%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEU526^2 : TPTP v6.1.0. Released v3.7.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n100.star.cs.uiowa.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory   : 32286.75MB
% OS       : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:32:22 EDT 2014

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

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEU526^2 : TPTP v6.1.0. Released v3.7.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n100.star.cs.uiowa.edu
% % Model    : x86_64 x86_64
% % CPU      : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory   : 32286.75MB
% % OS       : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 10:31:06 CDT 2014
% % CPUTime  : 300.00 
% 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 0x1dfed40>, <kernel.DependentProduct object at 0x2238e60>) of role type named in_type
% Using role type
% Declaring in:(fofType->(fofType->Prop))
% FOF formula (<kernel.Constant object at 0x21bb8c0>, <kernel.Single object at 0x1dfee60>) of role type named emptyset_type
% Using role type
% Declaring emptyset:fofType
% FOF formula (<kernel.Constant object at 0x1dfe518>, <kernel.Sort object at 0x1cc8998>) of role type named emptysetE_type
% Using role type
% Declaring emptysetE:Prop
% FOF formula (((eq Prop) emptysetE) (forall (Xx:fofType), (((in Xx) emptyset)->(forall (Xphi:Prop), Xphi)))) of role definition named emptysetE
% A new definition: (((eq Prop) emptysetE) (forall (Xx:fofType), (((in Xx) emptyset)->(forall (Xphi:Prop), Xphi))))
% Defined: emptysetE:=(forall (Xx:fofType), (((in Xx) emptyset)->(forall (Xphi:Prop), Xphi)))
% FOF formula (<kernel.Constant object at 0x1dfed40>, <kernel.Sort object at 0x1cc8998>) of role type named setext_type
% Using role type
% Declaring setext:Prop
% FOF formula (((eq Prop) setext) (forall (A:fofType) (B:fofType), ((forall (Xx:fofType), (((in Xx) A)->((in Xx) B)))->((forall (Xx:fofType), (((in Xx) B)->((in Xx) A)))->(((eq fofType) A) B))))) of role definition named setext
% A new definition: (((eq Prop) setext) (forall (A:fofType) (B:fofType), ((forall (Xx:fofType), (((in Xx) A)->((in Xx) B)))->((forall (Xx:fofType), (((in Xx) B)->((in Xx) A)))->(((eq fofType) A) B)))))
% Defined: setext:=(forall (A:fofType) (B:fofType), ((forall (Xx:fofType), (((in Xx) A)->((in Xx) B)))->((forall (Xx:fofType), (((in Xx) B)->((in Xx) A)))->(((eq fofType) A) B))))
% FOF formula (<kernel.Constant object at 0x1dfed40>, <kernel.DependentProduct object at 0x2238e60>) of role type named nonempty_type
% Using role type
% Declaring nonempty:(fofType->Prop)
% FOF formula (((eq (fofType->Prop)) nonempty) (fun (Xx:fofType)=> (not (((eq fofType) Xx) emptyset)))) of role definition named nonempty
% A new definition: (((eq (fofType->Prop)) nonempty) (fun (Xx:fofType)=> (not (((eq fofType) Xx) emptyset))))
% Defined: nonempty:=(fun (Xx:fofType)=> (not (((eq fofType) Xx) emptyset)))
% FOF formula (emptysetE->(setext->(forall (A:fofType), ((nonempty A)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) True))))))) of role conjecture named nonemptyImpWitness
% Conjecture to prove = (emptysetE->(setext->(forall (A:fofType), ((nonempty A)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) True))))))):Prop
% We need to prove ['(emptysetE->(setext->(forall (A:fofType), ((nonempty A)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) True)))))))']
% Parameter fofType:Type.
% Parameter in:(fofType->(fofType->Prop)).
% Parameter emptyset:fofType.
% Definition emptysetE:=(forall (Xx:fofType), (((in Xx) emptyset)->(forall (Xphi:Prop), Xphi))):Prop.
% Definition setext:=(forall (A:fofType) (B:fofType), ((forall (Xx:fofType), (((in Xx) A)->((in Xx) B)))->((forall (Xx:fofType), (((in Xx) B)->((in Xx) A)))->(((eq fofType) A) B)))):Prop.
% Definition nonempty:=(fun (Xx:fofType)=> (not (((eq fofType) Xx) emptyset))):(fofType->Prop).
% Trying to prove (emptysetE->(setext->(forall (A:fofType), ((nonempty A)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) True)))))))
% Found I:True
% Found I as proof of True
% Found I:True
% Found I as proof of True
% Found eq_ref00:=(eq_ref0 (fun (Xx:fofType)=> ((and ((in Xx) A)) True))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) A)) True))) (fun (Xx:fofType)=> ((and ((in Xx) A)) True)))
% Found (eq_ref0 (fun (Xx:fofType)=> ((and ((in Xx) A)) True))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) A)) True))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) A)) True))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) A)) True))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) A)) True))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) A)) True))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) A)) True))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) A)) True))) b)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and ((in x2) A)) True))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and ((in x2) A)) True))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and ((in x2) A)) True))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and ((in x2) A)) True))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) A)) True)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and ((in x2) A)) True))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and ((in x2) A)) True))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and ((in x2) A)) True))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and ((in x2) A)) True))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) A)) True)))
% Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) 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) (fun (Xx:fofType)=> ((and ((in Xx) A)) True)))
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and ((in Xx) A)) True)))
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and ((in Xx) A)) True)))
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and ((in Xx) A)) True)))
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and ((in Xx) A)) True)))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and ((in Xx) A)) True)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and ((in Xx) A)) True)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and ((in Xx) A)) True)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and ((in Xx) A)) True)))
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found eq_ref00:=(eq_ref0 A):(((eq fofType) A) A)
% Found (eq_ref0 A) as proof of (((eq fofType) A) b)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b)
% Found eq_ref00:=(eq_ref0 A):(((eq fofType) A) A)
% Found (eq_ref0 A) as proof of (((eq fofType) A) b)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b)
% Found x3:((in Xx) A0)
% Instantiate: A0:=A:fofType
% Found (fun (x3:((in Xx) A0))=> x3) as proof of ((in Xx) A)
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (((in Xx) A0)->((in Xx) A))
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) A)))
% Found x3:((in Xx) A)
% Instantiate: A0:=A:fofType
% Found (fun (x3:((in Xx) A))=> x3) as proof of ((in Xx) A0)
% Found (fun (Xx:fofType) (x3:((in Xx) A))=> x3) as proof of (((in Xx) A)->((in Xx) A0))
% Found (fun (Xx:fofType) (x3:((in Xx) A))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A)->((in Xx) A0)))
% Found x3:((in Xx) A0)
% Instantiate: A0:=A:fofType
% Found (fun (x3:((in Xx) A0))=> x3) as proof of ((in Xx) A)
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (((in Xx) A0)->((in Xx) A))
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) A)))
% Found x3:((in Xx) A)
% Instantiate: A0:=A:fofType
% Found (fun (x3:((in Xx) A))=> x3) as proof of ((in Xx) A0)
% Found (fun (Xx:fofType) (x3:((in Xx) A))=> x3) as proof of (((in Xx) A)->((in Xx) A0))
% Found (fun (Xx:fofType) (x3:((in Xx) A))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A)->((in Xx) A0)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A0)->((in Xx) A0))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A0)->((in Xx) A))
% Found ((eq_ref0 A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) A))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) A))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) A))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (((in Xx) A0)->((in Xx) A))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) A)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A)->((in Xx) A))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found ((eq_ref0 A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found (((eq_ref fofType) A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (((in Xx) A)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A)->((in Xx) A0)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A0)->((in Xx) A0))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A0)->((in Xx) A))
% Found ((eq_ref0 A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) A))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) A))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) A))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (((in Xx) A0)->((in Xx) A))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) A)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A)->((in Xx) A))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found ((eq_ref0 A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found (((eq_ref fofType) A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (((in Xx) A)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A)->((in Xx) A0)))
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b
% Found I:True
% Found I as proof of b
% Found eq_ref00:=(eq_ref0 emptyset):(((eq fofType) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% Found x3:((in Xx0) A0)
% Instantiate: A0:=emptyset:fofType
% Found (fun (x3:((in Xx0) A0))=> x3) as proof of ((in Xx0) emptyset)
% Found (fun (Xx0:fofType) (x3:((in Xx0) A0))=> x3) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found (fun (Xx0:fofType) (x3:((in Xx0) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) emptyset)))
% Found x3:((in Xx0) emptyset)
% Instantiate: A0:=emptyset:fofType
% Found (fun (x3:((in Xx0) emptyset))=> x3) as proof of ((in Xx0) A0)
% Found (fun (Xx0:fofType) (x3:((in Xx0) emptyset))=> x3) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (fun (Xx0:fofType) (x3:((in Xx0) emptyset))=> x3) as proof of (forall (Xx:fofType), (((in Xx) emptyset)->((in Xx) A0)))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found eq_ref000:=(eq_ref00 (in Xx0)):(((in Xx0) A0)->((in Xx0) A0))
% Found (eq_ref00 (in Xx0)) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found ((eq_ref0 A0) (in Xx0)) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found (((eq_ref fofType) A0) (in Xx0)) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found (((eq_ref fofType) A0) (in Xx0)) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) A0) (in Xx0))) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) A0) (in Xx0))) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) emptyset)))
% Found eq_ref000:=(eq_ref00 (in Xx0)):(((in Xx0) emptyset)->((in Xx0) emptyset))
% Found (eq_ref00 (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found ((eq_ref0 emptyset) (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (((eq_ref fofType) emptyset) (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) emptyset) (in Xx0))) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) emptyset) (in Xx0))) as proof of (forall (Xx:fofType), (((in Xx) emptyset)->((in Xx) A0)))
% Found eq_ref00:=(eq_ref0 f):(((eq (fofType->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (fofType->Prop)) f) b)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (fofType->Prop)) f) (fun (x:fofType)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (fofType->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (fofType->Prop)) f) b)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (fofType->Prop)) f) (fun (x:fofType)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (fofType->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (fofType->Prop)) f) b)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (fofType->Prop)) f) (fun (x:fofType)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (fofType->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% 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) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (fun (Xx:fofType)=> ((and ((in Xx) A)) True))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) A)) True))) (fun (Xx:fofType)=> ((and ((in Xx) A)) True)))
% Found (eq_ref0 (fun (Xx:fofType)=> ((and ((in Xx) A)) True))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) A)) True))) b0)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) A)) True))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) A)) True))) b0)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) A)) True))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) A)) True))) b0)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) A)) True))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) A)) True))) b0)
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) A)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) A)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) A)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) A)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) A)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) A)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) A)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) A)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found I:True
% Found I as proof of True
% Found I:True
% Found I as proof of True
% Found I:True
% Found I as proof of True
% Found eq_ref00:=(eq_ref0 f):(((eq (fofType->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (fofType->Prop)) f) b)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 f):(((eq (fofType->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (fofType->Prop)) f) b)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (fofType->Prop)) f) (fun (x:fofType)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (fofType->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 f):(((eq (fofType->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (fofType->Prop)) f) b)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: a:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of a
% Found I:True
% Found I as proof of a
% Found x:emptysetE
% Instantiate: b:=(forall (Xx:fofType), (((in Xx) emptyset)->(forall (Xphi:Prop), Xphi))):Prop
% Found x as proof of a
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) emptyset)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) emptyset)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) emptyset)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) emptyset)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found eq_ref00:=(eq_ref0 emptyset):(((eq fofType) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% Found I:True
% Found I as proof of True
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (fofType->Prop)) b0) (fun (x:fofType)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found x4:((in Xx0) A0)
% Instantiate: A0:=emptyset:fofType
% Found (fun (x4:((in Xx0) A0))=> x4) as proof of ((in Xx0) emptyset)
% Found (fun (Xx0:fofType) (x4:((in Xx0) A0))=> x4) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found (fun (Xx0:fofType) (x4:((in Xx0) A0))=> x4) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) emptyset)))
% Found x4:((in Xx0) emptyset)
% Instantiate: A0:=emptyset:fofType
% Found (fun (x4:((in Xx0) emptyset))=> x4) as proof of ((in Xx0) A0)
% Found (fun (Xx0:fofType) (x4:((in Xx0) emptyset))=> x4) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (fun (Xx0:fofType) (x4:((in Xx0) emptyset))=> x4) as proof of (forall (Xx:fofType), (((in Xx) emptyset)->((in Xx) A0)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (fofType->Prop)) b0) (fun (x:fofType)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) 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) f)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% 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 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found eq_ref000:=(eq_ref00 (in Xx0)):(((in Xx0) A0)->((in Xx0) A0))
% Found (eq_ref00 (in Xx0)) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found ((eq_ref0 A0) (in Xx0)) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found (((eq_ref fofType) A0) (in Xx0)) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found (((eq_ref fofType) A0) (in Xx0)) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) A0) (in Xx0))) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) A0) (in Xx0))) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) emptyset)))
