%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV221^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n108.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  : SEV221^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n108.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 08:32:01 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 0x941ea8>, <kernel.Type object at 0x941d40>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0xb15170>, <kernel.DependentProduct object at 0x941710>) of role type named cZ
% Using role type
% Declaring cZ:(a->Prop)
% FOF formula (<kernel.Constant object at 0x9419e0>, <kernel.DependentProduct object at 0x941cb0>) of role type named cW
% Using role type
% Declaring cW:((a->Prop)->Prop)
% FOF formula (forall (Xx:a), ((iff ((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (cZ Xx))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))))) of role conjecture named cTHM61_pme
% Conjecture to prove = (forall (Xx:a), ((iff ((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (cZ Xx))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (Xx:a), ((iff ((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (cZ Xx))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))))']
% Parameter a:Type.
% Parameter cZ:(a->Prop).
% Parameter cW:((a->Prop)->Prop).
% Trying to prove (forall (Xx:a), ((iff ((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (cZ Xx))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (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 eta_expansion_dep000:=(eta_expansion_dep00 (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) (fun (x:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x Xx))))
% Found (eta_expansion_dep00 (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x1:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x1:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x1:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x1:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) b)
% Found x2:(cZ Xx)
% Instantiate: x0:=cZ:(a->Prop)
% Found (fun (x2:(cZ Xx))=> x2) as proof of (x0 Xx)
% Found (fun (x1:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x2:(cZ Xx))=> x2) as proof of ((cZ Xx)->(x0 Xx))
% Found (fun (x1:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x2:(cZ Xx))=> x2) as proof of (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))->((cZ Xx)->(x0 Xx)))
% Found (and_rect00 (fun (x1:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x2:(cZ Xx))=> x2)) as proof of (x0 Xx)
% Found ((and_rect0 (x0 Xx)) (fun (x1:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x2:(cZ Xx))=> x2)) as proof of (x0 Xx)
% Found (((fun (P:Type) (x1:(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))->((cZ Xx)->P)))=> (((((and_rect ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (cZ Xx)) P) x1) x)) (x0 Xx)) (fun (x1:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x2:(cZ Xx))=> x2)) as proof of (x0 Xx)
% Found (((fun (P:Type) (x1:(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))->((cZ Xx)->P)))=> (((((and_rect ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (cZ Xx)) P) x1) x)) (x0 Xx)) (fun (x1:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x2:(cZ Xx))=> x2)) as proof of (x0 Xx)
% Found x1:(cZ Xx)
% Instantiate: x2:=cZ:(a->Prop)
% Found x1 as proof of (x2 Xx)
% Found x1 as proof of (x2 Xx)
% Found x1 as proof of (x2 Xx)
% Found x1:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))
% Instantiate: x0:=(fun (x3:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S x3))))):(a->Prop)
% Found x1 as proof of (x0 Xx)
% Found x1 as proof of (x0 Xx)
% Found x1 as proof of (x0 Xx)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x0 Xx)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x0 Xx)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x0 Xx)))
% Found (fun (x0:(a->Prop))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x0 Xx)))
% Found (fun (x0:(a->Prop))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x Xx))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x0 Xx)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x0 Xx)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x0 Xx)))
% Found (fun (x0:(a->Prop))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x0 Xx)))
% Found (fun (x0:(a->Prop))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x 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 eta_expansion_dep000:=(eta_expansion_dep00 (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) (fun (x:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x Xx))))
% Found (eta_expansion_dep00 (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x1:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x1:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x1:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x1:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) b)
