%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV268^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n184.star.cs.uiowa.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory   : 32286.75MB
% OS       : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:33:58 EDT 2014

% Result   : Timeout 300.06s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV268^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n184.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 08:40:21 CDT 2014
% % CPUTime  : 300.06 
% 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 0x2177998>, <kernel.Type object at 0x2177fc8>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (T:((a->Prop)->Prop)), (((and ((and ((and (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->(T R)))) (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))->(T R))))) (forall (K:((a->Prop)->Prop)) (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))->(T R))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S))))->(forall (S:(a->Prop)), ((iff (T S)) (forall (Xx:a), ((S Xx)->((ex (a->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))))))))) of role conjecture named cNBHD_THM_pme
% Conjecture to prove = (forall (T:((a->Prop)->Prop)), (((and ((and ((and (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->(T R)))) (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))->(T R))))) (forall (K:((a->Prop)->Prop)) (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))->(T R))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S))))->(forall (S:(a->Prop)), ((iff (T S)) (forall (Xx:a), ((S Xx)->((ex (a->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))))))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (T:((a->Prop)->Prop)), (((and ((and ((and (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->(T R)))) (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))->(T R))))) (forall (K:((a->Prop)->Prop)) (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))->(T R))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S))))->(forall (S:(a->Prop)), ((iff (T S)) (forall (Xx:a), ((S Xx)->((ex (a->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0))))))))))))']
% Parameter a:Type.
% Trying to prove (forall (T:((a->Prop)->Prop)), (((and ((and ((and (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->(T R)))) (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))->(T R))))) (forall (K:((a->Prop)->Prop)) (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))->(T R))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S))))->(forall (S:(a->Prop)), ((iff (T S)) (forall (Xx:a), ((S Xx)->((ex (a->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0))))))))))))
% Found eq_ref000:=(eq_ref00 x2):((x2 Xx0)->(x2 Xx0))
% Found (eq_ref00 x2) as proof of ((x2 Xx0)->(S Xx0))
% Found ((eq_ref0 Xx0) x2) as proof of ((x2 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x2) as proof of ((x2 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x2) as proof of ((x2 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x2)) as proof of ((x2 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x2)) as proof of (forall (Xx0:a), ((x2 Xx0)->(S Xx0)))
% Found eq_ref000:=(eq_ref00 K):((K Xx)->(K Xx))
% Found (eq_ref00 K) as proof of ((K Xx)->(T Xx))
% Found ((eq_ref0 Xx) K) as proof of ((K Xx)->(T Xx))
% Found (((eq_ref (a->Prop)) Xx) K) as proof of ((K Xx)->(T Xx))
% Found (((eq_ref (a->Prop)) Xx) K) as proof of ((K Xx)->(T Xx))
% Found (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K)) as proof of ((K Xx)->(T Xx))
% Found (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K)) as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 x2):((x2 Xx0)->(x2 Xx0))
% Found (eq_ref00 x2) as proof of ((x2 Xx0)->(S Xx0))
% Found ((eq_ref0 Xx0) x2) as proof of ((x2 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x2) as proof of ((x2 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x2) as proof of ((x2 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x2)) as proof of ((x2 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x2)) as proof of (forall (Xx0:a), ((x2 Xx0)->(S Xx0)))
% Found eq_ref000:=(eq_ref00 x4):((x4 Xx0)->(x4 Xx0))
% Found (eq_ref00 x4) as proof of ((x4 Xx0)->(S Xx0))
% Found ((eq_ref0 Xx0) x4) as proof of ((x4 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x4) as proof of ((x4 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x4) as proof of ((x4 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x4)) as proof of ((x4 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x4)) as proof of (forall (Xx0:a), ((x4 Xx0)->(S Xx0)))
% Found x4:(forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->(T R)))
% Instantiate: K:=(fun (x8:(a->Prop))=> (((eq (a->Prop)) x8) (fun (Xx:a)=> False))):((a->Prop)->Prop)
% Found x4 as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion_dep00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))):(((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) (fun (x:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(x Xx0))))) (N Xx))))) (forall (Xx0:a), ((x Xx0)->(S Xx0))))))