% Found eq_ref000:=(eq_ref00 (in Xx0)):(((in Xx0) emptyset)->((in Xx0) emptyset))
% Found (eq_ref00 (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found ((eq_ref0 emptyset) (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (((eq_ref fofType) emptyset) (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) emptyset) (in Xx0))) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) emptyset) (in Xx0))) as proof of (forall (Xx:fofType), (((in Xx) emptyset)->((in Xx) A0)))
% Found eq_ref00:=(eq_ref0 emptyset):(((eq fofType) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% Found eq_ref00:=(eq_ref0 emptyset):(((eq fofType) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% Found x4:((in Xx0) A0)
% Instantiate: A0:=emptyset:fofType
% Found (fun (x4:((in Xx0) A0))=> x4) as proof of ((in Xx0) emptyset)
% Found (fun (Xx0:fofType) (x4:((in Xx0) A0))=> x4) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found (fun (Xx0:fofType) (x4:((in Xx0) A0))=> x4) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) emptyset)))
% Found x4:((in Xx0) emptyset)
% Instantiate: A0:=emptyset:fofType
% Found (fun (x4:((in Xx0) emptyset))=> x4) as proof of ((in Xx0) A0)
% Found (fun (Xx0:fofType) (x4:((in Xx0) emptyset))=> x4) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (fun (Xx0:fofType) (x4:((in Xx0) emptyset))=> x4) as proof of (forall (Xx:fofType), (((in Xx) emptyset)->((in Xx) A0)))
% Found eq_ref00:=(eq_ref0 A):(((eq fofType) A) A)
% Found (eq_ref0 A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found x4:((in Xx0) A0)
% Instantiate: A0:=emptyset:fofType
% Found (fun (x4:((in Xx0) A0))=> x4) as proof of ((in Xx0) emptyset)
% Found (fun (Xx0:fofType) (x4:((in Xx0) A0))=> x4) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found (fun (Xx0:fofType) (x4:((in Xx0) A0))=> x4) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) emptyset)))
% Found x4:((in Xx0) emptyset)
% Instantiate: A0:=emptyset:fofType
% Found (fun (x4:((in Xx0) emptyset))=> x4) as proof of ((in Xx0) A0)
% Found (fun (Xx0:fofType) (x4:((in Xx0) emptyset))=> x4) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (fun (Xx0:fofType) (x4:((in Xx0) emptyset))=> x4) as proof of (forall (Xx:fofType), (((in Xx) emptyset)->((in Xx) A0)))
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found eq_ref00:=(eq_ref0 A0):(((eq fofType) A0) A0)
% Found (eq_ref0 A0) as proof of (((eq fofType) A0) b)
% Found ((eq_ref fofType) A0) as proof of (((eq fofType) A0) b)
% Found ((eq_ref fofType) A0) as proof of (((eq fofType) A0) b)
% Found ((eq_ref fofType) A0) as proof of (((eq fofType) A0) b)
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b0)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b0)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b0)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b0)
% Found eq_ref00:=(eq_ref0 A0):(((eq fofType) A0) A0)
% Found (eq_ref0 A0) as proof of (((eq fofType) A0) b)
% Found ((eq_ref fofType) A0) as proof of (((eq fofType) A0) b)
% Found ((eq_ref fofType) A0) as proof of (((eq fofType) A0) b)
% Found ((eq_ref fofType) A0) as proof of (((eq fofType) A0) b)
% Found x4:((in Xx0) A0)
% Instantiate: A0:=emptyset:fofType
% Found (fun (x4:((in Xx0) A0))=> x4) as proof of ((in Xx0) emptyset)
% Found (fun (Xx0:fofType) (x4:((in Xx0) A0))=> x4) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found (fun (Xx0:fofType) (x4:((in Xx0) A0))=> x4) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) emptyset)))
% Found eq_ref000:=(eq_ref00 (in Xx0)):(((in Xx0) emptyset)->((in Xx0) emptyset))
% Found (eq_ref00 (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found ((eq_ref0 emptyset) (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (((eq_ref fofType) emptyset) (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) emptyset) (in Xx0))) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) emptyset) (in Xx0))) as proof of (forall (Xx:fofType), (((in Xx) emptyset)->((in Xx) A0)))
% Found eq_ref00:=(eq_ref0 A0):(((eq fofType) A0) A0)
% Found (eq_ref0 A0) as proof of (((eq fofType) A0) b)
% Found ((eq_ref fofType) A0) as proof of (((eq fofType) A0) b)
% Found ((eq_ref fofType) A0) as proof of (((eq fofType) A0) b)
% Found ((eq_ref fofType) A0) as proof of (((eq fofType) A0) b)
% Found x3:((in Xx) A1)
% Instantiate: A0:=A1:fofType
% Found (fun (x3:((in Xx) A1))=> x3) as proof of ((in Xx) A0)
% Found (fun (Xx:fofType) (x3:((in Xx) A1))=> x3) as proof of (((in Xx) A1)->((in Xx) A0))
% Found (fun (Xx:fofType) (x3:((in Xx) A1))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A1)->((in Xx) A0)))
% Found x3:((in Xx) A0)
% Instantiate: A1:=A0:fofType
% Found (fun (x3:((in Xx) A0))=> x3) as proof of ((in Xx) A1)
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (((in Xx) A0)->((in Xx) A1))
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) A1)))
% Found x3:((in Xx) A0)
% Found (fun (x3:((in Xx) A0))=> x3) as proof of ((in Xx) A1)
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (((in Xx) A0)->((in Xx) A1))
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) A1)))
% Found eq_ref00:=(eq_ref0 A0):(((eq fofType) A0) A0)
% Found (eq_ref0 A0) as proof of (((eq fofType) A0) b)
% Found ((eq_ref fofType) A0) as proof of (((eq fofType) A0) b)
% Found ((eq_ref fofType) A0) as proof of (((eq fofType) A0) b)
% Found ((eq_ref fofType) A0) as proof of (((eq fofType) A0) b)
% Found x4:((in Xx0) A0)
% Instantiate: A0:=emptyset:fofType
% Found (fun (x4:((in Xx0) A0))=> x4) as proof of ((in Xx0) emptyset)
% Found (fun (Xx0:fofType) (x4:((in Xx0) A0))=> x4) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found (fun (Xx0:fofType) (x4:((in Xx0) A0))=> x4) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) emptyset)))
% Found eq_ref000:=(eq_ref00 (in Xx0)):(((in Xx0) emptyset)->((in Xx0) emptyset))
% Found (eq_ref00 (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found ((eq_ref0 emptyset) (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (((eq_ref fofType) emptyset) (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) emptyset) (in Xx0))) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) emptyset) (in Xx0))) as proof of (forall (Xx:fofType), (((in Xx) emptyset)->((in Xx) A0)))
% Found x3:((in Xx) A1)
% Instantiate: A0:=A1:fofType
% Found (fun (x3:((in Xx) A1))=> x3) as proof of ((in Xx) A0)
% Found (fun (Xx:fofType) (x3:((in Xx) A1))=> x3) as proof of (((in Xx) A1)->((in Xx) A0))
% Found (fun (Xx:fofType) (x3:((in Xx) A1))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A1)->((in Xx) A0)))
% Found x3:((in Xx) A0)
% Instantiate: A1:=A0:fofType
% Found (fun (x3:((in Xx) A0))=> x3) as proof of ((in Xx) A1)
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (((in Xx) A0)->((in Xx) A1))
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) A1)))
% Found x3:((in Xx) A0)
% Found (fun (x3:((in Xx) A0))=> x3) as proof of ((in Xx) A1)
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (((in Xx) A0)->((in Xx) A1))
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) A1)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A1)->((in Xx) A1))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A1)->((in Xx) A0))
% Found ((eq_ref0 A1) (in Xx)) as proof of (((in Xx) A1)->((in Xx) A0))
% Found (((eq_ref fofType) A1) (in Xx)) as proof of (((in Xx) A1)->((in Xx) A0))
% Found (((eq_ref fofType) A1) (in Xx)) as proof of (((in Xx) A1)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A1) (in Xx))) as proof of (((in Xx) A1)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A1) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A1)->((in Xx) A0)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A0)->((in Xx) A0))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A0)->((in Xx) A1))
% Found ((eq_ref0 A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) A1))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) A1))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (((in Xx) A0)->((in Xx) A1))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) A1)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A00)->((in Xx) A00))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A00)->((in Xx) A))
% Found ((eq_ref0 A00) (in Xx)) as proof of (((in Xx) A00)->((in Xx) A))
% Found (((eq_ref fofType) A00) (in Xx)) as proof of (((in Xx) A00)->((in Xx) A))
% Found (((eq_ref fofType) A00) (in Xx)) as proof of (((in Xx) A00)->((in Xx) A))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A00) (in Xx))) as proof of (((in Xx) A00)->((in Xx) A))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A00) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A00)->((in Xx) A)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A)->((in Xx) A))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A)->((in Xx) A00))
% Found ((eq_ref0 A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A00))
% Found (((eq_ref fofType) A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A00))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (((in Xx) A)->((in Xx) A00))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A)->((in Xx) A00)))
% Found x3:((in Xx) A0)
% Instantiate: b:=A0:fofType
% Found (fun (x3:((in Xx) A0))=> x3) as proof of ((in Xx) b)
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (((in Xx) A0)->((in Xx) b))
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) b)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) b)->((in Xx) b))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) b)->((in Xx) A0))
% Found ((eq_ref0 b) (in Xx)) as proof of (((in Xx) b)->((in Xx) A0))
% Found (((eq_ref fofType) b) (in Xx)) as proof of (((in Xx) b)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) b) (in Xx))) as proof of (((in Xx) b)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) b) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) b)->((in Xx) A0)))
% Found x3:((in Xx) A1)
% Instantiate: A0:=A1:fofType
% Found (fun (x3:((in Xx) A1))=> x3) as proof of ((in Xx) A0)
% Found (fun (Xx:fofType) (x3:((in Xx) A1))=> x3) as proof of (((in Xx) A1)->((in Xx) A0))
% Found (fun (Xx:fofType) (x3:((in Xx) A1))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A1)->((in Xx) A0)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A0)->((in Xx) A0))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A0)->((in Xx) A1))
% Found ((eq_ref0 A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) A1))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) A1))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (((in Xx) A0)->((in Xx) A1))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) A1)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A0)->((in Xx) A0))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A0)->((in Xx) b))
% Found ((eq_ref0 A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) b))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) b))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) b))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (((in Xx) A0)->((in Xx) b))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) b)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) b)->((in Xx) b))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) b)->((in Xx) A0))
% Found ((eq_ref0 b) (in Xx)) as proof of (((in Xx) b)->((in Xx) A0))
% Found (((eq_ref fofType) b) (in Xx)) as proof of (((in Xx) b)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) b) (in Xx))) as proof of (((in Xx) b)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) b) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) b)->((in Xx) A0)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A00)->((in Xx) A00))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A00)->((in Xx) A))
% Found ((eq_ref0 A00) (in Xx)) as proof of (((in Xx) A00)->((in Xx) A))
% Found (((eq_ref fofType) A00) (in Xx)) as proof of (((in Xx) A00)->((in Xx) A))
% Found (((eq_ref fofType) A00) (in Xx)) as proof of (((in Xx) A00)->((in Xx) A))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A00) (in Xx))) as proof of (((in Xx) A00)->((in Xx) A))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A00) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A00)->((in Xx) A)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A)->((in Xx) A))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A)->((in Xx) A00))
% Found ((eq_ref0 A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A00))
% Found (((eq_ref fofType) A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A00))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (((in Xx) A)->((in Xx) A00))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A)->((in Xx) A00)))
% Found x3:((in Xx) A0)
% Instantiate: b:=A0:fofType
% Found (fun (x3:((in Xx) A0))=> x3) as proof of ((in Xx) b)
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (((in Xx) A0)->((in Xx) b))
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) b)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) b)->((in Xx) b))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) b)->((in Xx) A0))
% Found ((eq_ref0 b) (in Xx)) as proof of (((in Xx) b)->((in Xx) A0))
% Found (((eq_ref fofType) b) (in Xx)) as proof of (((in Xx) b)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) b) (in Xx))) as proof of (((in Xx) b)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) b) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) b)->((in Xx) A0)))
% Found x3:((in Xx) A0)
% Instantiate: A0:=A:fofType
% Found (fun (x3:((in Xx) A0))=> x3) as proof of ((in Xx) b)
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (((in Xx) A0)->((in Xx) b))
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) b)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) b)->((in Xx) b))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) b)->((in Xx) A0))
% Found ((eq_ref0 b) (in Xx)) as proof of (((in Xx) b)->((in Xx) A0))
% Found (((eq_ref fofType) b) (in Xx)) as proof of (((in Xx) b)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) b) (in Xx))) as proof of (((in Xx) b)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) b) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) b)->((in Xx) A0)))
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found x3:(P2 (f x2))
% Instantiate: f0:=f:(fofType->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (f0 x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (f0 x2)))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:fofType), (((eq Prop) (f x)) (f0 x)))
% Found x3:(P2 (f x2))
% Instantiate: f0:=f:(fofType->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (f0 x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (f0 x2)))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:fofType), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found x3:(P2 (f x2))
% Instantiate: b:=f:(fofType->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (b x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (b x2)))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found x3:(P2 (f x2))
% Instantiate: b:=f:(fofType->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (b x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (b x2)))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 A):(((eq fofType) A) A)
% Found (eq_ref0 A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (fofType->Prop)) b0) (fun (x:fofType)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (fofType->Prop)) b0) (fun (x:fofType)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b0)
% 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) f)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) 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) f)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% 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) f)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% 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 x2):(((eq fofType) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq fofType) x2) b0)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b0)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b0)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b0)
% Found eq_ref00:=(eq_ref0 x2):(((eq fofType) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found eq_ref00:=(eq_ref0 x2):(((eq fofType) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found or_comm_i:=(fun (A:Prop) (B:Prop) (H:((or A) B))=> ((((((or_ind A) B) ((or B) A)) ((or_intror B) A)) ((or_introl B) A)) H)):(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A)))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A))):Prop
% Found or_comm_i as proof of b0