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of 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 x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))
% Instantiate: b:=(fun (S:(a->Prop))=> ((and (cW S)) (S Xx))):((a->Prop)->Prop)
% Found x0 as proof of (P b)
% Found eta_expansion000:=(eta_expansion00 (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) (fun (x:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x Xx))))
% Found (eta_expansion00 (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) b)
% Found x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))
% Instantiate: b:=(fun (S:(a->Prop))=> ((and (cW S)) (S Xx))):((a->Prop)->Prop)
% Found (fun (x1:(cZ Xx))=> x0) as proof of (P b)
% Found (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0) as proof of ((cZ Xx)->(P b))
% Found (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0) as proof of (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))->((cZ Xx)->(P b)))
% Found (and_rect00 (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0)) as proof of (P b)
% Found ((and_rect0 (P b)) (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0)) as proof of (P b)
% Found (((fun (P0:Type) (x0:(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))->((cZ Xx)->P0)))=> (((((and_rect ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (cZ Xx)) P0) x0) x)) (P b)) (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0)) as proof of (P b)
% Found (((fun (P0:Type) (x0:(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))->((cZ Xx)->P0)))=> (((((and_rect ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (cZ Xx)) P0) x0) x)) (P b)) (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0)) as proof of (P b)
% Found x2:(cZ Xx)
% Instantiate: x0:=cZ:(a->Prop)
% Found (fun (x2:(cZ Xx))=> x2) as proof of (x0 Xx)
% Found (fun (x1:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x2:(cZ Xx))=> x2) as proof of ((cZ Xx)->(x0 Xx))
% Found (fun (x1:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x2:(cZ Xx))=> x2) as proof of (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))->((cZ Xx)->(x0 Xx)))
% Found (and_rect00 (fun (x1:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x2:(cZ Xx))=> x2)) as proof of (x0 Xx)
% Found ((and_rect0 (x0 Xx)) (fun (x1:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x2:(cZ Xx))=> x2)) as proof of (x0 Xx)
% Found (((fun (P:Type) (x1:(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))->((cZ Xx)->P)))=> (((((and_rect ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (cZ Xx)) P) x1) x)) (x0 Xx)) (fun (x1:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x2:(cZ Xx))=> x2)) as proof of (x0 Xx)
% Found (((fun (P:Type) (x1:(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))->((cZ Xx)->P)))=> (((((and_rect ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (cZ Xx)) P) x1) x)) (x0 Xx)) (fun (x1:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x2:(cZ Xx))=> x2)) as proof of (x0 Xx)
% Found x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and (cW S)) (S Xx))):((a->Prop)->Prop)
% Found x0 as proof of (P f)
% Found x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and (cW S)) (S Xx))):((a->Prop)->Prop)
% Found x0 as proof of (P f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x0 Xx)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x0 Xx)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x0 Xx)))
% Found (fun (x0:(a->Prop))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x0 Xx)))
% Found (fun (x0:(a->Prop))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x Xx))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x0 Xx)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x0 Xx)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x0 Xx)))
% Found (fun (x0:(a->Prop))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x0 Xx)))
% Found (fun (x0:(a->Prop))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x Xx))))
% Found x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and (cW S)) (S Xx))):((a->Prop)->Prop)
% Found (fun (x1:(cZ Xx))=> x0) as proof of (P f)
% Found (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0) as proof of ((cZ Xx)->(P f))
% Found (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0) as proof of (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))->((cZ Xx)->(P f)))
% Found (and_rect00 (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0)) as proof of (P f)
% Found ((and_rect0 (P f)) (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0)) as proof of (P f)
% Found (((fun (P0:Type) (x0:(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))->((cZ Xx)->P0)))=> (((((and_rect ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (cZ Xx)) P0) x0) x)) (P f)) (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0)) as proof of (P f)
% Found (((fun (P0:Type) (x0:(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))->((cZ Xx)->P0)))=> (((((and_rect ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (cZ Xx)) P0) x0) x)) (P f)) (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0)) as proof of (P f)
