%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV222^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n117.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 13:33:54 EDT 2014

% Result   : Timeout 300.08s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV222^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n117.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 08:32:16 CDT 2014
% % CPUTime  : 300.08 
% 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 0x1831c68>, <kernel.Type object at 0x1831c20>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x1710d40>, <kernel.DependentProduct object at 0x1831830>) of role type named cZ
% Using role type
% Declaring cZ:(a->Prop)
% FOF formula (<kernel.Constant object at 0x1831a70>, <kernel.DependentProduct object at 0x1831d40>) of role type named cW
% Using role type
% Declaring cW:((a->Prop)->Prop)
% FOF formula (forall (Xx:a), ((iff ((or (forall (S:(a->Prop)), ((cW S)->(S Xx)))) (cZ Xx))) (forall (S:(a->Prop)), (((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz)))))))->(S Xx))))) of role conjecture named cTHM60_pme
% Conjecture to prove = (forall (Xx:a), ((iff ((or (forall (S:(a->Prop)), ((cW S)->(S Xx)))) (cZ Xx))) (forall (S:(a->Prop)), (((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz)))))))->(S Xx))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (Xx:a), ((iff ((or (forall (S:(a->Prop)), ((cW S)->(S Xx)))) (cZ Xx))) (forall (S:(a->Prop)), (((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz)))))))->(S Xx)))))']
% Parameter a:Type.
% Parameter cZ:(a->Prop).
% Parameter cW:((a->Prop)->Prop).
% Trying to prove (forall (Xx:a), ((iff ((or (forall (S:(a->Prop)), ((cW S)->(S Xx)))) (cZ Xx))) (forall (S:(a->Prop)), (((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz)))))))->(S Xx)))))
% Found eq_ref00:=(eq_ref0 (cZ Xx)):(((eq Prop) (cZ Xx)) (cZ Xx))
% Found (eq_ref0 (cZ Xx)) as proof of (((eq Prop) (cZ Xx)) b)
% Found ((eq_ref Prop) (cZ Xx)) as proof of (((eq Prop) (cZ Xx)) b)
% Found ((eq_ref Prop) (cZ Xx)) as proof of (((eq Prop) (cZ Xx)) b)
% Found ((eq_ref Prop) (cZ Xx)) as proof of (((eq Prop) (cZ Xx)) b)
% Found classic0:=(classic (forall (S:(a->Prop)), ((cW S)->(S Xx)))):((or (forall (S:(a->Prop)), ((cW S)->(S Xx)))) (not (forall (S:(a->Prop)), ((cW S)->(S Xx)))))
% Found (classic (forall (S:(a->Prop)), ((cW S)->(S Xx)))) as proof of (P b)
% Found (classic (forall (S:(a->Prop)), ((cW S)->(S Xx)))) as proof of (P b)
% Found (classic (forall (S:(a->Prop)), ((cW S)->(S Xx)))) as proof of (P b)
% Found x0:(cW S)
% Instantiate: x2:=S:(a->Prop)
% Found x0 as proof of (cW x2)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found x2:(P (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))))
% Instantiate: x1:=(fun (x4:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x4)))):(a->Prop)
% Found (fun (x2:(P (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))))=> x2) as proof of (P (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))
% Found (fun (P:((a->Prop)->Prop)) (x2:(P (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))))=> x2) as proof of ((P (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))))->(P (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz)))))
% Found (fun (P:((a->Prop)->Prop)) (x2:(P (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))))=> x2) as proof of (((eq (a->Prop)) (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))
% Found eq_ref00:=(eq_ref0 (cZ Xx)):(((eq Prop) (cZ Xx)) (cZ Xx))
% Found (eq_ref0 (cZ Xx)) as proof of (((eq Prop) (cZ Xx)) b)
% Found ((eq_ref Prop) (cZ Xx)) as proof of (((eq Prop) (cZ Xx)) b)
% Found ((eq_ref Prop) (cZ Xx)) as proof of (((eq Prop) (cZ Xx)) b)
% Found ((eq_ref Prop) (cZ Xx)) as proof of (((eq Prop) (cZ Xx)) b)
% Found classic0:=(classic (forall (S:(a->Prop)), ((cW S)->(S Xx)))):((or (forall (S:(a->Prop)), ((cW S)->(S Xx)))) (not (forall (S:(a->Prop)), ((cW S)->(S Xx)))))
% Found (classic (forall (S:(a->Prop)), ((cW S)->(S Xx)))) as proof of (P b)
% Found (classic (forall (S:(a->Prop)), ((cW S)->(S Xx)))) as proof of (P b)
% Found (classic (forall (S:(a->Prop)), ((cW S)->(S Xx)))) as proof of (P b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))):(((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) (fun (x:(a->Prop))=> ((and (cW x)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% Found (eta_expansion00 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x1)))) (cZ x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x1)))) (cZ x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x1)))) (cZ x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x1)))) (cZ x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x1)))) (cZ x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) Xx)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xx)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xx)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) Xx)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xx)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xx)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) Xx)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xx)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xx)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xx)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found x0:(cW S)
% Instantiate: x2:=S:(a->Prop)
% Found x0 as proof of (cW x2)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), ((cW S)->(S Xx)))):(((eq Prop) (forall (S:(a->Prop)), ((cW S)->(S Xx)))) (forall (S:(a->Prop)), ((cW S)->(S Xx))))
% Found (eq_ref0 (forall (S:(a->Prop)), ((cW S)->(S Xx)))) as proof of (((eq Prop) (forall (S:(a->Prop)), ((cW S)->(S Xx)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), ((cW S)->(S Xx)))) as proof of (((eq Prop) (forall (S:(a->Prop)), ((cW S)->(S Xx)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), ((cW S)->(S Xx)))) as proof of (((eq Prop) (forall (S:(a->Prop)), ((cW S)->(S Xx)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), ((cW S)->(S Xx)))) as proof of (((eq Prop) (forall (S:(a->Prop)), ((cW S)->(S Xx)))) b)
% Found classic0:=(classic (cZ Xx)):((or (cZ Xx)) (not (cZ Xx)))
% Found (classic (cZ Xx)) as proof of (P b)
% Found (classic (cZ Xx)) as proof of (P b)
% Found (classic (cZ Xx)) as proof of (P 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)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) (fun (x1:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x1)))) (cZ x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% 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)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) (fun (x1:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x1)))) (cZ x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% Found x2:(P (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))))