% Found or_comm_i:=(fun (A:Prop) (B:Prop) (H:((or A) B))=> ((((((or_ind A) B) ((or B) A)) ((or_intror B) A)) ((or_introl B) A)) H)):(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A)))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A))):Prop
% Found or_comm_i as proof of b0
% Found eq_ref00:=(eq_ref0 A):(((eq fofType) A) A)
% Found (eq_ref0 A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found x3:((in Xx) x2)
% Instantiate: A0:=x2:fofType
% Found (fun (x3:((in Xx) x2))=> x3) as proof of ((in Xx) A0)
% Found (fun (Xx:fofType) (x3:((in Xx) x2))=> x3) as proof of (((in Xx) x2)->((in Xx) A0))
% Found (fun (Xx:fofType) (x3:((in Xx) x2))=> x3) as proof of (forall (Xx:fofType), (((in Xx) x2)->((in Xx) A0)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A0)->((in Xx) A0))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A0)->((in Xx) x2))
% Found ((eq_ref0 A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) x2))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (((in Xx) A0)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) x2)))
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found eq_ref00:=(eq_ref0 A0):(((eq fofType) A0) A0)
% Found (eq_ref0 A0) as proof of (((eq fofType) A0) b)
% Found ((eq_ref fofType) A0) as proof of (((eq fofType) A0) b)
% Found ((eq_ref fofType) A0) as proof of (((eq fofType) A0) b)
% Found ((eq_ref fofType) A0) as proof of (((eq fofType) A0) b)
% Found x3:((in Xx) x2)
% Instantiate: A0:=x2:fofType
% Found (fun (x3:((in Xx) x2))=> x3) as proof of ((in Xx) A0)
% Found (fun (Xx:fofType) (x3:((in Xx) x2))=> x3) as proof of (((in Xx) x2)->((in Xx) A0))
% Found (fun (Xx:fofType) (x3:((in Xx) x2))=> x3) as proof of (forall (Xx:fofType), (((in Xx) x2)->((in Xx) A0)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A0)->((in Xx) A0))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A0)->((in Xx) x2))
% Found ((eq_ref0 A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) x2))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (((in Xx) A0)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) x2)))
% Found x3:((in Xx) x2)
% Instantiate: A0:=x2:fofType
% Found (fun (x3:((in Xx) x2))=> x3) as proof of ((in Xx) A0)
% Found (fun (Xx:fofType) (x3:((in Xx) x2))=> x3) as proof of (((in Xx) x2)->((in Xx) A0))
% Found (fun (Xx:fofType) (x3:((in Xx) x2))=> x3) as proof of (forall (Xx:fofType), (((in Xx) x2)->((in Xx) A0)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A0)->((in Xx) A0))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A0)->((in Xx) x2))
% Found ((eq_ref0 A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) x2))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (((in Xx) A0)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) x2)))
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b0)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b0)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b0)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b0)
% Found eq_ref00:=(eq_ref0 A0):(((eq fofType) A0) A0)
% Found (eq_ref0 A0) as proof of (((eq fofType) A0) b)
% Found ((eq_ref fofType) A0) as proof of (((eq fofType) A0) b)
% Found ((eq_ref fofType) A0) as proof of (((eq fofType) A0) b)
% Found ((eq_ref fofType) A0) as proof of (((eq fofType) A0) b)
% Found eq_ref00:=(eq_ref0 A):(((eq fofType) A) A)
% Found (eq_ref0 A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found eq_ref00:=(eq_ref0 A0):(((eq fofType) A0) A0)
% Found (eq_ref0 A0) as proof of (((eq fofType) A0) b)
% Found ((eq_ref fofType) A0) as proof of (((eq fofType) A0) b)
% Found ((eq_ref fofType) A0) as proof of (((eq fofType) A0) b)
% Found ((eq_ref fofType) A0) as proof of (((eq fofType) A0) b)
% Found x3:((in Xx) A1)
% Instantiate: A0:=A1:fofType
% Found (fun (x3:((in Xx) A1))=> x3) as proof of ((in Xx) A0)
% Found (fun (Xx:fofType) (x3:((in Xx) A1))=> x3) as proof of (((in Xx) A1)->((in Xx) A0))
% Found (fun (Xx:fofType) (x3:((in Xx) A1))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A1)->((in Xx) A0)))
% Found x3:((in Xx) A0)
% Instantiate: A1:=A0:fofType
% Found (fun (x3:((in Xx) A0))=> x3) as proof of ((in Xx) A1)
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (((in Xx) A0)->((in Xx) A1))
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) A1)))
% Found x3:((in Xx) A0)
% Found (fun (x3:((in Xx) A0))=> x3) as proof of ((in Xx) A1)
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (((in Xx) A0)->((in Xx) A1))
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) A1)))
% Found eq_ref00:=(eq_ref0 A):(((eq fofType) A) A)
% Found (eq_ref0 A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found eq_ref00:=(eq_ref0 A):(((eq fofType) A) A)
% Found (eq_ref0 A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found eq_ref00:=(eq_ref0 A0):(((eq fofType) A0) A0)
% Found (eq_ref0 A0) as proof of (((eq fofType) A0) b)
% Found ((eq_ref fofType) A0) as proof of (((eq fofType) A0) b)
% Found ((eq_ref fofType) A0) as proof of (((eq fofType) A0) b)
% Found ((eq_ref fofType) A0) as proof of (((eq fofType) A0) b)
% Found x3:((in Xx) A1)
% Instantiate: A0:=A1:fofType
% Found (fun (x3:((in Xx) A1))=> x3) as proof of ((in Xx) A0)
% Found (fun (Xx:fofType) (x3:((in Xx) A1))=> x3) as proof of (((in Xx) A1)->((in Xx) A0))
% Found (fun (Xx:fofType) (x3:((in Xx) A1))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A1)->((in Xx) A0)))
% Found x3:((in Xx) A0)
% Instantiate: A1:=A0:fofType
% Found (fun (x3:((in Xx) A0))=> x3) as proof of ((in Xx) A1)
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (((in Xx) A0)->((in Xx) A1))
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) A1)))
% Found x3:((in Xx) A0)
% Found (fun (x3:((in Xx) A0))=> x3) as proof of ((in Xx) A1)
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (((in Xx) A0)->((in Xx) A1))
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) A1)))
% Found eq_ref00:=(eq_ref0 A):(((eq fofType) A) A)
% Found (eq_ref0 A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found ex_intro0:=(ex_intro fofType):(forall (P:(fofType->Prop)) (x:fofType), ((P x)->((ex fofType) P)))
% Instantiate: b0:=(forall (P:(fofType->Prop)) (x:fofType), ((P x)->((ex fofType) P))):Prop
% Found ex_intro0 as proof of b0
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A1)->((in Xx) A1))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A1)->((in Xx) A0))
% Found ((eq_ref0 A1) (in Xx)) as proof of (((in Xx) A1)->((in Xx) A0))
% Found (((eq_ref fofType) A1) (in Xx)) as proof of (((in Xx) A1)->((in Xx) A0))
% Found (((eq_ref fofType) A1) (in Xx)) as proof of (((in Xx) A1)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A1) (in Xx))) as proof of (((in Xx) A1)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A1) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A1)->((in Xx) A0)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A0)->((in Xx) A0))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A0)->((in Xx) A1))
% Found ((eq_ref0 A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) A1))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) A1))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (((in Xx) A0)->((in Xx) A1))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) A1)))
% Found I:True
% Found I as proof of b0
% Found x3:((in Xx) A00)
% Instantiate: A00:=A:fofType
% Found (fun (x3:((in Xx) A00))=> x3) as proof of ((in Xx) A)
% Found (fun (Xx:fofType) (x3:((in Xx) A00))=> x3) as proof of (((in Xx) A00)->((in Xx) A))
% Found (fun (Xx:fofType) (x3:((in Xx) A00))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A00)->((in Xx) A)))
% Found x3:((in Xx) A)
% Instantiate: A00:=A:fofType
% Found (fun (x3:((in Xx) A))=> x3) as proof of ((in Xx) A00)
% Found (fun (Xx:fofType) (x3:((in Xx) A))=> x3) as proof of (((in Xx) A)->((in Xx) A00))
% Found (fun (Xx:fofType) (x3:((in Xx) A))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A)->((in Xx) A00)))
% Found x3:((in Xx) A00)
% Instantiate: A00:=A:fofType
% Found (fun (x3:((in Xx) A00))=> x3) as proof of ((in Xx) A)
% Found (fun (Xx:fofType) (x3:((in Xx) A00))=> x3) as proof of (((in Xx) A00)->((in Xx) A))
% Found (fun (Xx:fofType) (x3:((in Xx) A00))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A00)->((in Xx) A)))
% Found x3:((in Xx) A)
% Instantiate: A00:=A:fofType
% Found (fun (x3:((in Xx) A))=> x3) as proof of ((in Xx) A00)
% Found (fun (Xx:fofType) (x3:((in Xx) A))=> x3) as proof of (((in Xx) A)->((in Xx) A00))
% Found (fun (Xx:fofType) (x3:((in Xx) A))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A)->((in Xx) A00)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A0)->((in Xx) A0))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A0)->((in Xx) b))
% Found ((eq_ref0 A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) b))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) b))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) b))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (((in Xx) A0)->((in Xx) b))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) b)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) b)->((in Xx) b))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) b)->((in Xx) A0))
% Found ((eq_ref0 b) (in Xx)) as proof of (((in Xx) b)->((in Xx) A0))
% Found (((eq_ref fofType) b) (in Xx)) as proof of (((in Xx) b)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) b) (in Xx))) as proof of (((in Xx) b)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) b) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) b)->((in Xx) A0)))
% Found x3:((in Xx) A0)
% Instantiate: A0:=A:fofType
% Found (fun (x3:((in Xx) A0))=> x3) as proof of ((in Xx) A)
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (((in Xx) A0)->((in Xx) A))
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) A)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A)->((in Xx) A))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found ((eq_ref0 A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found (((eq_ref fofType) A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (((in Xx) A)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A)->((in Xx) A0)))
% Found x3:((in Xx) A1)
% Instantiate: A0:=A1:fofType
% Found (fun (x3:((in Xx) A1))=> x3) as proof of ((in Xx) A0)
% Found (fun (Xx:fofType) (x3:((in Xx) A1))=> x3) as proof of (((in Xx) A1)->((in Xx) A0))
% Found (fun (Xx:fofType) (x3:((in Xx) A1))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A1)->((in Xx) A0)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A0)->((in Xx) A0))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A0)->((in Xx) A1))
% Found ((eq_ref0 A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) A1))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) A1))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (((in Xx) A0)->((in Xx) A1))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) A1)))
% Found x3:((in Xx) A0)
% Instantiate: A0:=A:fofType
% Found (fun (x3:((in Xx) A0))=> x3) as proof of ((in Xx) b)
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (((in Xx) A0)->((in Xx) b))
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) b)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) b)->((in Xx) b))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) b)->((in Xx) A0))
% Found ((eq_ref0 b) (in Xx)) as proof of (((in Xx) b)->((in Xx) A0))
% Found (((eq_ref fofType) b) (in Xx)) as proof of (((in Xx) b)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) b) (in Xx))) as proof of (((in Xx) b)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) b) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) b)->((in Xx) A0)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (f0 x)))
% Found x3:((in Xx) A0)
% Instantiate: A0:=A:fofType
% Found (fun (x3:((in Xx) A0))=> x3) as proof of ((in Xx) A)
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (((in Xx) A0)->((in Xx) A))
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) A)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A)->((in Xx) A))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found ((eq_ref0 A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found (((eq_ref fofType) A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (((in Xx) A)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A)->((in Xx) A0)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A0)->((in Xx) A0))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A0)->((in Xx) A))
% Found ((eq_ref0 A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) A))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) A))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) A))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (((in Xx) A0)->((in Xx) A))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) A)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A)->((in Xx) A))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found ((eq_ref0 A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found (((eq_ref fofType) A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (((in Xx) A)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A)->((in Xx) A0)))
% Found x3:((in Xx) A00)
% Instantiate: A00:=A:fofType
% Found (fun (x3:((in Xx) A00))=> x3) as proof of ((in Xx) A)
% Found (fun (Xx:fofType) (x3:((in Xx) A00))=> x3) as proof of (((in Xx) A00)->((in Xx) A))
% Found (fun (Xx:fofType) (x3:((in Xx) A00))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A00)->((in Xx) A)))
% Found x3:((in Xx) A)
% Instantiate: A00:=A:fofType
% Found (fun (x3:((in Xx) A))=> x3) as proof of ((in Xx) A00)
% Found (fun (Xx:fofType) (x3:((in Xx) A))=> x3) as proof of (((in Xx) A)->((in Xx) A00))
% Found (fun (Xx:fofType) (x3:((in Xx) A))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A)->((in Xx) A00)))
% Found x3:((in Xx) A00)
% Instantiate: A00:=A:fofType
% Found (fun (x3:((in Xx) A00))=> x3) as proof of ((in Xx) A)
% Found (fun (Xx:fofType) (x3:((in Xx) A00))=> x3) as proof of (((in Xx) A00)->((in Xx) A))
% Found (fun (Xx:fofType) (x3:((in Xx) A00))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A00)->((in Xx) A)))
% Found x3:((in Xx) A)
% Instantiate: A00:=A:fofType
% Found (fun (x3:((in Xx) A))=> x3) as proof of ((in Xx) A00)
% Found (fun (Xx:fofType) (x3:((in Xx) A))=> x3) as proof of (((in Xx) A)->((in Xx) A00))
% Found (fun (Xx:fofType) (x3:((in Xx) A))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A)->((in Xx) A00)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A0)->((in Xx) A0))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A0)->((in Xx) b))
% Found ((eq_ref0 A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) b))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) b))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) b))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (((in Xx) A0)->((in Xx) b))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) b)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) b)->((in Xx) b))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) b)->((in Xx) A0))
% Found ((eq_ref0 b) (in Xx)) as proof of (((in Xx) b)->((in Xx) A0))
% Found (((eq_ref fofType) b) (in Xx)) as proof of (((in Xx) b)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) b) (in Xx))) as proof of (((in Xx) b)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) b) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) b)->((in Xx) A0)))
% Found x3:((in Xx) A0)
% Instantiate: A0:=A:fofType
% Found (fun (x3:((in Xx) A0))=> x3) as proof of ((in Xx) b)
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (((in Xx) A0)->((in Xx) b))
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) b)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) b)->((in Xx) b))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) b)->((in Xx) A0))
% Found ((eq_ref0 b) (in Xx)) as proof of (((in Xx) b)->((in Xx) A0))
% Found (((eq_ref fofType) b) (in Xx)) as proof of (((in Xx) b)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) b) (in Xx))) as proof of (((in Xx) b)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) b) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) b)->((in Xx) A0)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A0)->((in Xx) A0))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A0)->((in Xx) A))
% Found ((eq_ref0 A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) A))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) A))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) A))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (((in Xx) A0)->((in Xx) A))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) A)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A)->((in Xx) A))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found ((eq_ref0 A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found (((eq_ref fofType) A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (((in Xx) A)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A)->((in Xx) A0)))