% Found (eta_expansion_dep00 (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found ((eta_expansion_dep0 (fun (x3:(a->Prop))=> Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (S x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (S x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (S x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (S x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (S x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (S x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (S x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (S x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (S x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (S x)))
% Found eq_ref000:=(eq_ref00 K):((K Xx)->(K Xx))
% Found (eq_ref00 K) as proof of ((K Xx)->(T Xx))
% Found ((eq_ref0 Xx) K) as proof of ((K Xx)->(T Xx))
% Found (((eq_ref (a->Prop)) Xx) K) as proof of ((K Xx)->(T Xx))
% Found (((eq_ref (a->Prop)) Xx) K) as proof of ((K Xx)->(T Xx))
% Found (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K)) as proof of ((K Xx)->(T Xx))
% Found (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K)) as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 K):((K Xx)->(K Xx))
% Found (eq_ref00 K) as proof of ((K Xx)->(T Xx))
% Found ((eq_ref0 Xx) K) as proof of ((K Xx)->(T Xx))
% Found (((eq_ref (a->Prop)) Xx) K) as proof of ((K Xx)->(T Xx))
% Found (((eq_ref (a->Prop)) Xx) K) as proof of ((K Xx)->(T Xx))
% Found (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K)) as proof of ((K Xx)->(T Xx))
% Found (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K)) as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion_dep00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion_dep0 (fun (x8:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eta_expansion000:=(eta_expansion00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion0 Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found ((eta_expansion_dep0 (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found eq_ref000:=(eq_ref00 x4):((x4 Xx0)->(x4 Xx0))
% Found (eq_ref00 x4) as proof of ((x4 Xx0)->(S Xx0))
% Found ((eq_ref0 Xx0) x4) as proof of ((x4 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x4) as proof of ((x4 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x4) as proof of ((x4 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x4)) as proof of ((x4 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x4)) as proof of (forall (Xx0:a), ((x4 Xx0)->(S Xx0)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion_dep00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion_dep00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eq_ref000:=(eq_ref00 x6):((x6 Xx0)->(x6 Xx0))
% Found (eq_ref00 x6) as proof of ((x6 Xx0)->(S Xx0))
% Found ((eq_ref0 Xx0) x6) as proof of ((x6 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x6) as proof of ((x6 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x6) as proof of ((x6 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x6)) as proof of ((x6 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x6)) as proof of (forall (Xx0:a), ((x6 Xx0)->(S Xx0)))
% Found eq_ref000:=(eq_ref00 K):((K Xx)->(K Xx))
% Found (eq_ref00 K) as proof of ((K Xx)->(T Xx))
% Found ((eq_ref0 Xx) K) as proof of ((K Xx)->(T Xx))
% Found (((eq_ref (a->Prop)) Xx) K) as proof of ((K Xx)->(T Xx))
% Found (((eq_ref (a->Prop)) Xx) K) as proof of ((K Xx)->(T Xx))
% Found (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K)) as proof of ((K Xx)->(T Xx))
% Found (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K)) as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 K):((K Xx)->(K Xx))
% Found (eq_ref00 K) as proof of ((K Xx)->(T Xx))
% Found ((eq_ref0 Xx) K) as proof of ((K Xx)->(T Xx))
% Found (((eq_ref (a->Prop)) Xx) K) as proof of ((K Xx)->(T Xx))
% Found (((eq_ref (a->Prop)) Xx) K) as proof of ((K Xx)->(T Xx))
% Found (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K)) as proof of ((K Xx)->(T Xx))
% Found (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K)) as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found x6:(forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))->(T R)))
% Instantiate: K:=(fun (x8:(a->Prop))=> (((eq (a->Prop)) x8) (fun (Xx:a)=> (False->False)))):((a->Prop)->Prop)
% Found x6 as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 x4):((x4 Xx0)->(x4 Xx0))
% Found (eq_ref00 x4) as proof of ((x4 Xx0)->(S Xx0))
% Found ((eq_ref0 Xx0) x4) as proof of ((x4 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x4) as proof of ((x4 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x4) as proof of ((x4 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x4)) as proof of ((x4 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x4)) as proof of (forall (Xx0:a), ((x4 Xx0)->(S Xx0)))
