType->Prop)) U) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found eta_expansion_dep000:=(eta_expansion_dep00 emptyset):(((eq (fofType->Prop)) emptyset) (fun (x:fofType)=> (emptyset x)))
% Found (eta_expansion_dep00 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% 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) emptyset)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (fofType->Prop)) b0) emptyset)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) emptyset)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) emptyset)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) emptyset)
% Found x0:(P emptyset)
% Instantiate: f:=emptyset:(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (U x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (U x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (U x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (U x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (U x)))
% Found x0:(P emptyset)
% Instantiate: f:=emptyset:(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (U x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (U x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (U x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (U x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (U x)))
% Found x0:=(x (fun (x0:(fofType->Prop))=> (x0 U0))):((U U0)->(emptyset U0))
% Found (x (fun (x0:(fofType->Prop))=> (x0 U0))) as proof of ((U U0)->(emptyset U0))
% Found (fun (U0:fofType)=> (x (fun (x0:(fofType->Prop))=> (x0 U0)))) as proof of ((U U0)->(emptyset U0))
% Found (fun (x:(((eq (fofType->Prop)) U) emptyset)) (U0:fofType)=> (x (fun (x0:(fofType->Prop))=> (x0 U0)))) as proof of ((powerset emptyset) U)
% Found (fun (x:(((eq (fofType->Prop)) U) emptyset)) (U0:fofType)=> (x (fun (x0:(fofType->Prop))=> (x0 U0)))) as proof of ((((eq (fofType->Prop)) U) emptyset)->((powerset emptyset) U))
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found x1:(P (U x0))
% Instantiate: b:=(U x0):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found x1:(P (U x0))
% Instantiate: b:=(U x0):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found eq_ref00:=(eq_ref0 emptyset):(((eq (fofType->Prop)) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found x00:(P emptyset)
% Found (fun (x00:(P emptyset))=> x00) as proof of (P emptyset)
% Found (fun (x00:(P emptyset))=> x00) as proof of (P0 emptyset)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found eq_ref00:=(eq_ref0 emptyset):(((eq (fofType->Prop)) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found eq_ref00:=(eq_ref0 emptyset):(((eq (fofType->Prop)) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found x00:(P emptyset)
% Found (fun (x00:(P emptyset))=> x00) as proof of (P emptyset)
% Found (fun (x00:(P emptyset))=> x00) as proof of (P0 emptyset)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x10:(P (U x0))
% Found (fun (x10:(P (U x0)))=> x10) as proof of (P (U x0))
% Found (fun (x10:(P (U x0)))=> x10) as proof of (P0 (U x0))
% Found x10:(P (U x0))
% Found (fun (x10:(P (U x0)))=> x10) as proof of (P (U x0))
% Found (fun (x10:(P (U x0)))=> x10) as proof of (P0 (U x0))
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found x00:(P1 emptyset)
% Found (fun (x00:(P1 emptyset))=> x00) as proof of (P1 emptyset)
% Found (fun (x00:(P1 emptyset))=> x00) as proof of (P2 emptyset)
% Found x00:(P1 emptyset)
% Found (fun (x00:(P1 emptyset))=> x00) as proof of (P1 emptyset)
% Found (fun (x00:(P1 emptyset))=> x00) as proof of (P2 emptyset)
% Found x00:(P1 emptyset)
% Found (fun (x00:(P1 emptyset))=> x00) as proof of (P1 emptyset)
% Found (fun (x00:(P1 emptyset))=> x00) as proof of (P2 emptyset)
% Found x00:(P1 emptyset)
% Found (fun (x00:(P1 emptyset))=> x00) as proof of (P1 emptyset)
% Found (fun (x00:(P1 emptyset))=> x00) as proof of (P2 emptyset)
% Found x00:(P emptyset)
% Found (fun (x00:(P emptyset))=> x00) as proof of (P emptyset)
% Found (fun (x00:(P emptyset))=> x00) as proof of (P0 emptyset)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found eta_expansion000:=(eta_expansion00 emptyset):(((eq (fofType->Prop)) emptyset) (fun (x:fofType)=> (emptyset x)))
% Found (eta_expansion00 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eta_expansion0 Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found x10:(P1 (U x0))
% Found (fun (x10:(P1 (U x0)))=> x10) as proof of (P1 (U x0))
% Found (fun (x10:(P1 (U x0)))=> x10) as proof of (P2 (U x0))
% Found x10:(P1 (U x0))
% Found (fun (x10:(P1 (U x0)))=> x10) as proof of (P1 (U x0))
% Found (fun (x10:(P1 (U x0)))=> x10) as proof of (P2 (U x0))
% Found x10:(P1 (U x0))