% Found x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and (cW S)) (S Xx))):((a->Prop)->Prop)
% Found (fun (x1:(cZ Xx))=> x0) as proof of (P f)
% Found (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0) as proof of ((cZ Xx)->(P f))
% Found (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0) as proof of (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))->((cZ Xx)->(P f)))
% Found (and_rect00 (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0)) as proof of (P f)
% Found ((and_rect0 (P f)) (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0)) as proof of (P f)
% Found (((fun (P0:Type) (x0:(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))->((cZ Xx)->P0)))=> (((((and_rect ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (cZ Xx)) P0) x0) x)) (P f)) (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0)) as proof of (P f)
% Found (((fun (P0:Type) (x0:(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))->((cZ Xx)->P0)))=> (((((and_rect ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (cZ Xx)) P0) x0) x)) (P f)) (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0)) as proof of (P f)
% 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 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 ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x2) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x2 Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x2) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x2 Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x2) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x2 Xx)))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x2) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x2 Xx)))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x Xx))))
% 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 ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x2) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x2 Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x2) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x2 Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x2) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x2 Xx)))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x2) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x2 Xx)))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x Xx))))
% Found iff_refl:=(fun (A:Prop)=> ((((conj (A->A)) (A->A)) (fun (H:A)=> H)) (fun (H:A)=> H))):(forall (P:Prop), ((iff P) P))
% Instantiate: b:=(forall (P:Prop), ((iff P) P)):Prop
% Found iff_refl as proof of 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 x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))
% Instantiate: b:=(fun (S:(a->Prop))=> ((and (cW S)) (S Xx))):((a->Prop)->Prop)
% Found x0 as proof of (P b)
% Found eq_ref00:=(eq_ref0 (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Found (eq_ref0 (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) b)
% Found x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))
% Instantiate: b:=(fun (S:(a->Prop))=> ((and (cW S)) (S Xx))):((a->Prop)->Prop)
% Found (fun (x1:(cZ Xx))=> x0) as proof of (P b)
% Found (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0) as proof of ((cZ Xx)->(P b))
% Found (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0) as proof of (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))->((cZ Xx)->(P b)))
% Found (and_rect00 (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0)) as proof of (P b)
% Found ((and_rect0 (P b)) (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0)) as proof of (P b)
% Found (((fun (P0:Type) (x0:(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))->((cZ Xx)->P0)))=> (((((and_rect ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (cZ Xx)) P0) x0) x)) (P b)) (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0)) as proof of (P b)
% Found (((fun (P0:Type) (x0:(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))->((cZ Xx)->P0)))=> (((((and_rect ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (cZ Xx)) P0) x0) x)) (P b)) (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0)) as proof of (P b)
% Found x:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Instantiate: b:=(fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))):((a->Prop)->Prop)
% Found x as proof of (P b)
% Found x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and (cW S)) (S Xx))):((a->Prop)->Prop)
% Found x0 as proof of (P f)
% Found x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and (cW S)) (S Xx))):((a->Prop)->Prop)
% Found x0 as proof of (P f)
% Found x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and (cW S)) (S Xx))):((a->Prop)->Prop)
% Found (fun (x1:(cZ Xx))=> x0) as proof of (P f)
% Found (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0) as proof of ((cZ Xx)->(P f))