% Instantiate: x1:=(fun (x4:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x4)))):(a->Prop)
% Found (fun (x2:(P (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))))=> x2) as proof of (P (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))
% Found (fun (P:((a->Prop)->Prop)) (x2:(P (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))))=> x2) as proof of ((P (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))))->(P (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz)))))
% Found (fun (P:((a->Prop)->Prop)) (x2:(P (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))))=> x2) as proof of (((eq (a->Prop)) (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))):(((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) (fun (x:(a->Prop))=> ((and (cW x)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% Found (eta_expansion_dep00 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x1)))) (cZ x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found ((eta_expansion_dep0 (fun (x2:(a->Prop))=> Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x1)))) (cZ x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x1)))) (cZ x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x1)))) (cZ x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x1)))) (cZ x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found x0:(cW S)
% Instantiate: x2:=S:(a->Prop)
% Found x0 as proof of (cW x2)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) Xx)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xx)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xx)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) Xx)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xx)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xx)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) Xx)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xx)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xx)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xx)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))):(((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz)))))))
% Found (eq_ref0 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))):(((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) (fun (x:(a->Prop))=> ((and (cW x)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% Found (eta_expansion_dep00 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x1))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found ((eta_expansion_dep0 (fun (x2:(a->Prop))=> Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x1))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x1))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x1))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x1))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), ((cW S)->(S Xx)))):(((eq Prop) (forall (S:(a->Prop)), ((cW S)->(S Xx)))) (forall (S:(a->Prop)), ((cW S)->(S Xx))))
% Found (eq_ref0 (forall (S:(a->Prop)), ((cW S)->(S Xx)))) as proof of (((eq Prop) (forall (S:(a->Prop)), ((cW S)->(S Xx)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), ((cW S)->(S Xx)))) as proof of (((eq Prop) (forall (S:(a->Prop)), ((cW S)->(S Xx)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), ((cW S)->(S Xx)))) as proof of (((eq Prop) (forall (S:(a->Prop)), ((cW S)->(S Xx)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), ((cW S)->(S Xx)))) as proof of (((eq Prop) (forall (S:(a->Prop)), ((cW S)->(S Xx)))) b)
% Found classic0:=(classic (cZ Xx)):((or (cZ Xx)) (not (cZ Xx)))
% Found (classic (cZ Xx)) as proof of (P b)
% Found (classic (cZ Xx)) as proof of (P b)
% Found (classic (cZ Xx)) as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))):(((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) (fun (x:(a->Prop))=> ((and (cW x)) (((eq (a->Prop)) (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% Found (eta_expansion_dep00 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> ((or (cZ x1)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x1)))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found ((eta_expansion_dep0 (fun (x2:(a->Prop))=> Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> ((or (cZ x1)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x1)))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> ((or (cZ x1)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x1)))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> ((or (cZ x1)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x1)))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> ((or (cZ x1)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x1)))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) 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)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) (fun (x1:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x1)))) (cZ x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% 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)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) (fun (x1:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x1)))) (cZ x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))):(((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) (fun (x:(a->Prop))=> ((and (cW x)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% Found (eta_expansion00 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> ((cW S)->(S x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> ((cW S)->(S x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> ((cW S)->(S x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> ((cW S)->(S x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> ((cW S)->(S x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))):(((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) (fun (x:(a->Prop))=> ((and (cW x)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% Found (eta_expansion_dep00 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found ((eta_expansion_dep0 (fun (x3:(a->Prop))=> Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) 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)) ((and (cW x1)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% 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)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) (fun (x1:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x1))))) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% 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)) ((and (cW x1)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% 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)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) (fun (x1:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x1))))) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% 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)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) (fun (x1:a)=> ((or (cZ x1)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x1)))))) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% 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)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) (fun (x1:a)=> ((or (cZ x1)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x1)))))) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% Found x0:(cW S)