% Found (fun (x10:(P1 (U x0)))=> x10) as proof of (P1 (U x0))
% Found (fun (x10:(P1 (U x0)))=> x10) as proof of (P2 (U x0))
% Found x10:(P1 (U x0))
% Found (fun (x10:(P1 (U x0)))=> x10) as proof of (P1 (U x0))
% Found (fun (x10:(P1 (U x0)))=> x10) as proof of (P2 (U x0))
% Found eq_ref00:=(eq_ref0 (((powerset emptyset) U)->(((eq (fofType->Prop)) U) emptyset))):(((eq Prop) (((powerset emptyset) U)->(((eq (fofType->Prop)) U) emptyset))) (((powerset emptyset) U)->(((eq (fofType->Prop)) U) emptyset)))
% Found (eq_ref0 (((powerset emptyset) U)->(((eq (fofType->Prop)) U) emptyset))) as proof of (((eq Prop) (((powerset emptyset) U)->(((eq (fofType->Prop)) U) emptyset))) b)
% Found ((eq_ref Prop) (((powerset emptyset) U)->(((eq (fofType->Prop)) U) emptyset))) as proof of (((eq Prop) (((powerset emptyset) U)->(((eq (fofType->Prop)) U) emptyset))) b)
% Found ((eq_ref Prop) (((powerset emptyset) U)->(((eq (fofType->Prop)) U) emptyset))) as proof of (((eq Prop) (((powerset emptyset) U)->(((eq (fofType->Prop)) U) emptyset))) b)
% Found ((eq_ref Prop) (((powerset emptyset) U)->(((eq (fofType->Prop)) U) emptyset))) as proof of (((eq Prop) (((powerset emptyset) U)->(((eq (fofType->Prop)) U) emptyset))) b)
% 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 emptyset):(((eq (fofType->Prop)) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b0)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b0)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b0)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b0)
% Found x10:(P (U x0))
% Found (fun (x10:(P (U x0)))=> x10) as proof of (P (U x0))
% Found (fun (x10:(P (U x0)))=> x10) as proof of (P0 (U x0))
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found x10:(P (U x0))
% Found (fun (x10:(P (U x0)))=> x10) as proof of (P (U x0))
% Found (fun (x10:(P (U x0)))=> x10) as proof of (P0 (U x0))
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found x1:(P0 b)
% Instantiate: b:=(U x0):Prop
% Found (fun (x1:(P0 b))=> x1) as proof of (P0 (U x0))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of ((P0 b)->(P0 (U x0)))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found x1:(P0 b)
% Instantiate: b:=(U x0):Prop
% Found (fun (x1:(P0 b))=> x1) as proof of (P0 (U x0))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of ((P0 b)->(P0 (U x0)))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of (P b)
% Found eta_expansion000:=(eta_expansion00 U):(((eq (fofType->Prop)) U) (fun (x:fofType)=> (U x)))
% Found (eta_expansion00 U) as proof of (((eq (fofType->Prop)) U) b)
% Found ((eta_expansion0 Prop) U) as proof of (((eq (fofType->Prop)) U) b)
% Found (((eta_expansion fofType) Prop) U) as proof of (((eq (fofType->Prop)) U) b)
% Found (((eta_expansion fofType) Prop) U) as proof of (((eq (fofType->Prop)) U) b)
% Found (((eta_expansion fofType) Prop) U) as proof of (((eq (fofType->Prop)) U) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found eta_expansion_dep000:=(eta_expansion_dep00 emptyset):(((eq (fofType->Prop)) emptyset) (fun (x:fofType)=> (emptyset x)))
% Found (eta_expansion_dep00 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found x1:(P0 (b x0))
% Instantiate: b:=U:(fofType->Prop)
% Found (fun (x1:(P0 (b x0)))=> x1) as proof of (P0 (U x0))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 (b x0)))=> x1) as proof of ((P0 (b x0))->(P0 (U x0)))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 (b x0)))=> x1) as proof of (P b)
% Found eta_expansion000:=(eta_expansion00 emptyset):(((eq (fofType->Prop)) emptyset) (fun (x:fofType)=> (emptyset x)))
% Found (eta_expansion00 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eta_expansion0 Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found x1:(P0 (b x0))
% Instantiate: b:=U:(fofType->Prop)
% Found (fun (x1:(P0 (b x0)))=> x1) as proof of (P0 (U x0))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 (b x0)))=> x1) as proof of ((P0 (b x0))->(P0 (U x0)))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 (b x0)))=> x1) as proof of (P b)
% Found eq_ref00:=(eq_ref0 emptyset):(((eq (fofType->Prop)) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b0)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b0)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b0)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (fofType->Prop)) b0) (fun (x:fofType)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found x1:(P (emptyset x0))
% Instantiate: b:=(emptyset x0):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found x1:(P (emptyset x0))
% Instantiate: b:=(emptyset x0):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found x00:(P U)