% Found eq_ref00:=(eq_ref0 (fun (Xx:fofType)=> ((and ((in Xx) A)) True))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) A)) True))) (fun (Xx:fofType)=> ((and ((in Xx) A)) True)))
% Found (eq_ref0 (fun (Xx:fofType)=> ((and ((in Xx) A)) True))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) A)) True))) b0)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) A)) True))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) A)) True))) b0)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) A)) True))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) A)) True))) b0)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) A)) True))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) A)) True))) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found x3:(P2 (f x2))
% Instantiate: f0:=f:(fofType->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (f0 x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (f0 x2)))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:fofType), (((eq Prop) (f x)) (f0 x)))
% Found x3:(P2 (f x2))
% Instantiate: f0:=f:(fofType->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (f0 x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (f0 x2)))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:fofType), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found x3:(P2 (f x2))
% Instantiate: b:=f:(fofType->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (b x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (b x2)))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found x3:(P2 (f x2))
% Instantiate: b:=f:(fofType->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (b x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (b x2)))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found x3:(P2 (f x2))
% Instantiate: b:=f:(fofType->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (b x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (b x2)))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found x3:(P2 (f x2))
% Instantiate: b:=f:(fofType->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (b x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (b x2)))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 A):(((eq fofType) A) A)
% Found (eq_ref0 A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (f0 x)))
% Found x3:(P2 (f x2))
% Instantiate: f0:=f:(fofType->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (f0 x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (f0 x2)))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:fofType), (((eq Prop) (f x)) (f0 x)))
% Found x3:(P2 (f x2))
% Instantiate: f0:=f:(fofType->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (f0 x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (f0 x2)))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:fofType), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b0)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b0)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b0)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b0)
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found or_comm_i:=(fun (A:Prop) (B:Prop) (H:((or A) B))=> ((((((or_ind A) B) ((or B) A)) ((or_intror B) A)) ((or_introl B) A)) H)):(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A)))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A))):Prop
% Found or_comm_i as proof of b0
% Found or_comm_i:=(fun (A:Prop) (B:Prop) (H:((or A) B))=> ((((((or_ind A) B) ((or B) A)) ((or_intror B) A)) ((or_introl B) A)) H)):(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A)))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A))):Prop
% Found or_comm_i as proof of b0
% Found I:True
% Found I as proof of True
% Found I:True
% Found I as proof of True
% Found eq_ref00:=(eq_ref0 A):(((eq fofType) A) A)
% Found (eq_ref0 A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found I:True
% Found I as proof of True
% Found I:True
% Found I as proof of True
% Found eq_ref00:=(eq_ref0 A):(((eq fofType) A) A)
% Found (eq_ref0 A) as proof of (((eq fofType) A) b)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b)
% Found eq_ref00:=(eq_ref0 A):(((eq fofType) A) A)
% Found (eq_ref0 A) as proof of (((eq fofType) A) b)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b)
% Found eq_ref00:=(eq_ref0 A):(((eq fofType) A) A)
% Found (eq_ref0 A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found eq_ref00:=(eq_ref0 emptyset):(((eq fofType) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% Found eq_ref00:=(eq_ref0 A):(((eq fofType) A) A)
% Found (eq_ref0 A) as proof of (((eq fofType) A) b)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b)
% Found eq_ref00:=(eq_ref0 A):(((eq fofType) A) A)
% Found (eq_ref0 A) as proof of (((eq fofType) A) b)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b)
% Found ex_intro0:=(ex_intro fofType):(forall (P:(fofType->Prop)) (x:fofType), ((P x)->((ex fofType) P)))
% Instantiate: b0:=(forall (P:(fofType->Prop)) (x:fofType), ((P x)->((ex fofType) P))):Prop
% Found ex_intro0 as proof of b0
% Found x3:((in Xx) A00)
% Instantiate: A00:=A:fofType
% Found (fun (x3:((in Xx) A00))=> x3) as proof of ((in Xx) A)
% Found (fun (Xx:fofType) (x3:((in Xx) A00))=> x3) as proof of (((in Xx) A00)->((in Xx) A))
% Found (fun (Xx:fofType) (x3:((in Xx) A00))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A00)->((in Xx) A)))
% Found x3:((in Xx) A)
% Instantiate: A00:=A:fofType
% Found (fun (x3:((in Xx) A))=> x3) as proof of ((in Xx) A00)
% Found (fun (Xx:fofType) (x3:((in Xx) A))=> x3) as proof of (((in Xx) A)->((in Xx) A00))
% Found (fun (Xx:fofType) (x3:((in Xx) A))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A)->((in Xx) A00)))
% Found x3:((in Xx) A00)
% Instantiate: A00:=A:fofType
% Found (fun (x3:((in Xx) A00))=> x3) as proof of ((in Xx) A)
% Found (fun (Xx:fofType) (x3:((in Xx) A00))=> x3) as proof of (((in Xx) A00)->((in Xx) A))
% Found (fun (Xx:fofType) (x3:((in Xx) A00))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A00)->((in Xx) A)))
% Found x3:((in Xx) A)
% Instantiate: A00:=A:fofType
% Found (fun (x3:((in Xx) A))=> x3) as proof of ((in Xx) A00)
% Found (fun (Xx:fofType) (x3:((in Xx) A))=> x3) as proof of (((in Xx) A)->((in Xx) A00))
% Found (fun (Xx:fofType) (x3:((in Xx) A))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A)->((in Xx) A00)))
% Found x30:=(fun (x4:((in Xx0) emptyset))=> ((x3 x4) Xphi0)):(((in Xx0) emptyset)->Xphi0)
% Instantiate: A0:=emptyset:fofType;Xx:=Xx1:fofType
% Found x30 as proof of (((in Xx1) A0)->((in Xx1) emptyset))
% Found x4:((in Xx1) A0)
% Instantiate: A0:=emptyset:fofType
% Found (fun (x4:((in Xx1) A0))=> x4) as proof of ((in Xx1) emptyset)
% Found (fun (Xx1:fofType) (x4:((in Xx1) A0))=> x4) as proof of (((in Xx1) A0)->((in Xx1) emptyset))
% Found (fun (Xx1:fofType) (x4:((in Xx1) A0))=> x4) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) emptyset)))
% Found x4:((in Xx1) emptyset)
% Instantiate: A0:=emptyset:fofType
% Found (fun (x4:((in Xx1) emptyset))=> x4) as proof of ((in Xx1) A0)
% Found (fun (Xx1:fofType) (x4:((in Xx1) emptyset))=> x4) as proof of (((in Xx1) emptyset)->((in Xx1) A0))
% Found (fun (Xx1:fofType) (x4:((in Xx1) emptyset))=> x4) as proof of (forall (Xx:fofType), (((in Xx) emptyset)->((in Xx) A0)))
% Found I:True
% Found I as proof of True
% Found I:True
% Found I as proof of b0
% Found x3:((in Xx) A00)
% Instantiate: A00:=A:fofType
% Found (fun (x3:((in Xx) A00))=> x3) as proof of ((in Xx) A)
% Found (fun (Xx:fofType) (x3:((in Xx) A00))=> x3) as proof of (((in Xx) A00)->((in Xx) A))
% Found (fun (Xx:fofType) (x3:((in Xx) A00))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A00)->((in Xx) A)))
% Found x3:((in Xx) A)
% Instantiate: A00:=A:fofType
% Found (fun (x3:((in Xx) A))=> x3) as proof of ((in Xx) A00)
% Found (fun (Xx:fofType) (x3:((in Xx) A))=> x3) as proof of (((in Xx) A)->((in Xx) A00))
% Found (fun (Xx:fofType) (x3:((in Xx) A))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A)->((in Xx) A00)))
% Found x3:((in Xx) A00)
% Instantiate: A00:=A:fofType
% Found (fun (x3:((in Xx) A00))=> x3) as proof of ((in Xx) A)
% Found (fun (Xx:fofType) (x3:((in Xx) A00))=> x3) as proof of (((in Xx) A00)->((in Xx) A))
% Found (fun (Xx:fofType) (x3:((in Xx) A00))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A00)->((in Xx) A)))
% Found x3:((in Xx) A)
% Instantiate: A00:=A:fofType
% Found (fun (x3:((in Xx) A))=> x3) as proof of ((in Xx) A00)
% Found (fun (Xx:fofType) (x3:((in Xx) A))=> x3) as proof of (((in Xx) A)->((in Xx) A00))
% Found (fun (Xx:fofType) (x3:((in Xx) A))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A)->((in Xx) A00)))
% Found x3:((in Xx) A0)
% Instantiate: A0:=A:fofType
% Found (fun (x3:((in Xx) A0))=> x3) as proof of ((in Xx) A)
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (((in Xx) A0)->((in Xx) A))
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) A)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A)->((in Xx) A))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found ((eq_ref0 A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found (((eq_ref fofType) A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (((in Xx) A)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A)->((in Xx) A0)))
% Found eq_ref00:=(eq_ref0 A0):(((eq fofType) A0) A0)
% Found (eq_ref0 A0) as proof of (((eq fofType) A0) b)
% Found ((eq_ref fofType) A0) as proof of (((eq fofType) A0) b)
% Found ((eq_ref fofType) A0) as proof of (((eq fofType) A0) b)
% Found ((eq_ref fofType) A0) as proof of (((eq fofType) A0) b)
% Found I:True
% Found I as proof of True
% Found I:True
% Found I as proof of True
% Found eq_ref00:=(eq_ref0 A0):(((eq fofType) A0) A0)
% Found (eq_ref0 A0) as proof of (((eq fofType) A0) b)
% Found ((eq_ref fofType) A0) as proof of (((eq fofType) A0) b)
% Found ((eq_ref fofType) A0) as proof of (((eq fofType) A0) b)
% Found ((eq_ref fofType) A0) as proof of (((eq fofType) A0) b)
% Found x3:((in Xx) A0)
% Instantiate: A0:=A:fofType
% Found (fun (x3:((in Xx) A0))=> x3) as proof of ((in Xx) A)
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (((in Xx) A0)->((in Xx) A))
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) A)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A)->((in Xx) A))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found ((eq_ref0 A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found (((eq_ref fofType) A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (((in Xx) A)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A)->((in Xx) A0)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A0)->((in Xx) A0))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A0)->((in Xx) A))
% Found ((eq_ref0 A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) A))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) A))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) A))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (((in Xx) A0)->((in Xx) A))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) A)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A)->((in Xx) A))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found ((eq_ref0 A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found (((eq_ref fofType) A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (((in Xx) A)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A)->((in Xx) A0)))
% Found eq_ref00:=(eq_ref0 emptyset):(((eq fofType) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq fofType) emptyset) b0)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b0)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b0)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b0)
% Found x3:((in Xx) A0)
% Instantiate: A0:=A:fofType
% Found (fun (x3:((in Xx) A0))=> x3) as proof of ((in Xx) A)
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (((in Xx) A0)->((in Xx) A))
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) A)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A)->((in Xx) A))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found ((eq_ref0 A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found (((eq_ref fofType) A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (((in Xx) A)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A)->((in Xx) A0)))
% Found x4:((in Xx1) A0)
% Instantiate: A0:=emptyset:fofType
% Found (fun (x4:((in Xx1) A0))=> x4) as proof of ((in Xx1) emptyset)
% Found (fun (Xx1:fofType) (x4:((in Xx1) A0))=> x4) as proof of (((in Xx1) A0)->((in Xx1) emptyset))
% Found (fun (Xx1:fofType) (x4:((in Xx1) A0))=> x4) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) emptyset)))
% Found x30:=(fun (x4:((in Xx0) emptyset))=> ((x3 x4) Xphi0)):(((in Xx0) emptyset)->Xphi0)
% Instantiate: Xx:=Xx1:fofType;Xx0:=Xx1:fofType
% Found x30 as proof of (((in Xx1) emptyset)->((in Xx1) A0))
% Found x4:((in Xx1) emptyset)
% Instantiate: A0:=emptyset:fofType
% Found (fun (x4:((in Xx1) emptyset))=> x4) as proof of ((in Xx1) A0)
% Found (fun (Xx1:fofType) (x4:((in Xx1) emptyset))=> x4) as proof of (((in Xx1) emptyset)->((in Xx1) A0))
% Found (fun (Xx1:fofType) (x4:((in Xx1) emptyset))=> x4) as proof of (forall (Xx:fofType), (((in Xx) emptyset)->((in Xx) A0)))
% Found x3:((in Xx) A00)
% Instantiate: A00:=A:fofType
% Found (fun (x3:((in Xx) A00))=> x3) as proof of ((in Xx) A)
% Found (fun (Xx:fofType) (x3:((in Xx) A00))=> x3) as proof of (((in Xx) A00)->((in Xx) A))
% Found (fun (Xx:fofType) (x3:((in Xx) A00))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A00)->((in Xx) A)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A)->((in Xx) A))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A)->((in Xx) A00))
% Found ((eq_ref0 A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A00))
% Found (((eq_ref fofType) A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A00))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (((in Xx) A)->((in Xx) A00))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A)->((in Xx) A00)))
% Found eq_ref00:=(eq_ref0 emptyset):(((eq fofType) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq fofType) emptyset) b0)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b0)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b0)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b0)
% Found x3:((in Xx0) A1)
% Instantiate: A0:=A1:fofType
% Found (fun (x3:((in Xx0) A1))=> x3) as proof of ((in Xx0) A0)
% Found (fun (Xx0:fofType) (x3:((in Xx0) A1))=> x3) as proof of (((in Xx0) A1)->((in Xx0) A0))
% Found (fun (Xx0:fofType) (x3:((in Xx0) A1))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A1)->((in Xx) A0)))
% Found x3:((in Xx0) A0)
% Instantiate: A1:=A0:fofType
% Found (fun (x3:((in Xx0) A0))=> x3) as proof of ((in Xx0) A1)
% Found (fun (Xx0:fofType) (x3:((in Xx0) A0))=> x3) as proof of (((in Xx0) A0)->((in Xx0) A1))
% Found (fun (Xx0:fofType) (x3:((in Xx0) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) A1)))
% Found x3:((in Xx0) A0)
% Found (fun (x3:((in Xx0) A0))=> x3) as proof of ((in Xx0) A1)
% Found (fun (Xx0:fofType) (x3:((in Xx0) A0))=> x3) as proof of (((in Xx0) A0)->((in Xx0) A1))
% Found (fun (Xx0:fofType) (x3:((in Xx0) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) A1)))
% Found eq_ref00:=(eq_ref0 emptyset):(((eq fofType) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% Found eq_ref00:=(eq_ref0 emptyset):(((eq fofType) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A0)->((in Xx) A0))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A0)->((in Xx) A))
% Found ((eq_ref0 A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) A))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) A))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) A))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (((in Xx) A0)->((in Xx) A))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) A)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A)->((in Xx) A))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found ((eq_ref0 A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found (((eq_ref fofType) A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (((in Xx) A)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A)->((in Xx) A0)))
% Found x3:((in Xx) A0)
% Instantiate: A0:=A:fofType
% Found (fun (x3:((in Xx) A0))=> x3) as proof of ((in Xx) A)
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (((in Xx) A0)->((in Xx) A))
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) A)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A)->((in Xx) A))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found ((eq_ref0 A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found (((eq_ref fofType) A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (((in Xx) A)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A)->((in Xx) A0)))