% Instantiate: x2:=S:(a->Prop)
% Found x0 as proof of (cW x2)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))):(((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) (fun (x:(a->Prop))=> ((and (cW x)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% Found (eta_expansion_dep00 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found ((eta_expansion_dep0 (fun (x2:(a->Prop))=> Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))):(((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) (fun (x:(a->Prop))=> ((and (cW x)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% Found (eta_expansion00 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x1))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x1))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x1))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x1))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x1))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found eq_ref00:=(eq_ref0 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))):(((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz)))))))
% Found (eq_ref0 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> ((or (cZ x1)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x1)))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> ((or (cZ x1)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x1)))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> ((or (cZ x1)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x1)))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> ((or (cZ x1)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x1)))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) 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)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) (fun (x1:a)=> ((cW S)->(S x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% 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)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) (fun (x1:a)=> ((cW S)->(S x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))):(((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz)))))))
% Found (eq_ref0 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> ((cW S)->(S x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> ((cW S)->(S x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> ((cW S)->(S x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> ((cW S)->(S x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))):(((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz)))))))
% Found (eq_ref0 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))):(((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) (fun (x:(a->Prop))=> ((and (cW x)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% Found (eta_expansion00 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))):(((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) (fun (x:(a->Prop))=> ((and (cW x)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% Found (eta_expansion_dep00 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x1))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found ((eta_expansion_dep0 (fun (x2:(a->Prop))=> Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x1))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x1))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x1))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x1))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) 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)) ((and (cW x1)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% 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)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) (fun (x1:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x1))))) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% 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)) ((and (cW x1)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% 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)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) (fun (x1:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x1))))) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% 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)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) (fun (x1:a)=> ((or (cZ x1)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x1)))))) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% 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)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((or (cZ x10)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) (fun (x1:a)=> ((or (cZ x1)) (forall (S0:(a->Prop)), ((cW S0)->(S0 x1)))))) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))):(((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) (fun (x:(a->Prop))=> ((and (cW x)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% Found (eta_expansion00 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> ((cW S)->(S x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> ((cW S)->(S x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> ((cW S)->(S x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> ((cW S)->(S x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> ((cW S)->(S x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) 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)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) (fun (x1:a)=> ((cW S)->(S x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% 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)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) (fun (x1:a)=> ((cW S)->(S x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))):(((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) (fun (x:(a->Prop))=> ((and (cW x)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% Found (eta_expansion_dep00 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found ((eta_expansion_dep0 (fun (x3:(a->Prop))=> Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% 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)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) (fun (x1:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x1))))) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% 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)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) (fun (x1:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x1))))) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% 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)) ((and (cW x1)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% 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)) ((and (cW x1)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))):(((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz)))))))