% Found (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0) as proof of (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))->((cZ Xx)->(P f)))
% Found (and_rect00 (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0)) as proof of (P f)
% Found ((and_rect0 (P f)) (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0)) as proof of (P f)
% Found (((fun (P0:Type) (x0:(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))->((cZ Xx)->P0)))=> (((((and_rect ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (cZ Xx)) P0) x0) x)) (P f)) (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0)) as proof of (P f)
% Found (((fun (P0:Type) (x0:(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))->((cZ Xx)->P0)))=> (((((and_rect ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (cZ Xx)) P0) x0) x)) (P f)) (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0)) as proof of (P f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) (fun (x:(a->Prop))=> ((and (cW x)) (x Xx))))
% Found (eta_expansion_dep00 (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x1:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x1:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x1:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x1:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and (cW S)) (S Xx))):((a->Prop)->Prop)
% Found (fun (x1:(cZ Xx))=> x0) as proof of (P f)
% Found (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0) as proof of ((cZ Xx)->(P f))
% Found (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0) as proof of (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))->((cZ Xx)->(P f)))
% Found (and_rect00 (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0)) as proof of (P f)
% Found ((and_rect0 (P f)) (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0)) as proof of (P f)
% Found (((fun (P0:Type) (x0:(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))->((cZ Xx)->P0)))=> (((((and_rect ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (cZ Xx)) P0) x0) x)) (P f)) (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0)) as proof of (P f)
% Found (((fun (P0:Type) (x0:(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))->((cZ Xx)->P0)))=> (((((and_rect ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (cZ Xx)) P0) x0) x)) (P f)) (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0)) as proof of (P f)
% 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 eq_sym:=(fun (T:Type) (a:T) (b:T) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq T) x) a))) ((eq_ref T) a))):(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a)))
% Instantiate: b:=(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a))):Prop
% Found eq_sym as proof of b
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x2) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x2 Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x2) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x2 Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x2) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x2 Xx)))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x2) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x2 Xx)))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x Xx))))
% 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 ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x2) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x2 Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x2) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x2 Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x2) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x2 Xx)))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x2) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x2 Xx)))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x Xx))))
% Found x:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))):((a->Prop)->Prop)
% Found x as proof of (P f)
% Found x:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))):((a->Prop)->Prop)
% Found x as proof of (P f)
% 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 x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))
% Instantiate: b:=(fun (S:(a->Prop))=> ((and (cW S)) (S Xx))):((a->Prop)->Prop)
% Found x0 as proof of (P 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 (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Found (eq_ref0 (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) b)
% Found eq_sym:=(fun (T:Type) (a:T) (b:T) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq T) x) a))) ((eq_ref T) a))):(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a)))
% Instantiate: b:=(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a))):Prop
% Found eq_sym as proof of b
% Found x:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Instantiate: b:=(fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))):((a->Prop)->Prop)
% Found x as proof of (P b)