% Found x3:((in Xx) A00)
% Instantiate: A00:=A:fofType
% Found (fun (x3:((in Xx) A00))=> x3) as proof of ((in Xx) A)
% Found (fun (Xx:fofType) (x3:((in Xx) A00))=> x3) as proof of (((in Xx) A00)->((in Xx) A))
% Found (fun (Xx:fofType) (x3:((in Xx) A00))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A00)->((in Xx) A)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A)->((in Xx) A))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A)->((in Xx) A00))
% Found ((eq_ref0 A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A00))
% Found (((eq_ref fofType) A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A00))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (((in Xx) A)->((in Xx) A00))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A)->((in Xx) A00)))
% Found eq_ref00:=(eq_ref0 emptyset):(((eq fofType) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% Found eq_ref00:=(eq_ref0 emptyset):(((eq fofType) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% 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) b1)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b0)
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b0)
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b0)
% Found x3:((in Xx0) A1)
% Instantiate: A0:=A1:fofType
% Found (fun (x3:((in Xx0) A1))=> x3) as proof of ((in Xx0) A0)
% Found (fun (Xx0:fofType) (x3:((in Xx0) A1))=> x3) as proof of (((in Xx0) A1)->((in Xx0) A0))
% Found (fun (Xx0:fofType) (x3:((in Xx0) A1))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A1)->((in Xx) A0)))
% Found eq_ref000:=(eq_ref00 (in Xx0)):(((in Xx0) A0)->((in Xx0) A0))
% Found (eq_ref00 (in Xx0)) as proof of (((in Xx0) A0)->((in Xx0) A1))
% Found ((eq_ref0 A0) (in Xx0)) as proof of (((in Xx0) A0)->((in Xx0) A1))
% Found (((eq_ref fofType) A0) (in Xx0)) as proof of (((in Xx0) A0)->((in Xx0) A1))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) A0) (in Xx0))) as proof of (((in Xx0) A0)->((in Xx0) A1))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) A0) (in Xx0))) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) A1)))
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) emptyset)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) emptyset)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) emptyset)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) emptyset)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) 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) b00)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b00)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b00)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b00)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b00)
% Found x3:((in Xx0) A0)
% Instantiate: b:=A0:fofType
% Found (fun (x3:((in Xx0) A0))=> x3) as proof of ((in Xx0) b)
% Found (fun (Xx0:fofType) (x3:((in Xx0) A0))=> x3) as proof of (((in Xx0) A0)->((in Xx0) b))
% Found (fun (Xx0:fofType) (x3:((in Xx0) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) b)))
% Found eq_ref000:=(eq_ref00 (in Xx0)):(((in Xx0) b)->((in Xx0) b))
% Found (eq_ref00 (in Xx0)) as proof of (((in Xx0) b)->((in Xx0) A0))
% Found ((eq_ref0 b) (in Xx0)) as proof of (((in Xx0) b)->((in Xx0) A0))
% Found (((eq_ref fofType) b) (in Xx0)) as proof of (((in Xx0) b)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) b) (in Xx0))) as proof of (((in Xx0) b)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) b) (in Xx0))) as proof of (forall (Xx:fofType), (((in Xx) b)->((in Xx) A0)))
% Found x3:((in Xx0) A0)
% Instantiate: A0:=emptyset:fofType
% Found (fun (x3:((in Xx0) A0))=> x3) as proof of ((in Xx0) emptyset)
% Found (fun (Xx0:fofType) (x3:((in Xx0) A0))=> x3) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found (fun (Xx0:fofType) (x3:((in Xx0) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) emptyset)))
% Found eq_ref000:=(eq_ref00 (in Xx0)):(((in Xx0) emptyset)->((in Xx0) emptyset))
% Found (eq_ref00 (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found ((eq_ref0 emptyset) (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (((eq_ref fofType) emptyset) (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) emptyset) (in Xx0))) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) emptyset) (in Xx0))) as proof of (forall (Xx:fofType), (((in Xx) emptyset)->((in Xx) A0)))
% Found eq_ref000:=(eq_ref00 (in Xx0)):(((in Xx0) A0)->((in Xx0) A0))
% Found (eq_ref00 (in Xx0)) as proof of (((in Xx0) A0)->((in Xx0) b))
% Found ((eq_ref0 A0) (in Xx0)) as proof of (((in Xx0) A0)->((in Xx0) b))
% Found (((eq_ref fofType) A0) (in Xx0)) as proof of (((in Xx0) A0)->((in Xx0) b))
% Found (((eq_ref fofType) A0) (in Xx0)) as proof of (((in Xx0) A0)->((in Xx0) b))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) A0) (in Xx0))) as proof of (((in Xx0) A0)->((in Xx0) b))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) A0) (in Xx0))) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) b)))
% Found eq_ref000:=(eq_ref00 (in Xx0)):(((in Xx0) b)->((in Xx0) b))
% Found (eq_ref00 (in Xx0)) as proof of (((in Xx0) b)->((in Xx0) A0))
% Found ((eq_ref0 b) (in Xx0)) as proof of (((in Xx0) b)->((in Xx0) A0))
% Found (((eq_ref fofType) b) (in Xx0)) as proof of (((in Xx0) b)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) b) (in Xx0))) as proof of (((in Xx0) b)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) b) (in Xx0))) as proof of (forall (Xx:fofType), (((in Xx) b)->((in Xx) A0)))
% Found x3:((in Xx0) A0)
% Instantiate: A0:=emptyset:fofType
% Found (fun (x3:((in Xx0) A0))=> x3) as proof of ((in Xx0) emptyset)
% Found (fun (Xx0:fofType) (x3:((in Xx0) A0))=> x3) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found (fun (Xx0:fofType) (x3:((in Xx0) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) emptyset)))
% Found eq_ref000:=(eq_ref00 (in Xx0)):(((in Xx0) emptyset)->((in Xx0) emptyset))
% Found (eq_ref00 (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found ((eq_ref0 emptyset) (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (((eq_ref fofType) emptyset) (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) emptyset) (in Xx0))) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) emptyset) (in Xx0))) as proof of (forall (Xx:fofType), (((in Xx) emptyset)->((in Xx) A0)))
% Found eq_ref000:=(eq_ref00 (in Xx0)):(((in Xx0) A0)->((in Xx0) A0))
% Found (eq_ref00 (in Xx0)) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found ((eq_ref0 A0) (in Xx0)) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found (((eq_ref fofType) A0) (in Xx0)) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found (((eq_ref fofType) A0) (in Xx0)) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) A0) (in Xx0))) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) A0) (in Xx0))) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) emptyset)))
% Found eq_ref000:=(eq_ref00 (in Xx0)):(((in Xx0) emptyset)->((in Xx0) emptyset))
% Found (eq_ref00 (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found ((eq_ref0 emptyset) (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (((eq_ref fofType) emptyset) (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) emptyset) (in Xx0))) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) emptyset) (in Xx0))) as proof of (forall (Xx:fofType), (((in Xx) emptyset)->((in Xx) A0)))
% Found x3:((in Xx0) A0)
% Instantiate: A0:=emptyset:fofType
% Found (fun (x3:((in Xx0) A0))=> x3) as proof of ((in Xx0) emptyset)
% Found (fun (Xx0:fofType) (x3:((in Xx0) A0))=> x3) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found (fun (Xx0:fofType) (x3:((in Xx0) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) emptyset)))
% Found eq_ref000:=(eq_ref00 (in Xx0)):(((in Xx0) emptyset)->((in Xx0) emptyset))
% Found (eq_ref00 (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found ((eq_ref0 emptyset) (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (((eq_ref fofType) emptyset) (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) emptyset) (in Xx0))) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) emptyset) (in Xx0))) as proof of (forall (Xx:fofType), (((in Xx) emptyset)->((in Xx) A0)))
% 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 Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b0)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b0 x)))
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found x3:((in Xx0) A0)
% Instantiate: A0:=emptyset:fofType
% Found (fun (x3:((in Xx0) A0))=> x3) as proof of ((in Xx0) emptyset)
% Found (fun (Xx0:fofType) (x3:((in Xx0) A0))=> x3) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found (fun (Xx0:fofType) (x3:((in Xx0) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) emptyset)))
% Found eq_ref000:=(eq_ref00 (in Xx0)):(((in Xx0) emptyset)->((in Xx0) emptyset))
% Found (eq_ref00 (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found ((eq_ref0 emptyset) (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (((eq_ref fofType) emptyset) (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) emptyset) (in Xx0))) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) emptyset) (in Xx0))) as proof of (forall (Xx:fofType), (((in Xx) emptyset)->((in Xx) A0)))
% Found x3:((in Xx0) A0)
% Instantiate: A0:=emptyset:fofType
% Found (fun (x3:((in Xx0) A0))=> x3) as proof of ((in Xx0) emptyset)
% Found (fun (Xx0:fofType) (x3:((in Xx0) A0))=> x3) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found (fun (Xx0:fofType) (x3:((in Xx0) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) emptyset)))
% Found eq_ref000:=(eq_ref00 (in Xx0)):(((in Xx0) emptyset)->((in Xx0) emptyset))
% Found (eq_ref00 (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found ((eq_ref0 emptyset) (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (((eq_ref fofType) emptyset) (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) emptyset) (in Xx0))) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) emptyset) (in Xx0))) as proof of (forall (Xx:fofType), (((in Xx) emptyset)->((in Xx) A0)))
% 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 Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b0)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b0 x)))
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (fofType->Prop)) b0) (fun (x:fofType)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) 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) a)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) 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) a)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) emptyset)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) emptyset)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) emptyset)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) emptyset)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found x3:(P2 (f x2))
% Instantiate: b:=f:(fofType->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (b x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (b x2)))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found x3:(P2 (f x2))
% Instantiate: b:=f:(fofType->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (b x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (b x2)))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) emptyset)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) emptyset)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) emptyset)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) emptyset)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f0):(((eq (fofType->Prop)) f0) (fun (x:fofType)=> (f0 x)))
% Found (eta_expansion_dep00 f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found eq_ref00:=(eq_ref0 f0):(((eq (fofType->Prop)) f0) f0)
% Found (eq_ref0 f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found eq_ref00:=(eq_ref0 f):(((eq (fofType->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (fofType->Prop)) f) (fun (x:fofType)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found 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 (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 (f1 x2)):(((eq Prop) (f1 x2)) (f1 x2))
% Found (eq_ref0 (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x2)):(((eq Prop) (f1 x2)) (f1 x2))
% Found (eq_ref0 (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found eq_ref00:=(eq_ref0 f0):(((eq (fofType->Prop)) f0) f0)
% Found (eq_ref0 f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found eta_expansion000:=(eta_expansion00 f0):(((eq (fofType->Prop)) f0) (fun (x:fofType)=> (f0 x)))
% Found (eta_expansion00 f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eta_expansion0 Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found eta_expansion000:=(eta_expansion00 f0):(((eq (fofType->Prop)) f0) (fun (x:fofType)=> (f0 x)))
% Found (eta_expansion00 f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eta_expansion0 Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found eta_expansion000:=(eta_expansion00 f0):(((eq (fofType->Prop)) f0) (fun (x:fofType)=> (f0 x)))
% Found (eta_expansion00 f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eta_expansion0 Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) 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) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (fofType->Prop)) f) (fun (x:fofType)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (fofType->Prop)) f) (fun (x:fofType)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found eq_ref00:=(eq_ref0 f):(((eq (fofType->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found eq_ref00:=(eq_ref0 f):(((eq (fofType->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% 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 (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 (f1 x2)):(((eq Prop) (f1 x2)) (f1 x2))
% Found (eq_ref0 (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x2)):(((eq Prop) (f1 x2)) (f1 x2))
% Found (eq_ref0 (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x2)):(((eq Prop) (f1 x2)) (f1 x2))
% Found (eq_ref0 (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x2)):(((eq Prop) (f1 x2)) (f1 x2))
% Found (eq_ref0 (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b0 x)))
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found eq_ref00:=(eq_ref0 f0):(((eq (fofType->Prop)) f0) f0)
% Found (eq_ref0 f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found eta_expansion000:=(eta_expansion00 f0):(((eq (fofType->Prop)) f0) (fun (x:fofType)=> (f0 x)))
% Found (eta_expansion00 f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eta_expansion0 Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found eq_ref00:=(eq_ref0 (f1 x2)):(((eq Prop) (f1 x2)) (f1 x2))
% Found (eq_ref0 (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x2)):(((eq Prop) (f1 x2)) (f1 x2))
% Found (eq_ref0 (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b0 x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) A0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) A0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) A0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) A0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) A0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) A0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) A0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) A0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) A0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) A0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) A0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) A0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) A0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) A0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) A0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) A0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% 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) b1)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A0)->((in Xx) A0))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A0)->((in Xx) a))
% Found ((eq_ref0 A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) a))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) a))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) a))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (((in Xx) A0)->((in Xx) a))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) a)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) a)->((in Xx) a))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) a)->((in Xx) A0))
% Found ((eq_ref0 a) (in Xx)) as proof of (((in Xx) a)->((in Xx) A0))