% Found (eq_ref0 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found eq_ref00:=(eq_ref0 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))):(((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz)))))))
% Found (eq_ref0 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x1))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x1))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x1))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x1))))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) 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)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) (fun (x1:a)=> ((cW S)->(S x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% 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)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) (fun (x1:a)=> ((cW S)->(S x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))):(((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) (fun (x:(a->Prop))=> ((and (cW x)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% Found (eta_expansion_dep00 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> ((cW S)->(S x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found ((eta_expansion_dep0 (fun (x2:(a->Prop))=> Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> ((cW S)->(S x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> ((cW S)->(S x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> ((cW S)->(S x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x1:a)=> ((cW S)->(S x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) 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 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% Found eq_ref00:=(eq_ref0 (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))):(((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10))))
% Found (eq_ref0 (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) as proof of (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) b)
% Found ((eq_ref (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) as proof of (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) b)
% Found ((eq_ref (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) as proof of (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) b)
% Found ((eq_ref (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) as proof of (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) 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 (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))):(((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) (fun (x:(a->Prop))=> ((and (cW x)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% Found (eta_expansion_dep00 (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found ((eta_expansion_dep0 (fun (x3:(a->Prop))=> Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz))))))) 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)) ((and (cW x1)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% 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)) ((and (cW x1)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) cZ) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% 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)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) (fun (x1:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x1))))) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% 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)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) (fun (x1:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x1))))) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10))))->(P (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))))
% Found (eq_ref00 P) as proof of (P0 (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10))))
% Found ((eq_ref0 (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) P) as proof of (P0 (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10))))
% Found (((eq_ref (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) P) as proof of (P0 (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10))))
% Found (((eq_ref (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) P) as proof of (P0 (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) 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)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) (fun (x1:a)=> ((cW S)->(S x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% 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)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cW x1)) (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) (fun (x1:a)=> ((cW S)->(S x1)))) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% 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_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 (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))
% 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 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (S x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (S x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (S x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (S x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (S x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (S x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (S x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (S x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (S x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (S x)))
% 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_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 (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))
% Found classic0:=(classic (forall (S:(a->Prop)), ((cW S)->(S Xx)))):((or (forall (S:(a->Prop)), ((cW S)->(S Xx)))) (not (forall (S:(a->Prop)), ((cW S)->(S Xx)))))
% Found (classic (forall (S:(a->Prop)), ((cW S)->(S Xx)))) as proof of (P a0)
% Found (classic (forall (S:(a->Prop)), ((cW S)->(S Xx)))) as proof of (P a0)