% Found eq_ref000:=(eq_ref00 x4):((x4 Xx0)->(x4 Xx0))
% Found (eq_ref00 x4) as proof of ((x4 Xx0)->(S Xx0))
% Found ((eq_ref0 Xx0) x4) as proof of ((x4 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x4) as proof of ((x4 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x4) as proof of ((x4 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x4)) as proof of ((x4 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x4)) as proof of (forall (Xx0:a), ((x4 Xx0)->(S Xx0)))
% Found x5:(forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))->(T R)))
% Instantiate: K:=(fun (x8:(a->Prop))=> (((eq (a->Prop)) x8) (fun (Xx:a)=> (False->False)))):((a->Prop)->Prop)
% Found x5 as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% Found x6:(forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))->(T R)))
% Instantiate: K:=(fun (x8:(a->Prop))=> (((eq (a->Prop)) x8) (fun (Xx:a)=> (False->False)))):((a->Prop)->Prop)
% Found x6 as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% Found x5:(forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))->(T R)))
% Instantiate: K:=(fun (x8:(a->Prop))=> (((eq (a->Prop)) x8) (fun (Xx:a)=> (False->False)))):((a->Prop)->Prop)
% Found x5 as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% Found x5:(forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))->(T R)))
% Instantiate: K:=(fun (x8:(a->Prop))=> (((eq (a->Prop)) x8) (fun (Xx:a)=> (False->False)))):((a->Prop)->Prop)
% Found x5 as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% Found x5:(forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))->(T R)))
% Instantiate: K:=(fun (x8:(a->Prop))=> (((eq (a->Prop)) x8) (fun (Xx:a)=> (False->False)))):((a->Prop)->Prop)
% Found x5 as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 x2):((x2 Xx0)->(x2 Xx0))
% Found (eq_ref00 x2) as proof of ((x2 Xx0)->(S Xx0))
% Found ((eq_ref0 Xx0) x2) as proof of ((x2 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x2) as proof of ((x2 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x2) as proof of ((x2 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x2)) as proof of ((x2 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x2)) as proof of (forall (Xx0:a), ((x2 Xx0)->(S Xx0)))
% Found eta_expansion000:=(eta_expansion00 (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))):(((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) (fun (x:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(x Xx0))))) (N Xx))))) (forall (Xx0:a), ((x Xx0)->(S Xx0))))))
% Found (eta_expansion00 (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found ((eta_expansion0 Prop) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))):(((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) (fun (x:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(x Xx0))))) (N Xx))))) (forall (Xx0:a), ((x Xx0)->(S Xx0))))))
% Found (eta_expansion_dep00 (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found ((eta_expansion_dep0 (fun (x3:(a->Prop))=> Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (S x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (S x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (S x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (S x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (S x)))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (S x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (S x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (S x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (S x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (S x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (S x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (S x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (S x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (S x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (S x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (S x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (S x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (S x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (S x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (S x)))
% Found eq_ref000:=(eq_ref00 K):((K Xx)->(K Xx))
% Found (eq_ref00 K) as proof of ((K Xx)->(T Xx))
% Found ((eq_ref0 Xx) K) as proof of ((K Xx)->(T Xx))
% Found (((eq_ref (a->Prop)) Xx) K) as proof of ((K Xx)->(T Xx))
% Found (((eq_ref (a->Prop)) Xx) K) as proof of ((K Xx)->(T Xx))
% Found (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K)) as proof of ((K Xx)->(T Xx))
% Found (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K)) as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found x6:(forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))->(T R)))
% Instantiate: K:=(fun (x8:(a->Prop))=> (((eq (a->Prop)) x8) (fun (Xx:a)=> (False->False)))):((a->Prop)->Prop)