% Found (fun (x00:(P U))=> x00) as proof of (P U)
% Found (fun (x00:(P U))=> x00) as proof of (P0 U)
% Found x00:(P emptyset)
% Found (fun (x00:(P emptyset))=> x00) as proof of (P emptyset)
% Found (fun (x00:(P emptyset))=> x00) as proof of (P0 emptyset)
% Found x00:(P U)
% Found (fun (x00:(P U))=> x00) as proof of (P U)
% Found (fun (x00:(P U))=> x00) as proof of (P0 U)
% Found x00:(P U)
% Found (fun (x00:(P U))=> x00) as proof of (P U)
% Found (fun (x00:(P U))=> x00) as proof of (P0 U)
% Found x1:(P (emptyset x0))
% Instantiate: b:=(emptyset x0):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found x1:(P (emptyset x0))
% Instantiate: b:=(emptyset x0):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 U):(((eq (fofType->Prop)) U) U)
% Found (eq_ref0 U) as proof of (((eq (fofType->Prop)) U) b)
% Found ((eq_ref (fofType->Prop)) U) as proof of (((eq (fofType->Prop)) U) b)
% Found ((eq_ref (fofType->Prop)) U) as proof of (((eq (fofType->Prop)) U) b)
% Found ((eq_ref (fofType->Prop)) U) as proof of (((eq (fofType->Prop)) U) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found x00:(P U)
% Found (fun (x00:(P U))=> x00) as proof of (P U)
% Found (fun (x00:(P U))=> x00) as proof of (P0 U)
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found eta_expansion_dep000:=(eta_expansion_dep00 U):(((eq (fofType->Prop)) U) (fun (x:fofType)=> (U x)))
% Found (eta_expansion_dep00 U) as proof of (((eq (fofType->Prop)) U) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) U) as proof of (((eq (fofType->Prop)) U) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) U) as proof of (((eq (fofType->Prop)) U) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) U) as proof of (((eq (fofType->Prop)) U) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b)
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b0)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b0)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b0)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b0)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b0)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b0)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found x0:(P U)
% Instantiate: b:=U:(fofType->Prop)
% Found x0 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 emptyset):(((eq (fofType->Prop)) emptyset) (fun (x:fofType)=> (emptyset x)))
% Found (eta_expansion_dep00 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found x00:(P U)
% Found (fun (x00:(P U))=> x00) as proof of (P U)
% Found (fun (x00:(P U))=> x00) as proof of (P0 U)
% Found x00:(P U)
% Found (fun (x00:(P U))=> x00) as proof of (P U)
% Found (fun (x00:(P U))=> x00) as proof of (P0 U)
% Found x00:(P emptyset)
% Found (fun (x00:(P emptyset))=> x00) as proof of (P emptyset)
% Found (fun (x00:(P emptyset))=> x00) as proof of (P0 emptyset)
% Found x00:(P emptyset)
% Found (fun (x00:(P emptyset))=> x00) as proof of (P emptyset)
% Found (fun (x00:(P emptyset))=> x00) as proof of (P0 emptyset)
% Found x00:(P emptyset)
% Found (fun (x00:(P emptyset))=> x00) as proof of (P emptyset)
% Found (fun (x00:(P emptyset))=> x00) as proof of (P0 emptyset)
% Found x00:(P emptyset)
% Found (fun (x00:(P emptyset))=> x00) as proof of (P emptyset)
% Found (fun (x00:(P emptyset))=> x00) as proof of (P0 emptyset)
% Found x10:(P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P0 (emptyset x0))
% Found x10:(P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P0 (emptyset x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x10:(P (b x0))
% Found (fun (x10:(P (b x0)))=> x10) as proof of (P (b x0))
% Found (fun (x10:(P (b x0)))=> x10) as proof of (P0 (b x0))
% Found x10:(P (b x0))
% Found (fun (x10:(P (b x0)))=> x10) as proof of (P (b x0))
% Found (fun (x10:(P (b x0)))=> x10) as proof of (P0 (b x0))
% Found x0:(P U)
% Instantiate: f:=U:(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (emptyset x)))
% Found x0:(P U)
% Instantiate: f:=U:(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (emptyset x)))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found eq_ref00:=(eq_ref0 emptyset):(((eq (fofType->Prop)) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found eq_ref00:=(eq_ref0 emptyset):(((eq (fofType->Prop)) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found eq_ref00:=(eq_ref0 U):(((eq (fofType->Prop)) U) U)
% Found (eq_ref0 U) as proof of (((eq (fofType->Prop)) U) b)
% Found ((eq_ref (fofType->Prop)) U) as proof of (((eq (fofType->Prop)) U) b)
% Found ((eq_ref (fofType->Prop)) U) as proof of (((eq (fofType->Prop)) U) b)
% Found ((eq_ref (fofType->Prop)) U) as proof of (((eq (fofType->Prop)) U) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found x10:(P1 (emptyset x0))