% Found (classic (forall (S:(a->Prop)), ((cW S)->(S Xx)))) as proof of (P a0)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cZ Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cZ Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cZ Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cZ Xx))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x Xz)))))))
% 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 (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10))))->(P (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))))
% Found (eq_ref00 P) as proof of (P0 (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10))))
% Found ((eq_ref0 (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) P) as proof of (P0 (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10))))
% Found (((eq_ref (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) P) as proof of (P0 (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10))))
% Found (((eq_ref (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) P) as proof of (P0 (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10))))->(P (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))))
% Found (eq_ref00 P) as proof of (P0 (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10))))
% Found ((eq_ref0 (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) P) as proof of (P0 (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10))))
% Found (((eq_ref (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) P) as proof of (P0 (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10))))
% Found (((eq_ref (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) P) as proof of (P0 (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10))))
% 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 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))
% Found eta_expansion000:=(eta_expansion00 (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))):(((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (x:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x)))) (cZ x))))
% Found (eta_expansion00 (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) as proof of (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) b)
% Found ((eta_expansion0 Prop) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) as proof of (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) b)
% Found (((eta_expansion a) Prop) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) as proof of (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) b)
% Found (((eta_expansion a) Prop) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) as proof of (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) b)
% Found (((eta_expansion a) Prop) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) as proof of (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) 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_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) 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_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref000:=(eq_ref00 P):((P (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10))))->(P (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))))
% Found (eq_ref00 P) as proof of (P0 (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10))))
% Found ((eq_ref0 (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) P) as proof of (P0 (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10))))
% Found (((eq_ref (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) P) as proof of (P0 (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10))))
% Found (((eq_ref (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) P) as proof of (P0 (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 cZ):(((eq (a->Prop)) cZ) (fun (x:a)=> (cZ x)))
% Found (eta_expansion_dep00 cZ) as proof of (((eq (a->Prop)) cZ) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) cZ) as proof of (((eq (a->Prop)) cZ) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) cZ) as proof of (((eq (a->Prop)) cZ) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) cZ) as proof of (((eq (a->Prop)) cZ) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) cZ) as proof of (((eq (a->Prop)) cZ) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))):(((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (x:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x)))))
% Found (eta_expansion_dep00 (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) as proof of (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) as proof of (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) as proof of (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) as proof of (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) as proof of (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) b)
% Found classic0:=(classic (forall (S:(a->Prop)), ((cW S)->(S Xx)))):((or (forall (S:(a->Prop)), ((cW S)->(S Xx)))) (not (forall (S:(a->Prop)), ((cW S)->(S Xx)))))
% Found (classic (forall (S:(a->Prop)), ((cW S)->(S Xx)))) as proof of (P a0)
% Found (classic (forall (S:(a->Prop)), ((cW S)->(S Xx)))) as proof of (P a0)
% Found (classic (forall (S:(a->Prop)), ((cW S)->(S Xx)))) as proof of (P a0)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cZ Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cZ Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cZ Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cZ Xx))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq ((a->Prop)->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found ((eq_ref ((a->Prop)->Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found ((eq_ref ((a->Prop)->Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found ((eq_ref ((a->Prop)->Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq ((a->Prop)->Prop)) b) (fun (x:(a->Prop))=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz)))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz)))))))
% Found (((eta_expansion (a->Prop)) Prop) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz)))))))
% Found (((eta_expansion (a->Prop)) Prop) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz)))))))
% Found (((eta_expansion (a->Prop)) Prop) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz)))))))
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% 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_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (S x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (S x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (S x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (S x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (S x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (S x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (S x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (S x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (S x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (S x)))