% Found x6 as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% Found eta_expansion000:=(eta_expansion00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion0 Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found ((eta_expansion_dep0 (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion_dep0 (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eta_expansion_dep000:=(eta_expansion_dep00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion_dep00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion_dep0 (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found eta_expansion000:=(eta_expansion00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion0 Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found eq_ref000:=(eq_ref00 x6):((x6 Xx0)->(x6 Xx0))
% Found (eq_ref00 x6) as proof of ((x6 Xx0)->(S Xx0))
% Found ((eq_ref0 Xx0) x6) as proof of ((x6 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x6) as proof of ((x6 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x6) as proof of ((x6 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x6)) as proof of ((x6 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x6)) as proof of (forall (Xx0:a), ((x6 Xx0)->(S Xx0)))
% Found eq_ref000:=(eq_ref00 K):((K Xx)->(K Xx))
% Found (eq_ref00 K) as proof of ((K Xx)->(T Xx))
% Found ((eq_ref0 Xx) K) as proof of ((K Xx)->(T Xx))
% Found (((eq_ref (a->Prop)) Xx) K) as proof of ((K Xx)->(T Xx))
% Found (((eq_ref (a->Prop)) Xx) K) as proof of ((K Xx)->(T Xx))
% Found (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K)) as proof of ((K Xx)->(T Xx))
% Found (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K)) as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% Found x6:(forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))->(T R)))
% Instantiate: K:=(fun (x8:(a->Prop))=> (((eq (a->Prop)) x8) (fun (Xx:a)=> (False->False)))):((a->Prop)->Prop)
% Found x6 as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 K):((K Xx)->(K Xx))
% Found (eq_ref00 K) as proof of ((K Xx)->(T Xx))
% Found ((eq_ref0 Xx) K) as proof of ((K Xx)->(T Xx))
% Found (((eq_ref (a->Prop)) Xx) K) as proof of ((K Xx)->(T Xx))
% Found (((eq_ref (a->Prop)) Xx) K) as proof of ((K Xx)->(T Xx))
% Found (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K)) as proof of ((K Xx)->(T Xx))
% Found (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K)) as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 x6):((x6 Xx0)->(x6 Xx0))
% Found (eq_ref00 x6) as proof of ((x6 Xx0)->(S Xx0))
% Found ((eq_ref0 Xx0) x6) as proof of ((x6 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x6) as proof of ((x6 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x6) as proof of ((x6 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x6)) as proof of ((x6 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x6)) as proof of (forall (Xx0:a), ((x6 Xx0)->(S Xx0)))
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 x8):((x8 Xx0)->(x8 Xx0))
% Found (eq_ref00 x8) as proof of ((x8 Xx0)->(S Xx0))
% Found ((eq_ref0 Xx0) x8) as proof of ((x8 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x8) as proof of ((x8 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x8) as proof of ((x8 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x8)) as proof of ((x8 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x8)) as proof of (forall (Xx0:a), ((x8 Xx0)->(S Xx0)))
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found ((eta_expansion_dep0 (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion_dep0 (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eq_ref000:=(eq_ref00 x6):((x6 Xx0)->(x6 Xx0))
% Found (eq_ref00 x6) as proof of ((x6 Xx0)->(S Xx0))
% Found ((eq_ref0 Xx0) x6) as proof of ((x6 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x6) as proof of ((x6 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x6) as proof of ((x6 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x6)) as proof of ((x6 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x6)) as proof of (forall (Xx0:a), ((x6 Xx0)->(S Xx0)))
% Found eq_ref000:=(eq_ref00 x6):((x6 Xx0)->(x6 Xx0))
% Found (eq_ref00 x6) as proof of ((x6 Xx0)->(S Xx0))
% Found ((eq_ref0 Xx0) x6) as proof of ((x6 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x6) as proof of ((x6 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x6) as proof of ((x6 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x6)) as proof of ((x6 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x6)) as proof of (forall (Xx0:a), ((x6 Xx0)->(S Xx0)))
% Found eq_ref000:=(eq_ref00 x4):((x4 Xx0)->(x4 Xx0))
% Found (eq_ref00 x4) as proof of ((x4 Xx0)->(S Xx0))
% Found ((eq_ref0 Xx0) x4) as proof of ((x4 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x4) as proof of ((x4 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x4) as proof of ((x4 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x4)) as proof of ((x4 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x4)) as proof of (forall (Xx0:a), ((x4 Xx0)->(S Xx0)))