% Found (fun (x10:(P1 (emptyset x0)))=> x10) as proof of (P1 (emptyset x0))
% Found (fun (x10:(P1 (emptyset x0)))=> x10) as proof of (P2 (emptyset x0))
% Found x10:(P1 (emptyset x0))
% Found (fun (x10:(P1 (emptyset x0)))=> x10) as proof of (P1 (emptyset x0))
% Found (fun (x10:(P1 (emptyset x0)))=> x10) as proof of (P2 (emptyset x0))
% Found x10:(P1 (emptyset x0))
% Found (fun (x10:(P1 (emptyset x0)))=> x10) as proof of (P1 (emptyset x0))
% Found (fun (x10:(P1 (emptyset x0)))=> x10) as proof of (P2 (emptyset x0))
% Found x10:(P1 (emptyset x0))
% Found (fun (x10:(P1 (emptyset x0)))=> x10) as proof of (P1 (emptyset x0))
% Found (fun (x10:(P1 (emptyset x0)))=> x10) as proof of (P2 (emptyset x0))
% Found x10:(P1 (emptyset x0))
% Found (fun (x10:(P1 (emptyset x0)))=> x10) as proof of (P1 (emptyset x0))
% Found (fun (x10:(P1 (emptyset x0)))=> x10) as proof of (P2 (emptyset x0))
% Found x10:(P1 (emptyset x0))
% Found (fun (x10:(P1 (emptyset x0)))=> x10) as proof of (P1 (emptyset x0))
% Found (fun (x10:(P1 (emptyset x0)))=> x10) as proof of (P2 (emptyset x0))
% Found x10:(P1 (emptyset x0))
% Found (fun (x10:(P1 (emptyset x0)))=> x10) as proof of (P1 (emptyset x0))
% Found (fun (x10:(P1 (emptyset x0)))=> x10) as proof of (P2 (emptyset x0))
% Found x10:(P1 (emptyset x0))
% Found (fun (x10:(P1 (emptyset x0)))=> x10) as proof of (P1 (emptyset x0))
% Found (fun (x10:(P1 (emptyset x0)))=> x10) as proof of (P2 (emptyset x0))
% 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) emptyset)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) emptyset)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) emptyset)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) emptyset)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) emptyset)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found x10:(P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P0 (emptyset x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found x10:(P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P0 (emptyset x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found x10:(P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P0 (emptyset x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found x10:(P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P0 (emptyset x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found x1:(P0 b)
% Instantiate: b:=False:Prop
% Found (fun (x1:(P0 b))=> x1) as proof of (P0 (emptyset x0))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of ((P0 b)->(P0 (emptyset x0)))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found x1:(P0 b)
% Instantiate: b:=False:Prop
% Found (fun (x1:(P0 b))=> x1) as proof of (P0 (emptyset x0))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of ((P0 b)->(P0 (emptyset x0)))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found eq_ref00:=(eq_ref0 emptyset):(((eq (fofType->Prop)) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found x00:(P emptyset)
% Found (fun (x00:(P emptyset))=> x00) as proof of (P emptyset)
% Found (fun (x00:(P emptyset))=> x00) as proof of (P0 emptyset)
% Found x00:(P emptyset)
% Found (fun (x00:(P emptyset))=> x00) as proof of (P emptyset)
% Found (fun (x00:(P emptyset))=> x00) as proof of (P0 emptyset)
% Found x00:(P emptyset)
% Found (fun (x00:(P emptyset))=> x00) as proof of (P emptyset)
% Found (fun (x00:(P emptyset))=> x00) as proof of (P0 emptyset)
% Found x00:(P emptyset)
% Found (fun (x00:(P emptyset))=> x00) as proof of (P emptyset)
% Found (fun (x00:(P emptyset))=> x00) as proof of (P0 emptyset)
% Found x1:(P (U x0))
% Instantiate: b:=(U x0):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found x1:(P (U x0))
% Instantiate: b:=(U x0):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found x00:(P U)
% Found (fun (x00:(P U))=> x00) as proof of (P U)
% Found (fun (x00:(P U))=> x00) as proof of (P0 U)
% Found eta_expansion_dep000:=(eta_expansion_dep00 U):(((eq (fofType->Prop)) U) (fun (x:fofType)=> (U x)))
% Found (eta_expansion_dep00 U) as proof of (((eq (fofType->Prop)) U) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) U) as proof of (((eq (fofType->Prop)) U) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) U) as proof of (((eq (fofType->Prop)) U) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) U) as proof of (((eq (fofType->Prop)) U) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) U) as proof of (((eq (fofType->Prop)) U) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found x10:(P (U x0))
% Found (fun (x10:(P (U x0)))=> x10) as proof of (P (U x0))