% Found eq_ref00:=(eq_ref0 (fun (x3:a)=> (forall (x20:(a->Prop)), (((and (cW x20)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x20 Xz)))))->(S x3))))):(((eq (a->Prop)) (fun (x3:a)=> (forall (x20:(a->Prop)), (((and (cW x20)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x20 Xz)))))->(S x3))))) (fun (x3:a)=> (forall (x20:(a->Prop)), (((and (cW x20)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x20 Xz)))))->(S x3)))))
% Found (eq_ref0 (fun (x3:a)=> (forall (x20:(a->Prop)), (((and (cW x20)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x20 Xz)))))->(S x3))))) as proof of (((eq (a->Prop)) (fun (x3:a)=> (forall (x20:(a->Prop)), (((and (cW x20)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x20 Xz)))))->(S x3))))) b)
% Found ((eq_ref (a->Prop)) (fun (x3:a)=> (forall (x20:(a->Prop)), (((and (cW x20)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x20 Xz)))))->(S x3))))) as proof of (((eq (a->Prop)) (fun (x3:a)=> (forall (x20:(a->Prop)), (((and (cW x20)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x20 Xz)))))->(S x3))))) b)
% Found ((eq_ref (a->Prop)) (fun (x3:a)=> (forall (x20:(a->Prop)), (((and (cW x20)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x20 Xz)))))->(S x3))))) as proof of (((eq (a->Prop)) (fun (x3:a)=> (forall (x20:(a->Prop)), (((and (cW x20)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x20 Xz)))))->(S x3))))) b)
% Found ((eq_ref (a->Prop)) (fun (x3:a)=> (forall (x20:(a->Prop)), (((and (cW x20)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x20 Xz)))))->(S x3))))) as proof of (((eq (a->Prop)) (fun (x3:a)=> (forall (x20:(a->Prop)), (((and (cW x20)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x20 Xz)))))->(S x3))))) 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 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))
% Found eq_ref00:=(eq_ref0 (fun (x10:a)=> ((cW S)->(S x10)))):(((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (x10:a)=> ((cW S)->(S x10))))
% Found (eq_ref0 (fun (x10:a)=> ((cW S)->(S x10)))) as proof of (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) b)
% Found ((eq_ref (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) as proof of (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) b)
% Found ((eq_ref (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) as proof of (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) b)
% Found ((eq_ref (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) as proof of (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) 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 (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))
% 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 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 (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10))))->(P (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))))
% Found (eq_ref00 P) as proof of (P0 (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10))))
% Found ((eq_ref0 (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) P) as proof of (P0 (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10))))
% Found (((eq_ref (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) P) as proof of (P0 (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10))))
% Found (((eq_ref (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) P) as proof of (P0 (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10))))->(P (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))))
% Found (eq_ref00 P) as proof of (P0 (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10))))
% Found ((eq_ref0 (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) P) as proof of (P0 (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10))))
% Found (((eq_ref (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) P) as proof of (P0 (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10))))
% Found (((eq_ref (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) P) as proof of (P0 (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10))))
% Found eq_ref000:=(eq_ref00 P):((P cZ)->(P cZ))
% Found (eq_ref00 P) as proof of (P0 cZ)
% Found ((eq_ref0 cZ) P) as proof of (P0 cZ)
% Found (((eq_ref (a->Prop)) cZ) P) as proof of (P0 cZ)
% Found (((eq_ref (a->Prop)) cZ) P) as proof of (P0 cZ)
% Found eq_ref000:=(eq_ref00 P):((P (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))->(P (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))))
% Found (eq_ref00 P) as proof of (P0 (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))
% Found ((eq_ref0 (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) P) as proof of (P0 (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))
% Found (((eq_ref (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) P) as proof of (P0 (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))
% Found (((eq_ref (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) P) as proof of (P0 (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))
% 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 classic0:=(classic (cZ Xx)):((or (cZ Xx)) (not (cZ Xx)))
% Found (classic (cZ Xx)) as proof of (P a0)
% Found (classic (cZ Xx)) as proof of (P a0)
% Found (classic (cZ Xx)) as proof of (P a0)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), ((cW S)->(S Xx))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), ((cW S)->(S Xx))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), ((cW S)->(S Xx))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), ((cW S)->(S Xx))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 (fun (x4:a)=> (((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz)))))->(S x4)))):(((eq (a->Prop)) (fun (x4:a)=> (((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz)))))->(S x4)))) (fun (x4:a)=> (((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz)))))->(S x4))))
% Found (eq_ref0 (fun (x4:a)=> (((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz)))))->(S x4)))) as proof of (((eq (a->Prop)) (fun (x4:a)=> (((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz)))))->(S x4)))) b)
% Found ((eq_ref (a->Prop)) (fun (x4:a)=> (((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz)))))->(S x4)))) as proof of (((eq (a->Prop)) (fun (x4:a)=> (((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz)))))->(S x4)))) b)
% Found ((eq_ref (a->Prop)) (fun (x4:a)=> (((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz)))))->(S x4)))) as proof of (((eq (a->Prop)) (fun (x4:a)=> (((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz)))))->(S x4)))) b)