% Found (((eq_ref fofType) a) (in Xx)) as proof of (((in Xx) a)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) a) (in Xx))) as proof of (((in Xx) a)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) a) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) a)->((in Xx) A0)))
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found x3:(P2 (f x2))
% Instantiate: f0:=f:(fofType->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (f0 x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (f0 x2)))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:fofType), (((eq Prop) (f x)) (f0 x)))
% Found x3:(P2 (f x2))
% Instantiate: f0:=f:(fofType->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (f0 x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (f0 x2)))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:fofType), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A0)->((in Xx) A0))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A0)->((in Xx) a))
% Found ((eq_ref0 A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) a))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) a))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) a))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (((in Xx) A0)->((in Xx) a))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) a)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) a)->((in Xx) a))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) a)->((in Xx) A0))
% Found ((eq_ref0 a) (in Xx)) as proof of (((in Xx) a)->((in Xx) A0))
% Found (((eq_ref fofType) a) (in Xx)) as proof of (((in Xx) a)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) a) (in Xx))) as proof of (((in Xx) a)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) a) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) a)->((in Xx) A0)))
% Found x3:((in Xx) A0)
% Instantiate: A0:=A:fofType
% Found (fun (x3:((in Xx) A0))=> x3) as proof of ((in Xx) a)
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (((in Xx) A0)->((in Xx) a))
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) a)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) a)->((in Xx) a))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) a)->((in Xx) A0))
% Found ((eq_ref0 a) (in Xx)) as proof of (((in Xx) a)->((in Xx) A0))
% Found (((eq_ref fofType) a) (in Xx)) as proof of (((in Xx) a)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) a) (in Xx))) as proof of (((in Xx) a)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) a) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) a)->((in Xx) A0)))
% Found eq_ref00:=(eq_ref0 emptyset):(((eq fofType) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq fofType) emptyset) b0)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b0)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b0)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b0)
% Found x3:((in Xx) A0)
% Instantiate: a:=A0:fofType
% Found (fun (x3:((in Xx) A0))=> x3) as proof of ((in Xx) a)
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (((in Xx) A0)->((in Xx) a))
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) a)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) a)->((in Xx) a))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) a)->((in Xx) A0))
% Found ((eq_ref0 a) (in Xx)) as proof of (((in Xx) a)->((in Xx) A0))
% Found (((eq_ref fofType) a) (in Xx)) as proof of (((in Xx) a)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) a) (in Xx))) as proof of (((in Xx) a)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) a) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) a)->((in Xx) A0)))
% Found x3:((in Xx) A0)
% Instantiate: a:=A0:fofType
% Found (fun (x3:((in Xx) A0))=> x3) as proof of ((in Xx) a)
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (((in Xx) A0)->((in Xx) a))
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) a)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) a)->((in Xx) a))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) a)->((in Xx) A0))
% Found ((eq_ref0 a) (in Xx)) as proof of (((in Xx) a)->((in Xx) A0))
% Found (((eq_ref fofType) a) (in Xx)) as proof of (((in Xx) a)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) a) (in Xx))) as proof of (((in Xx) a)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) a) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) a)->((in Xx) A0)))
% Found x3:((in Xx) A0)
% Instantiate: A0:=A:fofType
% Found (fun (x3:((in Xx) A0))=> x3) as proof of ((in Xx) a)
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (((in Xx) A0)->((in Xx) a))
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) a)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) a)->((in Xx) a))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) a)->((in Xx) A0))
% Found ((eq_ref0 a) (in Xx)) as proof of (((in Xx) a)->((in Xx) A0))
% Found (((eq_ref fofType) a) (in Xx)) as proof of (((in Xx) a)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) a) (in Xx))) as proof of (((in Xx) a)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) a) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) a)->((in Xx) A0)))
% Found eq_ref00:=(eq_ref0 f):(((eq (fofType->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (fofType->Prop)) f) (fun (x:fofType)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found eq_ref00:=(eq_ref0 f):(((eq (fofType->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of b0
% Found I:True
% Found I as proof of b0
% Found choice_operator:=(fun (A:Type) (a:A)=> ((((classical_choice (A->Prop)) A) (fun (x3:(A->Prop))=> x3)) a)):(forall (A:Type), (A->((ex ((A->Prop)->A)) (fun (co:((A->Prop)->A))=> (forall (P:(A->Prop)), (((ex A) (fun (x:A)=> (P x)))->(P (co P))))))))
% Instantiate: b:=(forall (A:Type), (A->((ex ((A->Prop)->A)) (fun (co:((A->Prop)->A))=> (forall (P:(A->Prop)), (((ex A) (fun (x:A)=> (P x)))->(P (co P)))))))):Prop
% Found choice_operator as proof of b
% Found choice_operator:=(fun (A:Type) (a:A)=> ((((classical_choice (A->Prop)) A) (fun (x3:(A->Prop))=> x3)) a)):(forall (A:Type), (A->((ex ((A->Prop)->A)) (fun (co:((A->Prop)->A))=> (forall (P:(A->Prop)), (((ex A) (fun (x:A)=> (P x)))->(P (co P))))))))
% Instantiate: b:=(forall (A:Type), (A->((ex ((A->Prop)->A)) (fun (co:((A->Prop)->A))=> (forall (P:(A->Prop)), (((ex A) (fun (x:A)=> (P x)))->(P (co P)))))))):Prop
% Found choice_operator as proof of b
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found I:True
% Found I as proof of True
% Found I:True
% Found I as proof of True
% Found I:True
% Found I as proof of True
% Found I:True
% Found I as proof of True
% Found I:True
% Found I as proof of b
% Found I:True
% Found I as proof of b
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found x3:(P2 (f x2))
% Instantiate: a:=f:(fofType->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (a x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (a x2)))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found x3:(P2 (f x2))
% Instantiate: a:=f:(fofType->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (a x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (a x2)))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found x3:(P2 (f x2))
% Instantiate: a:=f:(fofType->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (a x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (a x2)))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found x3:(P2 (f x2))
% Instantiate: a:=f:(fofType->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (a x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (a x2)))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found eq_ref00:=(eq_ref0 A0):(((eq fofType) A0) A0)
% Found (eq_ref0 A0) as proof of (((eq fofType) A0) b)
% Found ((eq_ref fofType) A0) as proof of (((eq fofType) A0) b)
% Found ((eq_ref fofType) A0) as proof of (((eq fofType) A0) b)
% Found ((eq_ref fofType) A0) as proof of (((eq fofType) A0) b)
% Found eq_ref00:=(eq_ref0 A0):(((eq fofType) A0) A0)
% Found (eq_ref0 A0) as proof of (((eq fofType) A0) b)
% Found ((eq_ref fofType) A0) as proof of (((eq fofType) A0) b)
% Found ((eq_ref fofType) A0) as proof of (((eq fofType) A0) b)
% Found ((eq_ref fofType) A0) as proof of (((eq fofType) A0) b)
% Found eq_ref00:=(eq_ref0 emptyset):(((eq fofType) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq fofType) emptyset) b0)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b0)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b0)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b0)
% Found I:True
% Found I as proof of True
% Found I:True
% Found I as proof of True
% Found eq_ref00:=(eq_ref0 emptyset):(((eq fofType) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq fofType) emptyset) b0)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b0)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b0)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b0)
% Found x3:((in Xx0) A1)
% Instantiate: A0:=A1:fofType
% Found (fun (x3:((in Xx0) A1))=> x3) as proof of ((in Xx0) A0)
% Found (fun (Xx0:fofType) (x3:((in Xx0) A1))=> x3) as proof of (((in Xx0) A1)->((in Xx0) A0))
% Found (fun (Xx0:fofType) (x3:((in Xx0) A1))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A1)->((in Xx) A0)))
% Found x3:((in Xx0) A0)
% Instantiate: A1:=A0:fofType
% Found (fun (x3:((in Xx0) A0))=> x3) as proof of ((in Xx0) A1)
% Found (fun (Xx0:fofType) (x3:((in Xx0) A0))=> x3) as proof of (((in Xx0) A0)->((in Xx0) A1))
% Found (fun (Xx0:fofType) (x3:((in Xx0) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) A1)))
% Found x3:((in Xx0) A0)
% Found (fun (x3:((in Xx0) A0))=> x3) as proof of ((in Xx0) A1)
% Found (fun (Xx0:fofType) (x3:((in Xx0) A0))=> x3) as proof of (((in Xx0) A0)->((in Xx0) A1))
% Found (fun (Xx0:fofType) (x3:((in Xx0) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) A1)))
% Found eq_ref00:=(eq_ref0 emptyset):(((eq fofType) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% Found eq_ref00:=(eq_ref0 emptyset):(((eq fofType) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% Found eq_ref00:=(eq_ref0 emptyset):(((eq fofType) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% Found eq_ref00:=(eq_ref0 emptyset):(((eq fofType) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b)
% Found eq_ref00:=(eq_ref0 x2):(((eq fofType) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq fofType) x2) b0)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b0)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b0)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b0)
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b0)
% Found eq_ref000:=(eq_ref00 (in Xx0)):(((in Xx0) A1)->((in Xx0) A1))
% Found (eq_ref00 (in Xx0)) as proof of (((in Xx0) A1)->((in Xx0) A0))
% Found ((eq_ref0 A1) (in Xx0)) as proof of (((in Xx0) A1)->((in Xx0) A0))
% Found (((eq_ref fofType) A1) (in Xx0)) as proof of (((in Xx0) A1)->((in Xx0) A0))
% Found (((eq_ref fofType) A1) (in Xx0)) as proof of (((in Xx0) A1)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) A1) (in Xx0))) as proof of (((in Xx0) A1)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) A1) (in Xx0))) as proof of (forall (Xx:fofType), (((in Xx) A1)->((in Xx) A0)))
% Found eq_ref000:=(eq_ref00 (in Xx0)):(((in Xx0) A0)->((in Xx0) A0))
% Found (eq_ref00 (in Xx0)) as proof of (((in Xx0) A0)->((in Xx0) A1))
% Found ((eq_ref0 A0) (in Xx0)) as proof of (((in Xx0) A0)->((in Xx0) A1))
% Found (((eq_ref fofType) A0) (in Xx0)) as proof of (((in Xx0) A0)->((in Xx0) A1))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) A0) (in Xx0))) as proof of (((in Xx0) A0)->((in Xx0) A1))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) A0) (in Xx0))) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) A1)))
% Found eq_ref00:=(eq_ref0 x2):(((eq fofType) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq fofType) x2) b0)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b0)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b0)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b0)
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b0)
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b0)
% Found eq_ref000:=(eq_ref00 (in Xx0)):(((in Xx0) A0)->((in Xx0) A0))
% Found (eq_ref00 (in Xx0)) as proof of (((in Xx0) A0)->((in Xx0) b))
% Found ((eq_ref0 A0) (in Xx0)) as proof of (((in Xx0) A0)->((in Xx0) b))
% Found (((eq_ref fofType) A0) (in Xx0)) as proof of (((in Xx0) A0)->((in Xx0) b))
% Found (((eq_ref fofType) A0) (in Xx0)) as proof of (((in Xx0) A0)->((in Xx0) b))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) A0) (in Xx0))) as proof of (((in Xx0) A0)->((in Xx0) b))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) A0) (in Xx0))) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) b)))
% Found eq_ref000:=(eq_ref00 (in Xx0)):(((in Xx0) b)->((in Xx0) b))
% Found (eq_ref00 (in Xx0)) as proof of (((in Xx0) b)->((in Xx0) A0))
% Found ((eq_ref0 b) (in Xx0)) as proof of (((in Xx0) b)->((in Xx0) A0))
% Found (((eq_ref fofType) b) (in Xx0)) as proof of (((in Xx0) b)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) b) (in Xx0))) as proof of (((in Xx0) b)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) b) (in Xx0))) as proof of (forall (Xx:fofType), (((in Xx) b)->((in Xx) A0)))
% Found x3:((in Xx0) A0)
% Instantiate: A0:=emptyset:fofType
% Found (fun (x3:((in Xx0) A0))=> x3) as proof of ((in Xx0) emptyset)
% Found (fun (Xx0:fofType) (x3:((in Xx0) A0))=> x3) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found (fun (Xx0:fofType) (x3:((in Xx0) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) emptyset)))
% Found eq_ref000:=(eq_ref00 (in Xx0)):(((in Xx0) emptyset)->((in Xx0) emptyset))
% Found (eq_ref00 (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found ((eq_ref0 emptyset) (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (((eq_ref fofType) emptyset) (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) emptyset) (in Xx0))) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) emptyset) (in Xx0))) as proof of (forall (Xx:fofType), (((in Xx) emptyset)->((in Xx) A0)))
% Found x3:((in Xx0) A00)
% Instantiate: A00:=emptyset:fofType
% Found (fun (x3:((in Xx0) A00))=> x3) as proof of ((in Xx0) emptyset)
% Found (fun (Xx0:fofType) (x3:((in Xx0) A00))=> x3) as proof of (((in Xx0) A00)->((in Xx0) emptyset))
% Found (fun (Xx0:fofType) (x3:((in Xx0) A00))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A00)->((in Xx) emptyset)))
% Found x3:((in Xx0) emptyset)
% Instantiate: A00:=emptyset:fofType
% Found (fun (x3:((in Xx0) emptyset))=> x3) as proof of ((in Xx0) A00)
% Found (fun (Xx0:fofType) (x3:((in Xx0) emptyset))=> x3) as proof of (((in Xx0) emptyset)->((in Xx0) A00))
% Found (fun (Xx0:fofType) (x3:((in Xx0) emptyset))=> x3) as proof of (forall (Xx:fofType), (((in Xx) emptyset)->((in Xx) A00)))
% Found x3:((in Xx0) A00)
% Instantiate: A00:=emptyset:fofType
% Found (fun (x3:((in Xx0) A00))=> x3) as proof of ((in Xx0) emptyset)
% Found (fun (Xx0:fofType) (x3:((in Xx0) A00))=> x3) as proof of (((in Xx0) A00)->((in Xx0) emptyset))
% Found (fun (Xx0:fofType) (x3:((in Xx0) A00))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A00)->((in Xx) emptyset)))
% Found x3:((in Xx0) emptyset)
% Instantiate: A00:=emptyset:fofType
% Found (fun (x3:((in Xx0) emptyset))=> x3) as proof of ((in Xx0) A00)
% Found (fun (Xx0:fofType) (x3:((in Xx0) emptyset))=> x3) as proof of (((in Xx0) emptyset)->((in Xx0) A00))
% Found (fun (Xx0:fofType) (x3:((in Xx0) emptyset))=> x3) as proof of (forall (Xx:fofType), (((in Xx) emptyset)->((in Xx) A00)))
% Found eq_ref000:=(eq_ref00 (in Xx0)):(((in Xx0) A0)->((in Xx0) A0))
% Found (eq_ref00 (in Xx0)) as proof of (((in Xx0) A0)->((in Xx0) b))
% Found ((eq_ref0 A0) (in Xx0)) as proof of (((in Xx0) A0)->((in Xx0) b))
% Found (((eq_ref fofType) A0) (in Xx0)) as proof of (((in Xx0) A0)->((in Xx0) b))
% Found (((eq_ref fofType) A0) (in Xx0)) as proof of (((in Xx0) A0)->((in Xx0) b))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) A0) (in Xx0))) as proof of (((in Xx0) A0)->((in Xx0) b))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) A0) (in Xx0))) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) b)))
% Found eq_ref000:=(eq_ref00 (in Xx0)):(((in Xx0) b)->((in Xx0) b))
% Found (eq_ref00 (in Xx0)) as proof of (((in Xx0) b)->((in Xx0) A0))
% Found ((eq_ref0 b) (in Xx0)) as proof of (((in Xx0) b)->((in Xx0) A0))
% Found (((eq_ref fofType) b) (in Xx0)) as proof of (((in Xx0) b)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) b) (in Xx0))) as proof of (((in Xx0) b)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) b) (in Xx0))) as proof of (forall (Xx:fofType), (((in Xx) b)->((in Xx) A0)))
% Found x3:((in Xx0) A0)
% Instantiate: A0:=emptyset:fofType
% Found (fun (x3:((in Xx0) A0))=> x3) as proof of ((in Xx0) emptyset)