% Found (fun (x10:(P (U x0)))=> x10) as proof of (P0 (U x0))
% Found x10:(P (U x0))
% Found (fun (x10:(P (U x0)))=> x10) as proof of (P (U x0))
% Found (fun (x10:(P (U x0)))=> x10) as proof of (P0 (U x0))
% Found eta_expansion_dep000:=(eta_expansion_dep00 emptyset):(((eq (fofType->Prop)) emptyset) (fun (x:fofType)=> (emptyset x)))
% Found (eta_expansion_dep00 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found x0:(P0 b)
% Instantiate: b:=U:(fofType->Prop)
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 U)
% Found (fun (P0:((fofType->Prop)->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 U))
% Found (fun (P0:((fofType->Prop)->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found x10:(P (U x0))
% Found (fun (x10:(P (U x0)))=> x10) as proof of (P (U x0))
% Found (fun (x10:(P (U x0)))=> x10) as proof of (P0 (U x0))
% Found x10:(P (U x0))
% Found (fun (x10:(P (U x0)))=> x10) as proof of (P (U x0))
% Found (fun (x10:(P (U x0)))=> x10) as proof of (P0 (U x0))
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) 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 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) 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 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found x0:(P emptyset)
% Instantiate: b:=emptyset:(fofType->Prop)
% Found x0 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 U):(((eq (fofType->Prop)) U) (fun (x:fofType)=> (U x)))
% Found (eta_expansion_dep00 U) as proof of (((eq (fofType->Prop)) U) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) U) as proof of (((eq (fofType->Prop)) U) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) U) as proof of (((eq (fofType->Prop)) U) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) U) as proof of (((eq (fofType->Prop)) U) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) U) as proof of (((eq (fofType->Prop)) U) b)
% Found x1:(P (U x0))
% Instantiate: b:=U:(fofType->Prop)
% Found x1 as proof of (P0 b)
% Found eta_expansion000:=(eta_expansion00 emptyset):(((eq (fofType->Prop)) emptyset) (fun (x:fofType)=> (emptyset x)))
% Found (eta_expansion00 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eta_expansion0 Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found x1:(P (U x0))
% Instantiate: b:=U:(fofType->Prop)
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 emptyset):(((eq (fofType->Prop)) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found x00:(P emptyset)
% Found (fun (x00:(P emptyset))=> x00) as proof of (P emptyset)
% Found (fun (x00:(P emptyset))=> x00) as proof of (P0 emptyset)
% Found x00:(P emptyset)
% Found (fun (x00:(P emptyset))=> x00) as proof of (P emptyset)
% Found (fun (x00:(P emptyset))=> x00) as proof of (P0 emptyset)
% Found eta_expansion000:=(eta_expansion00 emptyset):(((eq (fofType->Prop)) emptyset) (fun (x:fofType)=> (emptyset x)))
% Found (eta_expansion00 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eta_expansion0 Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found x1:(P0 (U x0))
% Instantiate: b:=U:(fofType->Prop)
% Found (fun (x1:(P0 (U x0)))=> x1) as proof of (P0 (b x0))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 (U x0)))=> x1) as proof of ((P0 (U x0))->(P0 (b x0)))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 (U x0)))=> x1) as proof of (P b)
% Found eq_ref00:=(eq_ref0 emptyset):(((eq (fofType->Prop)) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found x1:(P0 (U x0))
% Instantiate: b:=U:(fofType->Prop)
% Found (fun (x1:(P0 (U x0)))=> x1) as proof of (P0 (b x0))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 (U x0)))=> x1) as proof of ((P0 (U x0))->(P0 (b x0)))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 (U x0)))=> x1) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) 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 x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found x10:(P (U x0))
% Found (fun (x10:(P (U x0)))=> x10) as proof of (P (U x0))
% Found (fun (x10:(P (U x0)))=> x10) as proof of (P0 (U x0))
% Found x10:(P (U x0))
% Found (fun (x10:(P (U x0)))=> x10) as proof of (P (U x0))
% Found (fun (x10:(P (U x0)))=> x10) as proof of (P0 (U x0))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (U x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (U x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (U x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (U x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (U x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (U x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (U x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (U x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (U x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (U x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (U x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (U x0))