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eta_expansion000:=(eta_expansion00 (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))):(((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) (fun (x:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(x Xx0))))) (N Xx))))) (forall (Xx0:a), ((x Xx0)->(S Xx0))))))
% Found (eta_expansion00 (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found ((eta_expansion0 Prop) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found x5:(forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->(T R)))
% Instantiate: K:=(fun (x8:(a->Prop))=> (((eq (a->Prop)) x8) (fun (Xx:a)=> False))):((a->Prop)->Prop)
% Found x5 as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 x4):((x4 Xx0)->(x4 Xx0))
% Found (eq_ref00 x4) as proof of ((x4 Xx0)->(S Xx0))
% Found ((eq_ref0 Xx0) x4) as proof of ((x4 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x4) as proof of ((x4 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x4) as proof of ((x4 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x4)) as proof of ((x4 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x4)) as proof of (forall (Xx0:a), ((x4 Xx0)->(S Xx0)))
% Found x5:(forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->(T R)))
% Instantiate: K:=(fun (x8:(a->Prop))=> (((eq (a->Prop)) x8) (fun (Xx:a)=> False))):((a->Prop)->Prop)
% Found x5 as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% Found x5:(forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->(T R)))
% Instantiate: K:=(fun (x8:(a->Prop))=> (((eq (a->Prop)) x8) (fun (Xx:a)=> False))):((a->Prop)->Prop)
% Found x5 as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))):(((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) (fun (x:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(x Xx0))))) (N Xx))))) (forall (Xx0:a), ((x Xx0)->(S Xx0))))))
% Found (eta_expansion_dep00 (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found ((eta_expansion_dep0 (fun (x7:(a->Prop))=> Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x7:(a->Prop))=> Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x7:(a->Prop))=> Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x7:(a->Prop))=> Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 x2):((x2 Xx0)->(x2 Xx0))
% Found (eq_ref00 x2) as proof of ((x2 Xx0)->(S Xx0))
% Found ((eq_ref0 Xx0) x2) as proof of ((x2 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x2) as proof of ((x2 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x2) as proof of ((x2 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x2)) as proof of ((x2 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x2)) as proof of (forall (Xx0:a), ((x2 Xx0)->(S Xx0)))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (S x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (S x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (S x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (S x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (S x)))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (S x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (S x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (S x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (S x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (S x)))
% Found eq_ref00:=(eq_ref0 (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))):(((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0))))))
% Found (eq_ref0 (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found x5:(forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->(T R)))
% Instantiate: K:=(fun (x8:(a->Prop))=> (((eq (a->Prop)) x8) (fun (Xx:a)=> False))):((a->Prop)->Prop)
% Found x5 as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% Found x5:(forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->(T R)))
% Instantiate: K:=(fun (x8:(a->Prop))=> (((eq (a->Prop)) x8) (fun (Xx:a)=> False))):((a->Prop)->Prop)
% Found x5 as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% Found x5:(forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->(T R)))
% Instantiate: K:=(fun (x8:(a->Prop))=> (((eq (a->Prop)) x8) (fun (Xx:a)=> False))):((a->Prop)->Prop)
% Found x5 as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) (S x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (S x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (S x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) (S x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) (S x)))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) (S x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (S x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (S x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) (S x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) (S x)))
% Found eq_ref00:=(eq_ref0 (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))):(((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0))))))