% Found x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and (cW S)) (S Xx))):((a->Prop)->Prop)
% Found x0 as proof of (P f)
% Found x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and (cW S)) (S Xx))):((a->Prop)->Prop)
% Found x0 as proof of (P f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and (cW x0)) (x0 Xx)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cW x0)) (x0 Xx)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cW x0)) (x0 Xx)))
% Found (fun (x0:(a->Prop))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (cW x0)) (x0 Xx)))
% Found (fun (x0:(a->Prop))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (x Xx))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and (cW x0)) (x0 Xx)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cW x0)) (x0 Xx)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cW x0)) (x0 Xx)))
% Found (fun (x0:(a->Prop))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (cW x0)) (x0 Xx)))
% Found (fun (x0:(a->Prop))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (x Xx))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) (fun (x:(a->Prop))=> ((and (cW x)) (x Xx))))
% Found (eta_expansion_dep00 (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x1:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x1:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x1:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x1:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found x4:((and (cW x3)) (x3 Xx))
% Instantiate: x2:=(fun (x5:a)=> ((and (cW x3)) (x3 x5))):(a->Prop)
% Found (fun (x4:((and (cW x3)) (x3 Xx)))=> x4) as proof of (x2 Xx)
% Found x4:((and (cW x3)) (x3 Xx))
% Instantiate: x0:=(fun (x5:a)=> ((and (cW x3)) (x3 x5))):(a->Prop)
% Found (fun (x4:((and (cW x3)) (x3 Xx)))=> x4) as proof of (x0 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 eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x4) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4 Xx)))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x4) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4 Xx)))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x4) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4 Xx)))
% Found (fun (x4:(a->Prop))=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x4) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4 Xx)))
% Found (fun (x4:(a->Prop))=> ((eq_ref Prop) (f x4))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x Xx))))
% 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)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x4) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4 Xx)))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x4) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4 Xx)))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x4) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4 Xx)))
% Found (fun (x4:(a->Prop))=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x4) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4 Xx)))
% Found (fun (x4:(a->Prop))=> ((eq_ref Prop) (f x4))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x Xx))))
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b
% Found x:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))):((a->Prop)->Prop)
% Found x as proof of (P f)
% Found x:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))):((a->Prop)->Prop)
% Found x as proof of (P f)
% 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 eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (a->Prop)) x0) (fun (x:a)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (x1 Xx0))))
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) x0) as proof of (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (x1 Xx0))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) x0) as proof of (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (x1 Xx0))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) x0) as proof of (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (x1 Xx0))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) x0) as proof of (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (x1 Xx0))))
% 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:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))
% Instantiate: b:=(fun (S:(a->Prop))=> ((and (cW S)) (S Xx))):((a->Prop)->Prop)
% Found x0 as proof of (P b)
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b
% Found x:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Instantiate: b:=(fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))):((a->Prop)->Prop)
% Found x as proof of (P b)
% Found eta_expansion000:=(eta_expansion00 (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) (fun (x:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x Xx))))
% Found (eta_expansion00 (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) (fun (x:(a->Prop))=> ((and (cW x)) (x Xx))))