% Found x10:(P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P0 (emptyset x0))
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x0:(P emptyset)
% Instantiate: f:=emptyset:(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P emptyset)
% Instantiate: f:=emptyset:(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (U x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (U x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (U x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (U x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (U x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (U x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (U x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (U x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (U x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (U x)))
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found x1:(P (U x0))
% Instantiate: b:=(U x0):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found x1:(P (U x0))
% Instantiate: b:=(U x0):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 U):(((eq (fofType->Prop)) U) U)
% Found (eq_ref0 U) as proof of (((eq (fofType->Prop)) U) b0)
% Found ((eq_ref (fofType->Prop)) U) as proof of (((eq (fofType->Prop)) U) b0)
% Found ((eq_ref (fofType->Prop)) U) as proof of (((eq (fofType->Prop)) U) b0)
% Found ((eq_ref (fofType->Prop)) U) as proof of (((eq (fofType->Prop)) U) 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) emptyset)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) emptyset)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) emptyset)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) emptyset)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) emptyset)
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x10:(P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P0 (emptyset x0))
% Found x10:(P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P0 (emptyset x0))
% Found x10:(P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P0 (emptyset x0))
% Found x10:(P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P0 (emptyset x0))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found eta_expansion_dep000:=(eta_expansion_dep00 emptyset):(((eq (fofType->Prop)) emptyset) (fun (x:fofType)=> (emptyset x)))
% Found (eta_expansion_dep00 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found eta_expansion_dep000:=(eta_expansion_dep00 emptyset):(((eq (fofType->Prop)) emptyset) (fun (x:fofType)=> (emptyset x)))
% Found (eta_expansion_dep00 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found eta_expansion_dep000:=(eta_expansion_dep00 emptyset):(((eq (fofType->Prop)) emptyset) (fun (x:fofType)=> (emptyset x)))
% Found (eta_expansion_dep00 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((((eq (fofType->Prop)) U) emptyset)->((powerset emptyset) U))):(((eq Prop) ((((eq (fofType->Prop)) U) emptyset)->((powerset emptyset) U))) ((((eq (fofType->Prop)) U) emptyset)->((powerset emptyset) U)))
% Found (eq_ref0 ((((eq (fofType->Prop)) U) emptyset)->((powerset emptyset) U))) as proof of (((eq Prop) ((((eq (fofType->Prop)) U) emptyset)->((powerset emptyset) U))) b)
% Found ((eq_ref Prop) ((((eq (fofType->Prop)) U) emptyset)->((powerset emptyset) U))) as proof of (((eq Prop) ((((eq (fofType->Prop)) U) emptyset)->((powerset emptyset) U))) b)
% Found ((eq_ref Prop) ((((eq (fofType->Prop)) U) emptyset)->((powerset emptyset) U))) as proof of (((eq Prop) ((((eq (fofType->Prop)) U) emptyset)->((powerset emptyset) U))) b)
% Found ((eq_ref Prop) ((((eq (fofType->Prop)) U) emptyset)->((powerset emptyset) U))) as proof of (((eq Prop) ((((eq (fofType->Prop)) U) emptyset)->((powerset emptyset) U))) b)
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found x10:(P (U x0))
% Found (fun (x10:(P (U x0)))=> x10) as proof of (P (U x0))
% Found (fun (x10:(P (U x0)))=> x10) as proof of (P0 (U x0))
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found x10:(P (U x0))
% Found (fun (x10:(P (U x0)))=> x10) as proof of (P (U x0))
% Found (fun (x10:(P (U x0)))=> x10) as proof of (P0 (U x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found x00:(P emptyset)
% Found (fun (x00:(P emptyset))=> x00) as proof of (P emptyset)
% Found (fun (x00:(P emptyset))=> x00) as proof of (P0 emptyset)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) U)