% Found (eq_ref0 (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (S x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (S x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (S x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (S x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (S x)))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (S x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (S x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (S x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (S x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (S x)))
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (S x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (S x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (S x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (S x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (S x)))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (S x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (S x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (S x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (S x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (S x)))
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found x6:(forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))->(T R)))
% Instantiate: K:=(fun (x8:(a->Prop))=> (((eq (a->Prop)) x8) (fun (Xx:a)=> (False->False)))):((a->Prop)->Prop)
% Found x6 as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion_dep0 (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eta_expansion_dep000:=(eta_expansion_dep00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion_dep00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion_dep0 (fun (x8:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found ((eta_expansion_dep0 (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found eta_expansion000:=(eta_expansion00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion0 Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eta_expansion_dep000:=(eta_expansion_dep00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion_dep00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found x6:(forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))->(T R)))
% Instantiate: K:=(fun (x8:(a->Prop))=> (((eq (a->Prop)) x8) (fun (Xx:a)=> (False->False)))):((a->Prop)->Prop)
% Found x6 as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 K):((K Xx)->(K Xx))
% Found (eq_ref00 K) as proof of ((K Xx)->(T Xx))
% Found ((eq_ref0 Xx) K) as proof of ((K Xx)->(T Xx))
% Found (((eq_ref (a->Prop)) Xx) K) as proof of ((K Xx)->(T Xx))
% Found (((eq_ref (a->Prop)) Xx) K) as proof of ((K Xx)->(T Xx))
% Found (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K)) as proof of ((K Xx)->(T Xx))
% Found (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K)) as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 x6):((x6 Xx0)->(x6 Xx0))
% Found (eq_ref00 x6) as proof of ((x6 Xx0)->(S Xx0))
% Found ((eq_ref0 Xx0) x6) as proof of ((x6 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x6) as proof of ((x6 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x6) as proof of ((x6 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x6)) as proof of ((x6 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x6)) as proof of (forall (Xx0:a), ((x6 Xx0)->(S Xx0)))
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion_dep00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> (False->False))):(((eq (a->Prop)) (fun (Xx:a)=> (False->False))) (fun (x:a)=> (False->False)))
% Found (eta_expansion00 (fun (Xx:a)=> (False->False))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> (False->False))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> (False->False))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> (False->False))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> (False->False))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> (False->False))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> (False->False))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> (False->False))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> (False->False))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> (False->False))) b)
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eq_ref000:=(eq_ref00 x2):((x2 Xx0)->(x2 Xx0))
% Found (eq_ref00 x2) as proof of ((x2 Xx0)->(S Xx0))
% Found ((eq_ref0 Xx0) x2) as proof of ((x2 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x2) as proof of ((x2 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x2) as proof of ((x2 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x2)) as proof of ((x2 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x2)) as proof of (forall (Xx0:a), ((x2 Xx0)->(S Xx0)))
% Found eq_ref000:=(eq_ref00 x4):((x4 Xx0)->(x4 Xx0))
% Found (eq_ref00 x4) as proof of ((x4 Xx0)->(S Xx0))