% Found (eta_expansion00 (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and (cW x0)) (x0 Xx)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cW x0)) (x0 Xx)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cW x0)) (x0 Xx)))
% Found (fun (x0:(a->Prop))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (cW x0)) (x0 Xx)))
% Found (fun (x0:(a->Prop))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (x Xx))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and (cW x0)) (x0 Xx)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cW x0)) (x0 Xx)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cW x0)) (x0 Xx)))
% Found (fun (x0:(a->Prop))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (cW x0)) (x0 Xx)))
% Found (fun (x0:(a->Prop))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (x Xx))))
% Found x6:(x3 Xx)
% Instantiate: x2:=x3:(a->Prop)
% Found (fun (x6:(x3 Xx))=> x6) as proof of (x2 Xx)
% Found (fun (x5:(cW x3)) (x6:(x3 Xx))=> x6) as proof of ((x3 Xx)->(x2 Xx))
% Found (fun (x5:(cW x3)) (x6:(x3 Xx))=> x6) as proof of ((cW x3)->((x3 Xx)->(x2 Xx)))
% Found (and_rect10 (fun (x5:(cW x3)) (x6:(x3 Xx))=> x6)) as proof of (x2 Xx)
% Found ((and_rect1 (x2 Xx)) (fun (x5:(cW x3)) (x6:(x3 Xx))=> x6)) as proof of (x2 Xx)
% Found (((fun (P:Type) (x5:((cW x3)->((x3 Xx)->P)))=> (((((and_rect (cW x3)) (x3 Xx)) P) x5) x4)) (x2 Xx)) (fun (x5:(cW x3)) (x6:(x3 Xx))=> x6)) as proof of (x2 Xx)
% Found (((fun (P:Type) (x5:((cW x3)->((x3 Xx)->P)))=> (((((and_rect (cW x3)) (x3 Xx)) P) x5) x4)) (x2 Xx)) (fun (x5:(cW x3)) (x6:(x3 Xx))=> x6)) as proof of (x2 Xx)
% Found x6:(x3 Xx)
% Instantiate: x0:=x3:(a->Prop)
% Found (fun (x6:(x3 Xx))=> x6) as proof of (x0 Xx)
% Found (fun (x5:(cW x3)) (x6:(x3 Xx))=> x6) as proof of ((x3 Xx)->(x0 Xx))
% Found (fun (x5:(cW x3)) (x6:(x3 Xx))=> x6) as proof of ((cW x3)->((x3 Xx)->(x0 Xx)))
% Found (and_rect10 (fun (x5:(cW x3)) (x6:(x3 Xx))=> x6)) as proof of (x0 Xx)
% Found ((and_rect1 (x0 Xx)) (fun (x5:(cW x3)) (x6:(x3 Xx))=> x6)) as proof of (x0 Xx)
% Found (((fun (P:Type) (x5:((cW x3)->((x3 Xx)->P)))=> (((((and_rect (cW x3)) (x3 Xx)) P) x5) x4)) (x0 Xx)) (fun (x5:(cW x3)) (x6:(x3 Xx))=> x6)) as proof of (x0 Xx)
% Found (((fun (P:Type) (x5:((cW x3)->((x3 Xx)->P)))=> (((((and_rect (cW x3)) (x3 Xx)) P) x5) x4)) (x0 Xx)) (fun (x5:(cW x3)) (x6:(x3 Xx))=> x6)) as proof of (x0 Xx)
% Found x6:(x2 Xx)
% Instantiate: x4:=x2:(a->Prop)
% Found (fun (x6:(x2 Xx))=> x6) as proof of (x4 Xx)
% Found (fun (x5:(cW x2)) (x6:(x2 Xx))=> x6) as proof of ((x2 Xx)->(x4 Xx))
% Found (fun (x5:(cW x2)) (x6:(x2 Xx))=> x6) as proof of ((cW x2)->((x2 Xx)->(x4 Xx)))
% Found (and_rect10 (fun (x5:(cW x2)) (x6:(x2 Xx))=> x6)) as proof of (x4 Xx)
% Found ((and_rect1 (x4 Xx)) (fun (x5:(cW x2)) (x6:(x2 Xx))=> x6)) as proof of (x4 Xx)
% Found (((fun (P:Type) (x5:((cW x2)->((x2 Xx)->P)))=> (((((and_rect (cW x2)) (x2 Xx)) P) x5) x3)) (x4 Xx)) (fun (x5:(cW x2)) (x6:(x2 Xx))=> x6)) as proof of (x4 Xx)
% Found (((fun (P:Type) (x5:((cW x2)->((x2 Xx)->P)))=> (((((and_rect (cW x2)) (x2 Xx)) P) x5) x3)) (x4 Xx)) (fun (x5:(cW x2)) (x6:(x2 Xx))=> x6)) as proof of (x4 Xx)
% Found x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and (cW S)) (S Xx))):((a->Prop)->Prop)
% Found x0 as proof of (P f)
% Found x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and (cW S)) (S Xx))):((a->Prop)->Prop)
% Found x0 as proof of (P f)
% 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 x:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))):((a->Prop)->Prop)
% Found x as proof of (P f)
% Found x:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))):((a->Prop)->Prop)
% Found x as proof of (P f)
% Found conj1:=(conj ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))):(forall (B:Prop), (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))->(B->((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) B))))
% Instantiate: b:=(forall (B:Prop), (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))->(B->((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) B)))):Prop
% Found conj1 as proof of 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 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x4) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4 Xx)))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x4) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4 Xx)))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x4) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4 Xx)))
% Found (fun (x4:(a->Prop))=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x4) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4 Xx)))
% Found (fun (x4:(a->Prop))=> ((eq_ref Prop) (f x4))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x Xx))))
% 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)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x4) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4 Xx)))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x4) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4 Xx)))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x4) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4 Xx)))