% Found ((eq_ref (a->Prop)) (fun (x4:a)=> (((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz)))))->(S x4)))) as proof of (((eq (a->Prop)) (fun (x4:a)=> (((and (cW x2)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz)))))->(S x4)))) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) 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 (fun (x10:a)=> ((cW S)->(S x10))))->(P (fun (x10:a)=> ((cW S)->(S x10)))))
% Found (eq_ref00 P) as proof of (P0 (fun (x10:a)=> ((cW S)->(S x10))))
% Found ((eq_ref0 (fun (x10:a)=> ((cW S)->(S x10)))) P) as proof of (P0 (fun (x10:a)=> ((cW S)->(S x10))))
% Found (((eq_ref (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) P) as proof of (P0 (fun (x10:a)=> ((cW S)->(S x10))))
% Found (((eq_ref (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) P) as proof of (P0 (fun (x10:a)=> ((cW S)->(S x10))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) 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 (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_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 (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))
% 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 (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_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 (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))
% Found eq_ref00:=(eq_ref0 (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))):(((eq Prop) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz)))))
% Found (eq_ref0 (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))) as proof of (((eq Prop) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))) as proof of (((eq Prop) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))) as proof of (((eq Prop) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))) as proof of (((eq Prop) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz)))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz)))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz)))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10)))) (fun (Xz:a)=> ((or (cZ Xz)) (Xt Xz)))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq ((a->Prop)->Prop)) a0) (fun (x:(a->Prop))=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x2:(a->Prop))=> Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq ((a->Prop)->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found ((eq_ref ((a->Prop)->Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found ((eq_ref ((a->Prop)->Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found ((eq_ref ((a->Prop)->Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found eq_ref000:=(eq_ref00 P):((P cZ)->(P cZ))
% Found (eq_ref00 P) as proof of (P0 cZ)
% Found ((eq_ref0 cZ) P) as proof of (P0 cZ)
% Found (((eq_ref (a->Prop)) cZ) P) as proof of (P0 cZ)
% Found (((eq_ref (a->Prop)) cZ) P) as proof of (P0 cZ)
% Found eq_ref000:=(eq_ref00 P):((P cZ)->(P cZ))
% Found (eq_ref00 P) as proof of (P0 cZ)
% Found ((eq_ref0 cZ) P) as proof of (P0 cZ)
% Found (((eq_ref (a->Prop)) cZ) P) as proof of (P0 cZ)
% Found (((eq_ref (a->Prop)) cZ) P) as proof of (P0 cZ)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) 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 (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))->(P (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))))
% Found (eq_ref00 P) as proof of (P0 (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))
% Found ((eq_ref0 (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) P) as proof of (P0 (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))
% Found (((eq_ref (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) P) as proof of (P0 (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))
% Found (((eq_ref (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) P) as proof of (P0 (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))->(P (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))))
% Found (eq_ref00 P) as proof of (P0 (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))
% Found ((eq_ref0 (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) P) as proof of (P0 (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))
% Found (((eq_ref (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) P) as proof of (P0 (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))
% Found (((eq_ref (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) P) as proof of (P0 (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))))
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) 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_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))
% Found eq_ref00:=(eq_ref0 cZ):(((eq (a->Prop)) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq (a->Prop)) cZ) b)
% Found ((eq_ref (a->Prop)) cZ) as proof of (((eq (a->Prop)) cZ) b)
% Found ((eq_ref (a->Prop)) cZ) as proof of (((eq (a->Prop)) cZ) b)
% Found ((eq_ref (a->Prop)) cZ) as proof of (((eq (a->Prop)) cZ) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))
% Found eta_expansion000:=(eta_expansion00 (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))):(((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) (fun (x:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x)))))
% Found (eta_expansion00 (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) as proof of (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) b)
% Found ((eta_expansion0 Prop) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) as proof of (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) b)
% Found (((eta_expansion a) Prop) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) as proof of (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) b)
% Found (((eta_expansion a) Prop) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) as proof of (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) b)
% Found (((eta_expansion a) Prop) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) as proof of (((eq (a->Prop)) (fun (x10:a)=> (forall (S0:(a->Prop)), ((cW S0)->(S0 x10))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (x10:a)=> ((or (forall (S0:(a->Prop)), ((cW S0)->(S0 x10)))) (cZ x10))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz)))) (fun (x:a)=> ((or (cZ x)) (x1 x))))
% Found (eta_expansion00 (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz)))) b)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz)))) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) 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 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 eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz)))) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))