% Found (fun (Xx0:fofType) (x3:((in Xx0) A0))=> x3) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found (fun (Xx0:fofType) (x3:((in Xx0) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) emptyset)))
% Found eq_ref000:=(eq_ref00 (in Xx0)):(((in Xx0) emptyset)->((in Xx0) emptyset))
% Found (eq_ref00 (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found ((eq_ref0 emptyset) (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (((eq_ref fofType) emptyset) (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) emptyset) (in Xx0))) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) emptyset) (in Xx0))) as proof of (forall (Xx:fofType), (((in Xx) emptyset)->((in Xx) A0)))
% Found eq_ref000:=(eq_ref00 (in Xx0)):(((in Xx0) A0)->((in Xx0) A0))
% Found (eq_ref00 (in Xx0)) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found ((eq_ref0 A0) (in Xx0)) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found (((eq_ref fofType) A0) (in Xx0)) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found (((eq_ref fofType) A0) (in Xx0)) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) A0) (in Xx0))) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) A0) (in Xx0))) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) emptyset)))
% Found eq_ref000:=(eq_ref00 (in Xx0)):(((in Xx0) emptyset)->((in Xx0) emptyset))
% Found (eq_ref00 (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found ((eq_ref0 emptyset) (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (((eq_ref fofType) emptyset) (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) emptyset) (in Xx0))) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) emptyset) (in Xx0))) as proof of (forall (Xx:fofType), (((in Xx) emptyset)->((in Xx) A0)))
% Found x3:((in Xx0) A0)
% Instantiate: A0:=emptyset:fofType
% Found (fun (x3:((in Xx0) A0))=> x3) as proof of ((in Xx0) emptyset)
% Found (fun (Xx0:fofType) (x3:((in Xx0) A0))=> x3) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found (fun (Xx0:fofType) (x3:((in Xx0) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) emptyset)))
% Found eq_ref000:=(eq_ref00 (in Xx0)):(((in Xx0) emptyset)->((in Xx0) emptyset))
% Found (eq_ref00 (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found ((eq_ref0 emptyset) (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (((eq_ref fofType) emptyset) (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) emptyset) (in Xx0))) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) emptyset) (in Xx0))) as proof of (forall (Xx:fofType), (((in Xx) emptyset)->((in Xx) A0)))
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b0 x)))
% Found eq_ref000:=(eq_ref00 (in Xx0)):(((in Xx0) A0)->((in Xx0) A0))
% Found (eq_ref00 (in Xx0)) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found ((eq_ref0 A0) (in Xx0)) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found (((eq_ref fofType) A0) (in Xx0)) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found (((eq_ref fofType) A0) (in Xx0)) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) A0) (in Xx0))) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) A0) (in Xx0))) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) emptyset)))
% Found eq_ref000:=(eq_ref00 (in Xx0)):(((in Xx0) emptyset)->((in Xx0) emptyset))
% Found (eq_ref00 (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found ((eq_ref0 emptyset) (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (((eq_ref fofType) emptyset) (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) emptyset) (in Xx0))) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) emptyset) (in Xx0))) as proof of (forall (Xx:fofType), (((in Xx) emptyset)->((in Xx) A0)))
% Found x3:((in Xx0) A0)
% Instantiate: A0:=emptyset:fofType
% Found (fun (x3:((in Xx0) A0))=> x3) as proof of ((in Xx0) emptyset)
% Found (fun (Xx0:fofType) (x3:((in Xx0) A0))=> x3) as proof of (((in Xx0) A0)->((in Xx0) emptyset))
% Found (fun (Xx0:fofType) (x3:((in Xx0) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) emptyset)))
% Found eq_ref000:=(eq_ref00 (in Xx0)):(((in Xx0) emptyset)->((in Xx0) emptyset))
% Found (eq_ref00 (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found ((eq_ref0 emptyset) (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (((eq_ref fofType) emptyset) (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) emptyset) (in Xx0))) as proof of (((in Xx0) emptyset)->((in Xx0) A0))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) emptyset) (in Xx0))) as proof of (forall (Xx:fofType), (((in Xx) emptyset)->((in Xx) A0)))
% 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 Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) 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) a)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b0 x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:fofType)=> ((and ((in Xx) A)) True))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) A)) True))) (fun (x:fofType)=> ((and ((in x) A)) True)))
% Found (eta_expansion_dep00 (fun (Xx:fofType)=> ((and ((in Xx) A)) True))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) A)) True))) b0)
% Found ((eta_expansion_dep0 (fun (x4:fofType)=> Prop)) (fun (Xx:fofType)=> ((and ((in Xx) A)) True))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) A)) True))) b0)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) (fun (Xx:fofType)=> ((and ((in Xx) A)) True))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) A)) True))) b0)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) (fun (Xx:fofType)=> ((and ((in Xx) A)) True))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) A)) True))) b0)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) (fun (Xx:fofType)=> ((and ((in Xx) A)) True))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) A)) True))) 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 Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) 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) a)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (fofType->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found eq_ref000:=(eq_ref00 (in Xx0)):(((in Xx0) A00)->((in Xx0) A00))
% Found (eq_ref00 (in Xx0)) as proof of (((in Xx0) A00)->((in Xx0) emptyset))
% Found ((eq_ref0 A00) (in Xx0)) as proof of (((in Xx0) A00)->((in Xx0) emptyset))
% Found (((eq_ref fofType) A00) (in Xx0)) as proof of (((in Xx0) A00)->((in Xx0) emptyset))
% Found (((eq_ref fofType) A00) (in Xx0)) as proof of (((in Xx0) A00)->((in Xx0) emptyset))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) A00) (in Xx0))) as proof of (((in Xx0) A00)->((in Xx0) emptyset))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) A00) (in Xx0))) as proof of (forall (Xx:fofType), (((in Xx) A00)->((in Xx) emptyset)))
% Found eq_ref000:=(eq_ref00 (in Xx0)):(((in Xx0) emptyset)->((in Xx0) emptyset))
% Found (eq_ref00 (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A00))
% Found ((eq_ref0 emptyset) (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A00))
% Found (((eq_ref fofType) emptyset) (in Xx0)) as proof of (((in Xx0) emptyset)->((in Xx0) A00))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) emptyset) (in Xx0))) as proof of (((in Xx0) emptyset)->((in Xx0) A00))
% Found (fun (Xx0:fofType)=> (((eq_ref fofType) emptyset) (in Xx0))) as proof of (forall (Xx:fofType), (((in Xx) emptyset)->((in Xx) A00)))
% Found x3:((in Xx) x2)
% Instantiate: A0:=x2:fofType
% Found (fun (x3:((in Xx) x2))=> x3) as proof of ((in Xx) A0)
% Found (fun (Xx:fofType) (x3:((in Xx) x2))=> x3) as proof of (((in Xx) x2)->((in Xx) A0))
% Found (fun (Xx:fofType) (x3:((in Xx) x2))=> x3) as proof of (forall (Xx:fofType), (((in Xx) x2)->((in Xx) A0)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A0)->((in Xx) A0))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A0)->((in Xx) x2))
% Found ((eq_ref0 A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) x2))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (((in Xx) A0)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) x2)))
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: a:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of a
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of b0
% Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) x2)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) x2)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) x2)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) x2)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) x2)->((in Xx) x2))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) x2)->((in Xx) A0))
% Found ((eq_ref0 x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) A0))
% Found (((eq_ref fofType) x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) A0))
% Found (((eq_ref fofType) x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))) as proof of (((in Xx) x2)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) x2)->((in Xx) A0)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A0)->((in Xx) A0))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A0)->((in Xx) x2))
% Found ((eq_ref0 A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) x2))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (((in Xx) A0)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) x2)))
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of b0
% Found x0:setext
% Instantiate: b:=(forall (A:fofType) (B:fofType), ((forall (Xx:fofType), (((in Xx) A)->((in Xx) B)))->((forall (Xx:fofType), (((in Xx) B)->((in Xx) A)))->(((eq fofType) A) B)))):Prop
% Found x0 as proof of a
% Found eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of b0
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: a:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of a
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of b0
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: a:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of a
% 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 (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found I:True
% Found I as proof of b0
% Found I:True
% Found I as proof of a
% Found x0:setext
% Instantiate: b:=(forall (A:fofType) (B:fofType), ((forall (Xx:fofType), (((in Xx) A)->((in Xx) B)))->((forall (Xx:fofType), (((in Xx) B)->((in Xx) A)))->(((eq fofType) A) B)))):Prop
% Found x0 as proof of 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) b1)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: a:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of a
% 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) b1)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) x2)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x2)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x2)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x2)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) x2)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x2)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x2)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x2)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) 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) f)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% 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) f)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) f)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) f)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) f)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) f)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found eq_ref00:=(eq_ref0 A):(((eq fofType) A) A)
% Found (eq_ref0 A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found eq_ref00:=(eq_ref0 A):(((eq fofType) A) A)
% Found (eq_ref0 A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found eq_ref00:=(eq_ref0 A):(((eq fofType) A) A)
% Found (eq_ref0 A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found eq_ref00:=(eq_ref0 A):(((eq fofType) A) A)
% Found (eq_ref0 A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) 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) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 A):(((eq fofType) A) A)
% Found (eq_ref0 A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% 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 (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eta_expansion000:=(eta_expansion00 f0):(((eq (fofType->Prop)) f0) (fun (x:fofType)=> (f0 x)))
% Found (eta_expansion00 f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eta_expansion0 Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found eq_ref00:=(eq_ref0 f0):(((eq (fofType->Prop)) f0) f0)
% Found (eq_ref0 f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found eta_expansion000:=(eta_expansion00 f0):(((eq (fofType->Prop)) f0) (fun (x:fofType)=> (f0 x)))
% Found (eta_expansion00 f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eta_expansion0 Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found eq_ref00:=(eq_ref0 f0):(((eq (fofType->Prop)) f0) f0)
% Found (eq_ref0 f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) 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) b00)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b00)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b00)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b00)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b00)
% Found eq_ref00:=(eq_ref0 f):(((eq (fofType->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found eq_ref00:=(eq_ref0 f):(((eq (fofType->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (fofType->Prop)) f) (fun (x:fofType)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% 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 (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% 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 (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (fofType->Prop)) f) (fun (x:fofType)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% 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 (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 (f1 x2)):(((eq Prop) (f1 x2)) (f1 x2))
% Found (eq_ref0 (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x2)):(((eq Prop) (f1 x2)) (f1 x2))
% Found (eq_ref0 (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 A):(((eq fofType) A) A)
% Found (eq_ref0 A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found ((eq_ref fofType) A) as proof of (((eq fofType) A) b0)
% Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% 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) f)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found eta_expansion000:=(eta_expansion00 f0):(((eq (fofType->Prop)) f0) (fun (x:fofType)=> (f0 x)))
% Found (eta_expansion00 f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eta_expansion0 Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found eq_ref00:=(eq_ref0 f0):(((eq (fofType->Prop)) f0) f0)
% Found (eq_ref0 f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found eta_expansion000:=(eta_expansion00 f0):(((eq (fofType->Prop)) f0) (fun (x:fofType)=> (f0 x)))
% Found (eta_expansion00 f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eta_expansion0 Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f0):(((eq (fofType->Prop)) f0) (fun (x:fofType)=> (f0 x)))
% Found (eta_expansion_dep00 f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found eq_ref00:=(eq_ref0 f0):(((eq (fofType->Prop)) f0) f0)
% Found (eq_ref0 f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found eta_expansion000:=(eta_expansion00 f0):(((eq (fofType->Prop)) f0) (fun (x:fofType)=> (f0 x)))
% Found (eta_expansion00 f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eta_expansion0 Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found eta_expansion000:=(eta_expansion00 f0):(((eq (fofType->Prop)) f0) (fun (x:fofType)=> (f0 x)))
% Found (eta_expansion00 f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eta_expansion0 Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found eta_expansion000:=(eta_expansion00 f0):(((eq (fofType->Prop)) f0) (fun (x:fofType)=> (f0 x)))
% Found (eta_expansion00 f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eta_expansion0 Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) 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) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 f):(((eq (fofType->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (fofType->Prop)) f) (fun (x:fofType)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (fofType->Prop)) f) (fun (x:fofType)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (fofType->Prop)) f) (fun (x:fofType)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (fofType->Prop)) f) (fun (x:fofType)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (fofType->Prop)) f) (fun (x:fofType)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (fofType->Prop)) f) (fun (x:fofType)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (fofType->Prop)) f) (fun (x:fofType)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% 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 (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found 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 (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 (f1 x2)):(((eq Prop) (f1 x2)) (f1 x2))