% Found (fun (x4:(a->Prop))=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x4) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4 Xx)))
% Found (fun (x4:(a->Prop))=> ((eq_ref Prop) (f x4))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x Xx))))
% Found x2:((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x1) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x1 Xx))
% Instantiate: x0:=(fun (x3:a)=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x1) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x1 x3))):(a->Prop)
% Found (fun (x2:((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x1) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x1 Xx)))=> x2) as proof of (x0 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 eta_expansion000:=(eta_expansion00 x0):(((eq (a->Prop)) x0) (fun (x:a)=> (x0 x)))
% Found (eta_expansion00 x0) as proof of (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (x1 Xx0))))
% Found ((eta_expansion0 Prop) x0) as proof of (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (x1 Xx0))))
% Found (((eta_expansion a) Prop) x0) as proof of (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (x1 Xx0))))
% Found (((eta_expansion a) Prop) x0) as proof of (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (x1 Xx0))))
% Found (((eta_expansion a) Prop) x0) as proof of (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (x1 Xx0))))
% Found x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))
% Instantiate: b:=(fun (S:(a->Prop))=> ((and (cW S)) (S Xx))):((a->Prop)->Prop)
% Found x0 as proof of (P b)
% Found conj1:=(conj ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))):(forall (B:Prop), (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))->(B->((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) B))))
% Instantiate: b:=(forall (B:Prop), (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))->(B->((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) B)))):Prop
% Found conj1 as proof of b
% Found conj1:=(conj ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))):(forall (B:Prop), (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))->(B->((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) B))))
% Instantiate: b:=(forall (B:Prop), (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))->(B->((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) B)))):Prop
% Found conj1 as proof of b
% Found eta_expansion000:=(eta_expansion00 (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) (fun (x:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x Xx))))
% Found (eta_expansion00 (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) b)
% Found x:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Instantiate: b:=(fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))):((a->Prop)->Prop)
% Found x as proof of (P 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)) (x2 Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (x2 Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (x2 Xx)))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (x2 Xx)))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (x Xx))))
% 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)) (x2 Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (x2 Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (x2 Xx)))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (x2 Xx)))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (x Xx))))
% Found x:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Instantiate: b:=(fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))):((a->Prop)->Prop)
% Found x as proof of (P b)
% Found eta_expansion000:=(eta_expansion00 (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) (fun (x:(a->Prop))=> ((and (cW x)) (x Xx))))
% Found (eta_expansion00 (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and (cW S)) (S Xx))):((a->Prop)->Prop)
% Found x0 as proof of (P f)
% Found x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and (cW S)) (S Xx))):((a->Prop)->Prop)
% Found x0 as proof of (P f)
% Found eta_expansion000:=(eta_expansion00 (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) (fun (x:(a->Prop))=> ((and (cW x)) (x Xx))))
% Found (eta_expansion00 (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% 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 x6:(x3 Xx)
% Instantiate: x0:=x3:(a->Prop)
% Found (fun (x6:(x3 Xx))=> x6) as proof of (x0 Xx)
% Found (fun (x5:(cW x3)) (x6:(x3 Xx))=> x6) as proof of ((x3 Xx)->(x0 Xx))
% Found (fun (x5:(cW x3)) (x6:(x3 Xx))=> x6) as proof of ((cW x3)->((x3 Xx)->(x0 Xx)))
% Found (and_rect10 (fun (x5:(cW x3)) (x6:(x3 Xx))=> x6)) as proof of (x0 Xx)
% Found ((and_rect1 (x0 Xx)) (fun (x5:(cW x3)) (x6:(x3 Xx))=> x6)) as proof of (x0 Xx)
% Found (((fun (P:Type) (x5:((cW x3)->((x3 Xx)->P)))=> (((((and_rect (cW x3)) (x3 Xx)) P) x5) x4)) (x0 Xx)) (fun (x5:(cW x3)) (x6:(x3 Xx))=> x6)) as proof of (x0 Xx)