% Found (eq_ref0 (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz)))) b)
% Found eq_ref000:=(eq_ref00 P):((P (fun (x10:a)=> ((cW S)->(S x10))))->(P (fun (x10:a)=> ((cW S)->(S x10)))))
% Found (eq_ref00 P) as proof of (P0 (fun (x10:a)=> ((cW S)->(S x10))))
% Found ((eq_ref0 (fun (x10:a)=> ((cW S)->(S x10)))) P) as proof of (P0 (fun (x10:a)=> ((cW S)->(S x10))))
% Found (((eq_ref (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) P) as proof of (P0 (fun (x10:a)=> ((cW S)->(S x10))))
% Found (((eq_ref (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) P) as proof of (P0 (fun (x10:a)=> ((cW S)->(S x10))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (x10:a)=> ((cW S)->(S x10))))->(P (fun (x10:a)=> ((cW S)->(S x10)))))
% Found (eq_ref00 P) as proof of (P0 (fun (x10:a)=> ((cW S)->(S x10))))
% Found ((eq_ref0 (fun (x10:a)=> ((cW S)->(S x10)))) P) as proof of (P0 (fun (x10:a)=> ((cW S)->(S x10))))
% Found (((eq_ref (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) P) as proof of (P0 (fun (x10:a)=> ((cW S)->(S x10))))
% Found (((eq_ref (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) P) as proof of (P0 (fun (x10:a)=> ((cW S)->(S x10))))
% 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 eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz)))) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz))))
% Found (eq_ref0 (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cZ Xz)) (x2 Xz)))) b)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (forall (x20:(a->Prop)), (((and (cW x20)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x20 Xz)))))->(S x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (forall (x20:(a->Prop)), (((and (cW x20)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x20 Xz)))))->(S x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (forall (x20:(a->Prop)), (((and (cW x20)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x20 Xz)))))->(S x2))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (forall (x20:(a->Prop)), (((and (cW x20)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x20 Xz)))))->(S x2))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (forall (x20:(a->Prop)), (((and (cW x20)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x20 Xz)))))->(S x)))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (forall (x20:(a->Prop)), (((and (cW x20)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x20 Xz)))))->(S x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (forall (x20:(a->Prop)), (((and (cW x20)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x20 Xz)))))->(S x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (forall (x20:(a->Prop)), (((and (cW x20)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x20 Xz)))))->(S x2))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (forall (x20:(a->Prop)), (((and (cW x20)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x20 Xz)))))->(S x2))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (forall (x20:(a->Prop)), (((and (cW x20)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x20 Xz)))))->(S x)))))
% 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 (fun (x3:a)=> (forall (x20:(a->Prop)), (((and (cW x20)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x20 Xz)))))->(S x3))))):(((eq (a->Prop)) (fun (x3:a)=> (forall (x20:(a->Prop)), (((and (cW x20)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x20 Xz)))))->(S x3))))) (fun (x3:a)=> (forall (x20:(a->Prop)), (((and (cW x20)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x20 Xz)))))->(S x3)))))
% Found (eq_ref0 (fun (x3:a)=> (forall (x20:(a->Prop)), (((and (cW x20)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x20 Xz)))))->(S x3))))) as proof of (((eq (a->Prop)) (fun (x3:a)=> (forall (x20:(a->Prop)), (((and (cW x20)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x20 Xz)))))->(S x3))))) b)
% Found ((eq_ref (a->Prop)) (fun (x3:a)=> (forall (x20:(a->Prop)), (((and (cW x20)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x20 Xz)))))->(S x3))))) as proof of (((eq (a->Prop)) (fun (x3:a)=> (forall (x20:(a->Prop)), (((and (cW x20)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x20 Xz)))))->(S x3))))) b)
% Found ((eq_ref (a->Prop)) (fun (x3:a)=> (forall (x20:(a->Prop)), (((and (cW x20)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x20 Xz)))))->(S x3))))) as proof of (((eq (a->Prop)) (fun (x3:a)=> (forall (x20:(a->Prop)), (((and (cW x20)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x20 Xz)))))->(S x3))))) b)
% Found ((eq_ref (a->Prop)) (fun (x3:a)=> (forall (x20:(a->Prop)), (((and (cW x20)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x20 Xz)))))->(S x3))))) as proof of (((eq (a->Prop)) (fun (x3:a)=> (forall (x20:(a->Prop)), (((and (cW x20)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((or (cZ Xz)) (x20 Xz)))))->(S x3))))) 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_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cZ Xz)) (x1 Xz))))
% Found eta_expansion000:=(eta_expansion00 (fun (x10:a)=> ((cW S)->(S x10)))):(((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) (fun (x:a)=> ((cW S)->(S x))))
% Found (eta_expansion00 (fun (x10:a)=> ((cW S)->(S x10)))) as proof of (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) b)
% Found ((eta_expansion0 Prop) (fun (x10:a)=> ((cW S)->(S x10)))) as proof of (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) b)
% Found (((eta_expansion a) Prop) (fun (x10:a)=> ((cW S)->(S x10)))) as proof of (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) b)
% Found (((eta_expansion a) Prop) (fun (x10:a)=> ((cW S)->(S x10)))) as proof of (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) b)
% Found (((eta_expansion a) Prop) (fun (x10:a)=> ((cW S)->(S x10)))) as proof of (((eq (a->Prop)) (fun (x10:a)=> ((cW S)->(S x10)))) b)
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) (S x4))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (S x4))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (S x4))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) (S x4))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:a), (((eq Prop) (f x)) (S x)))
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) (S x4))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (S x4))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (S x4))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) (S x4))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:a), (((eq Prop) (f x)) (S x)))
% 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) 
% EOF
%------------------------------------------------------------------------------