% Found (eq_ref0 (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x2)):(((eq Prop) (f1 x2)) (f1 x2))
% Found (eq_ref0 (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x2)):(((eq Prop) (f1 x2)) (f1 x2))
% Found (eq_ref0 (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x2)):(((eq Prop) (f1 x2)) (f1 x2))
% Found (eq_ref0 (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b0 x)))
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found eta_expansion000:=(eta_expansion00 f0):(((eq (fofType->Prop)) f0) (fun (x:fofType)=> (f0 x)))
% Found (eta_expansion00 f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eta_expansion0 Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found eq_ref00:=(eq_ref0 f0):(((eq (fofType->Prop)) f0) f0)
% Found (eq_ref0 f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f0):(((eq (fofType->Prop)) f0) (fun (x:fofType)=> (f0 x)))
% Found (eta_expansion_dep00 f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f0):(((eq (fofType->Prop)) f0) (fun (x:fofType)=> (f0 x)))
% Found (eta_expansion_dep00 f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% Found eq_ref00:=(eq_ref0 (f1 x2)):(((eq Prop) (f1 x2)) (f1 x2))
% Found (eq_ref0 (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x2)):(((eq Prop) (f1 x2)) (f1 x2))
% Found (eq_ref0 (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b0 x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b0 x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b0 x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) A0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) A0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) A0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) A0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (b x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) A0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) A0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) A0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) A0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found x3:((in Xx) A0)
% Instantiate: A0:=A:fofType
% Found (fun (x3:((in Xx) A0))=> x3) as proof of ((in Xx) A)
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (((in Xx) A0)->((in Xx) A))
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) A)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A)->((in Xx) A))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found ((eq_ref0 A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found (((eq_ref fofType) A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (((in Xx) A)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A)->((in Xx) A0)))
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) A)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) A)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) A)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) A)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A0)->((in Xx) A0))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A0)->((in Xx) A))
% Found ((eq_ref0 A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) A))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) A))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) A))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (((in Xx) A0)->((in Xx) A))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) A)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A)->((in Xx) A))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found ((eq_ref0 A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found (((eq_ref fofType) A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (((in Xx) A)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A)->((in Xx) A0)))
% Found x3:((in Xx) A0)
% Instantiate: A0:=A:fofType
% Found (fun (x3:((in Xx) A0))=> x3) as proof of ((in Xx) A)
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (((in Xx) A0)->((in Xx) A))
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) A)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A)->((in Xx) A))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found ((eq_ref0 A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found (((eq_ref fofType) A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (((in Xx) A)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A)->((in Xx) A0)))
% Found x3:((in Xx) A0)
% Instantiate: A0:=A:fofType
% Found (fun (x3:((in Xx) A0))=> x3) as proof of ((in Xx) A)
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (((in Xx) A0)->((in Xx) A))
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) A)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A)->((in Xx) A))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found ((eq_ref0 A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found (((eq_ref fofType) A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (((in Xx) A)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A)->((in Xx) A0)))
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) A0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) A0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) A0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) A0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) A)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) A)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) A)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) A)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) A)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) A)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) A)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) A)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A0)->((in Xx) A0))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A0)->((in Xx) A))
% Found ((eq_ref0 A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) A))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) A))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) A))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (((in Xx) A0)->((in Xx) A))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) A)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A)->((in Xx) A))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found ((eq_ref0 A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found (((eq_ref fofType) A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (((in Xx) A)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A)->((in Xx) A0)))
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) A0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) A0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) A0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) A0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% 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) b1)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A0)->((in Xx) A0))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A0)->((in Xx) A))
% Found ((eq_ref0 A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) A))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) A))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) A))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (((in Xx) A0)->((in Xx) A))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) A)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A)->((in Xx) A))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found ((eq_ref0 A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found (((eq_ref fofType) A) (in Xx)) as proof of (((in Xx) A)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (((in Xx) A)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A)->((in Xx) A0)))
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found x3:((in Xx) A0)
% Instantiate: a:=A0:fofType
% Found (fun (x3:((in Xx) A0))=> x3) as proof of ((in Xx) a)
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (((in Xx) A0)->((in Xx) a))
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) a)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) a)->((in Xx) a))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) a)->((in Xx) A0))
% Found ((eq_ref0 a) (in Xx)) as proof of (((in Xx) a)->((in Xx) A0))
% Found (((eq_ref fofType) a) (in Xx)) as proof of (((in Xx) a)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) a) (in Xx))) as proof of (((in Xx) a)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) a) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) a)->((in Xx) A0)))
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) A)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) A)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) A)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) A)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found x3:((in Xx) A0)
% Instantiate: A0:=A:fofType
% Found (fun (x3:((in Xx) A0))=> x3) as proof of ((in Xx) a)
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (((in Xx) A0)->((in Xx) a))
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) a)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) a)->((in Xx) a))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) a)->((in Xx) A0))
% Found ((eq_ref0 a) (in Xx)) as proof of (((in Xx) a)->((in Xx) A0))
% Found (((eq_ref fofType) a) (in Xx)) as proof of (((in Xx) a)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) a) (in Xx))) as proof of (((in Xx) a)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) a) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) a)->((in Xx) A0)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A0)->((in Xx) A0))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A0)->((in Xx) a))
% Found ((eq_ref0 A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) a))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) a))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) a))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (((in Xx) A0)->((in Xx) a))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) a)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) a)->((in Xx) a))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) a)->((in Xx) A0))
% Found ((eq_ref0 a) (in Xx)) as proof of (((in Xx) a)->((in Xx) A0))
% Found (((eq_ref fofType) a) (in Xx)) as proof of (((in Xx) a)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) a) (in Xx))) as proof of (((in Xx) a)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) a) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) a)->((in Xx) A0)))
% Found eq_ref00:=(eq_ref0 emptyset):(((eq fofType) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq fofType) emptyset) b0)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b0)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b0)
% Found ((eq_ref fofType) emptyset) as proof of (((eq fofType) emptyset) b0)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A0)->((in Xx) A0))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A0)->((in Xx) a))
% Found ((eq_ref0 A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) a))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) a))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) a))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (((in Xx) A0)->((in Xx) a))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) a)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) a)->((in Xx) a))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) a)->((in Xx) A0))
% Found ((eq_ref0 a) (in Xx)) as proof of (((in Xx) a)->((in Xx) A0))
% Found (((eq_ref fofType) a) (in Xx)) as proof of (((in Xx) a)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) a) (in Xx))) as proof of (((in Xx) a)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) a) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) a)->((in Xx) A0)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) A0)->((in Xx) A0))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) A0)->((in Xx) a))
% Found ((eq_ref0 A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) a))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) a))
% Found (((eq_ref fofType) A0) (in Xx)) as proof of (((in Xx) A0)->((in Xx) a))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (((in Xx) A0)->((in Xx) a))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) A0) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) a)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) a)->((in Xx) a))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) a)->((in Xx) A0))
% Found ((eq_ref0 a) (in Xx)) as proof of (((in Xx) a)->((in Xx) A0))
% Found (((eq_ref fofType) a) (in Xx)) as proof of (((in Xx) a)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) a) (in Xx))) as proof of (((in Xx) a)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) a) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) a)->((in Xx) A0)))
% Found x3:((in Xx) A0)
% Instantiate: A0:=A:fofType
% Found (fun (x3:((in Xx) A0))=> x3) as proof of ((in Xx) a)
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (((in Xx) A0)->((in Xx) a))
% Found (fun (Xx:fofType) (x3:((in Xx) A0))=> x3) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) a)))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) a)->((in Xx) a))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) a)->((in Xx) A0))
% Found ((eq_ref0 a) (in Xx)) as proof of (((in Xx) a)->((in Xx) A0))
% Found (((eq_ref fofType) a) (in Xx)) as proof of (((in Xx) a)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) a) (in Xx))) as proof of (((in Xx) a)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) a) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) a)->((in Xx) A0)))
% Found eq_ref00:=(eq_ref0 (f1 x2)):(((eq Prop) (f1 x2)) (f1 x2))
% Found (eq_ref0 (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x2)):(((eq Prop) (f1 x2)) (f1 x2))
% Found (eq_ref0 (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x2)):(((eq Prop) (f1 x2)) (f1 x2))
% Found (eq_ref0 (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x2)):(((eq Prop) (f1 x2)) (f1 x2))
% Found (eq_ref0 (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x2)):(((eq Prop) (f1 x2)) (f1 x2))
% Found (eq_ref0 (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x2)):(((eq Prop) (f1 x2)) (f1 x2))
% Found (eq_ref0 (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x2)):(((eq Prop) (f1 x2)) (f1 x2))
% Found (eq_ref0 (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x2)):(((eq Prop) (f1 x2)) (f1 x2))
% Found (eq_ref0 (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found ((eq_ref Prop) (f1 x2)) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (((eq Prop) (f1 x2)) (f0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f1 x2))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) b00)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b00)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b00)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b00)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (fofType->Prop)) f) (fun (x:fofType)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (fofType->Prop)) f) (fun (x:fofType)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (fofType->Prop)) f) (fun (x:fofType)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (fofType->Prop)) f) (fun (x:fofType)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found eq_ref00:=(eq_ref0 f):(((eq (fofType->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (fofType->Prop)) f) (fun (x:fofType)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (fofType->Prop)) f) (fun (x:fofType)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (fofType->Prop)) f) b0)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b0)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found x3:(P2 (f x2))
% Instantiate: a:=f:(fofType->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (a x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (a x2)))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (a x2))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found x3:(P2 (f x2))
% Instantiate: a:=f:(fofType->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (a x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (a x2)))
% Found (fun (x2:
% EOF
%------------------------------------------------------------------------------