% Found ((eq_ref0 Xx0) x4) as proof of ((x4 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x4) as proof of ((x4 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x4) as proof of ((x4 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x4)) as proof of ((x4 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x4)) as proof of (forall (Xx0:a), ((x4 Xx0)->(S Xx0)))
% Found x5:(forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->(T R)))
% Instantiate: K:=(fun (x8:(a->Prop))=> (((eq (a->Prop)) x8) (fun (Xx:a)=> False))):((a->Prop)->Prop)
% Found x5 as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% Found x5:(forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->(T R)))
% Instantiate: K:=(fun (x8:(a->Prop))=> (((eq (a->Prop)) x8) (fun (Xx:a)=> False))):((a->Prop)->Prop)
% Found x5 as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion_dep0 (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found ((eta_expansion_dep0 (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion_dep0 (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion_dep0 (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found x5:(forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->(T R)))
% Instantiate: K:=(fun (x8:(a->Prop))=> (((eq (a->Prop)) x8) (fun (Xx:a)=> False))):((a->Prop)->Prop)
% Found x5 as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 x8):((x8 Xx0)->(x8 Xx0))
% Found (eq_ref00 x8) as proof of ((x8 Xx0)->(S Xx0))
% Found ((eq_ref0 Xx0) x8) as proof of ((x8 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x8) as proof of ((x8 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x8) as proof of ((x8 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x8)) as proof of ((x8 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x8)) as proof of (forall (Xx0:a), ((x8 Xx0)->(S Xx0)))
% Found x5:(forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->(T R)))
% Instantiate: K:=(fun (x8:(a->Prop))=> (((eq (a->Prop)) x8) (fun (Xx:a)=> False))):((a->Prop)->Prop)
% Found x5 as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% Found eta_expansion000:=(eta_expansion00 (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))):(((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) (fun (x:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(x Xx0))))) (N Xx))))) (forall (Xx0:a), ((x Xx0)->(S Xx0))))))
% Found (eta_expansion00 (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found ((eta_expansion0 Prop) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found eq_ref000:=(eq_ref00 x6):((x6 Xx0)->(x6 Xx0))
% Found (eq_ref00 x6) as proof of ((x6 Xx0)->(S Xx0))
% Found ((eq_ref0 Xx0) x6) as proof of ((x6 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x6) as proof of ((x6 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x6) as proof of ((x6 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x6)) as proof of ((x6 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x6)) as proof of (forall (Xx0:a), ((x6 Xx0)->(S Xx0)))
% Found eq_ref000:=(eq_ref00 x6):((x6 Xx0)->(x6 Xx0))
% Found (eq_ref00 x6) as proof of ((x6 Xx0)->(S Xx0))
% Found ((eq_ref0 Xx0) x6) as proof of ((x6 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x6) as proof of ((x6 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x6) as proof of ((x6 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x6)) as proof of ((x6 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x6)) as proof of (forall (Xx0:a), ((x6 Xx0)->(S Xx0)))
% Found eq_ref000:=(eq_ref00 x8):((x8 Xx0)->(x8 Xx0))
% Found (eq_ref00 x8) as proof of ((x8 Xx0)->(S Xx0))
% Found ((eq_ref0 Xx0) x8) as proof of ((x8 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x8) as proof of ((x8 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x8) as proof of ((x8 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x8)) as proof of ((x8 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x8)) as proof of (forall (Xx0:a), ((x8 Xx0)->(S Xx0)))
% Found eq_ref00:=(eq_ref0 (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))):(((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0))))))
% Found (eq_ref0 (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found ((eta_expansion_dep0 (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> (False->False)))
% Found (x50 (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b)) as proof of (P b)
% Found ((x5 b) (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b)) as proof of (P b)
% Found ((x5 b) (((eta_expansion_dep a) (fun (x8:a)=> Prop)) b)) as proof of (P b)
% Found eta_expansion000:=(eta_expansion00 (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))):(((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) (fun (x:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(x Xx0))))) (N Xx))))) (forall (Xx0:a), ((x Xx0)->(S Xx0))))))
% Found (eta_expansion00 (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found ((eta_expansion0 Prop) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (R:(a->Prop))=> ((
% EOF
%------------------------------------------------------------------------------