% Found eta_expansion_dep000:=(eta_expansion_dep00 emptyset):(((eq (fofType->Prop)) emptyset) (fun (x:fofType)=> (emptyset x)))
% Found (eta_expansion_dep00 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found x00:(P1 emptyset)
% Found (fun (x00:(P1 emptyset))=> x00) as proof of (P1 emptyset)
% Found (fun (x00:(P1 emptyset))=> x00) as proof of (P2 emptyset)
% Found x00:(P1 emptyset)
% Found (fun (x00:(P1 emptyset))=> x00) as proof of (P1 emptyset)
% Found (fun (x00:(P1 emptyset))=> x00) as proof of (P2 emptyset)
% Found x00:(P1 emptyset)
% Found (fun (x00:(P1 emptyset))=> x00) as proof of (P1 emptyset)
% Found (fun (x00:(P1 emptyset))=> x00) as proof of (P2 emptyset)
% Found x00:(P1 emptyset)
% Found (fun (x00:(P1 emptyset))=> x00) as proof of (P1 emptyset)
% Found (fun (x00:(P1 emptyset))=> x00) as proof of (P2 emptyset)
% Found x10:(P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P0 (emptyset x0))
% Found x10:(P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P0 (emptyset x0))
% Found x10:(P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P0 (emptyset x0))
% Found x10:(P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P0 (emptyset x0))
% Found x10:(P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P0 (emptyset x0))
% Found x10:(P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P0 (emptyset x0))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x0))
% 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 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b0)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b0)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b0)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b0)
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (U x0))
% 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 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b0)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b0)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b0)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b0)
% Found eta_expansion000:=(eta_expansion00 U):(((eq (fofType->Prop)) U) (fun (x:fofType)=> (U x)))
% Found (eta_expansion00 U) as proof of (((eq (fofType->Prop)) U) b)
% Found ((eta_expansion0 Prop) U) as proof of (((eq (fofType->Prop)) U) b)
% Found (((eta_expansion fofType) Prop) U) as proof of (((eq (fofType->Prop)) U) b)
% Found (((eta_expansion fofType) Prop) U) as proof of (((eq (fofType->Prop)) U) b)
% Found (((eta_expansion fofType) Prop) U) as proof of (((eq (fofType->Prop)) U) 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) emptyset)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found x10:(P1 (U x0))
% Found (fun (x10:(P1 (U x0)))=> x10) as proof of (P1 (U x0))
% Found (fun (x10:(P1 (U x0)))=> x10) as proof of (P2 (U x0))
% Found x10:(P1 (U x0))
% Found (fun (x10:(P1 (U x0)))=> x10) as proof of (P1 (U x0))
% Found (fun (x10:(P1 (U x0)))=> x10) as proof of (P2 (U x0))
% Found x10:(P1 (U x0))
% Found (fun (x10:(P1 (U x0)))=> x10) as proof of (P1 (U x0))
% Found (fun (x10:(P1 (U x0)))=> x10) as proof of (P2 (U x0))
% Found x10:(P1 (U x0))
% Found (fun (x10:(P1 (U x0)))=> x10) as proof of (P1 (U x0))
% Found (fun (x10:(P1 (U x0)))=> x10) as proof of (P2 (U x0))
% 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 emptyset):(((eq (fofType->Prop)) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b0)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b0)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b0)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b0)
% Found x0:(P U)
% Instantiate: b:=U:(fofType->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 emptyset):(((eq (fofType->Prop)) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found x10:(P (U x0))
% Found (fun (x10:(P (U x0)))=> x10) as proof of (P (U x0))
% Found (fun (x10:(P (U x0)))=> x10) as proof of (P0 (U x0))
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found x10:(P (U x0))
% Found (fun (x10:(P (U x0)))=> x10) as proof of (P (U x0))
% Found (fun (x10:(P (U x0)))=> x10) as proof of (P0 (U x0))
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found x00:(P U)
% Found (fun (x00:(P U))=> x00) as proof of (P U)
% Found (fun (x00:(P U))=> x00) as proof of (P0 U)
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found x1:(P0 b)
% Instantiate: b:=(U x0):Prop
% Found (fun (x1:(P0 b))=> x1) as proof of (P0 (U x0))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of ((P0 b)->(P0 (U x0)))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found x1:(P0 b)
% Instantiate: b:=(U x0):Prop
% Found (fun (x1:(P0 b))=> x1) as proof of (P0 (U x0))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of ((P0 b)->(P0 (U x0)))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b0)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b0)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b0)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) 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 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b0)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b0)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b0)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) 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 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found eta_expansion_dep000:=(eta_expansion_dep00 emptyset):(((eq (fofType->Prop)) emptyset) (fun (x:fofType)=> (emptyset x)))
% Found (eta_expansion_dep00 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found x1:(P0 (b x0))
% Instantiate: b:=U:(fofType->Prop)
% Found (fun (x1:(P0 (b x0)))=> x1) as proof of (P0 (U x0))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 (b x0)))=> x1) as proof of ((P0 (b x0))->(P0 (U x0)))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 (b x0)))=> x1) as proof of (P b)
% Found eta_expansion000:=(eta_expansion00 emptyset):(((eq (fofType->Prop)) emptyset) (fun (x:fofType)=> (emptyset x)))
% Found (eta_expansion00 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eta_expansion0 Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found x1:(P0 (b x0))
% Instantiate: b:=U:(fofType->Prop)
% Found (fun (x1:(P0 (b x0)))=> x1) as proof of (P0 (U x0))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 (b x0)))=> x1) as proof of ((P0 (b x0))->(P0 (U x0)))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 (b x0)))=> x1) as proof of (P b)
% Found eta_expansion000:=(eta_expansion00 emptyset):(((eq (fofType->Prop)) emptyset) (fun (x:fofType)=> (emptyset x)))
% Found (eta_expansion00 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b0)
% Found ((eta_expansion0 Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b0)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b0)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b0)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) 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 U)
% Instantiate: f:=U:(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (emptyset x)))
% Found x10:(P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P0 (emptyset x0))
% Found x10:(P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P (emptyset x0))
% Found (fun (x10:(P (emptyset x0)))=> x10) as proof of (P0 (emptyset x0))
% Found x0:(P U)
% Instantiate: f:=U:(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (emptyset x)))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((((eq (fofType->Prop)) U) emptyset)->((powerset emptyset) U)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((((eq (fofType->Prop)) U) emptyset)->((powerset emptyset) U)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((((eq (fofType->Prop)) U) emptyset)->((powerset emptyset) U)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((((eq (fofType->Prop)) U) emptyset)->((powerset emptyset) U)))
% Found x00:(P U)
% Found (fun (x00:(P U))=> x00) as proof of (P U)
% Found (fun (x00:(P U))=> x00) as proof of (P0 U)
% Found x00:(P U)
% Found (fun (x00:(P U))=> x00) as proof of (P U)
% Found (fun (x00:(P U))=> x00) as proof of (P0 U)
% Found x1:(P (emptyset x0))
% Instantiate: b:=(emptyset x0):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found x1:(P (emptyset x0))
% Instantiate: b:=(emptyset x0):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (U x0)):(((eq Prop) (U x0)) (U x0))
% Found (eq_ref0 (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found ((eq_ref Prop) (U x0)) as proof of (((eq Prop) (U x0)) b)
% Found x0:(P U)
% Instantiate: b:=U:(fofType->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 emptyset):(((eq (fofType->Prop)) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found eta_expansion000:=(eta_expansion00 U):(((eq (fofType->Prop)) U) (fun (x:fofType)=> (U x)))
% Found (eta_expansion00 U) as proof of (((eq (fofType->Prop)) U) b)
% Found ((eta_expansion0 Prop) U) as proof of (((eq (fofType->Prop)) U) b)
% Found (((eta_expansion fofType) Prop) U) as proof of (((eq (fofType->Prop)) U) b)
% Found (((eta_expansion fofType) Prop) U) as proof of (((eq (fofType->Prop)) U) b)
% Found (((eta_expansion fofType) Prop) U) as proof of (((eq (fofType->Prop)) U) 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) emptyset)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType
% EOF
%------------------------------------------------------------------------------