% Found (((fun (P:Type) (x5:((cW x3)->((x3 Xx)->P)))=> (((((and_rect (cW x3)) (x3 Xx)) P) x5) x4)) (x0 Xx)) (fun (x5:(cW x3)) (x6:(x3 Xx))=> x6)) as proof of (x0 Xx)
% Found x6:(x3 Xx)
% Instantiate: x2:=x3:(a->Prop)
% Found (fun (x6:(x3 Xx))=> x6) as proof of (x2 Xx)
% Found (fun (x5:(cW x3)) (x6:(x3 Xx))=> x6) as proof of ((x3 Xx)->(x2 Xx))
% Found (fun (x5:(cW x3)) (x6:(x3 Xx))=> x6) as proof of ((cW x3)->((x3 Xx)->(x2 Xx)))
% Found (and_rect10 (fun (x5:(cW x3)) (x6:(x3 Xx))=> x6)) as proof of (x2 Xx)
% Found ((and_rect1 (x2 Xx)) (fun (x5:(cW x3)) (x6:(x3 Xx))=> x6)) as proof of (x2 Xx)
% Found (((fun (P:Type) (x5:((cW x3)->((x3 Xx)->P)))=> (((((and_rect (cW x3)) (x3 Xx)) P) x5) x4)) (x2 Xx)) (fun (x5:(cW x3)) (x6:(x3 Xx))=> x6)) as proof of (x2 Xx)
% Found (((fun (P:Type) (x5:((cW x3)->((x3 Xx)->P)))=> (((((and_rect (cW x3)) (x3 Xx)) P) x5) x4)) (x2 Xx)) (fun (x5:(cW x3)) (x6:(x3 Xx))=> x6)) as proof of (x2 Xx)
% Found x6:(x2 Xx)
% Instantiate: x4:=x2:(a->Prop)
% Found (fun (x6:(x2 Xx))=> x6) as proof of (x4 Xx)
% Found (fun (x5:(cW x2)) (x6:(x2 Xx))=> x6) as proof of ((x2 Xx)->(x4 Xx))
% Found (fun (x5:(cW x2)) (x6:(x2 Xx))=> x6) as proof of ((cW x2)->((x2 Xx)->(x4 Xx)))
% Found (and_rect10 (fun (x5:(cW x2)) (x6:(x2 Xx))=> x6)) as proof of (x4 Xx)
% Found ((and_rect1 (x4 Xx)) (fun (x5:(cW x2)) (x6:(x2 Xx))=> x6)) as proof of (x4 Xx)
% Found (((fun (P:Type) (x5:((cW x2)->((x2 Xx)->P)))=> (((((and_rect (cW x2)) (x2 Xx)) P) x5) x3)) (x4 Xx)) (fun (x5:(cW x2)) (x6:(x2 Xx))=> x6)) as proof of (x4 Xx)
% Found (((fun (P:Type) (x5:((cW x2)->((x2 Xx)->P)))=> (((((and_rect (cW x2)) (x2 Xx)) P) x5) x3)) (x4 Xx)) (fun (x5:(cW x2)) (x6:(x2 Xx))=> x6)) as proof of (x4 Xx)
% Found x:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))):((a->Prop)->Prop)
% Found x as proof of (P f)
% Found x:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))):((a->Prop)->Prop)
% Found x as proof of (P f)
% Found x:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))):((a->Prop)->Prop)
% Found x as proof of (P f)
% Found x:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))):((a->Prop)->Prop)
% Found x as proof of (P f)
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b
% Found eq_ref00:=(eq_ref0 (f x6)):(((eq Prop) (f x6)) (f x6))
% Found (eq_ref0 (f x6)) as proof of (((eq Prop) (f x6)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x6) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x6 Xx)))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x6) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x6 Xx)))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x6) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x6 Xx)))
% Found (fun (x6:(a->Prop))=> ((eq_ref Prop) (f x6))) as proof of (((eq Prop) (f x6)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x6) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x6 Xx)))
% Found (fun (x6:(a->Prop))=> ((eq_ref Prop) (f x6))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x Xx))))
% Found eq_ref00:=(eq_ref0 (f x6)):(((eq Prop) (f x6)) (f x6))
% Found (eq_ref0 (f x6)) as proof of (((eq Prop) (f x6)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x6) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x6 Xx)))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x6) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x6 Xx)))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x6) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x6 Xx)))
% Found (fun (x6:(a->Prop))=> ((eq_ref Prop) (f x6))) as proof of (((eq Prop) (f x6)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x6) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x6 Xx)))
% Found (fun (x6:(a->Prop))=> ((eq_ref Prop) (f x6))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x 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 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 eta_expansion_dep000:=(eta_expansion_dep00 x2):(((eq (a->Prop)) x2) (fun (x:a)=> (x2 x)))
% Found (eta_expansion_dep00 x2) as proof of (((eq (a->Prop)) x2) (fun (Xx0:a)=> ((and (cZ Xx0)) (x3 Xx0))))
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) x2) as proof of (((eq (a->Prop)) x2) (fun (Xx0:a)=> ((and (cZ Xx0)) (x3 Xx0))))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) x2) as proof of (((eq (a->Prop)) x2) (fun (Xx0:a)=> ((and (cZ Xx0)) (x3 Xx0))))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) x2) as proof of (((eq (a->Prop)) x2) (fun (Xx0:a)=> ((and (cZ Xx0)) (x3 Xx0))))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) x2) as proof of (((eq (a->Prop)) x2) (fun (Xx0:a)=> ((and (cZ Xx0)) (x3 Xx0))))
% Found x2:((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x1) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x1 Xx))
% Instantiate: x0:=(fun (x3:a)=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x1) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x1 x3))):(a->Prop)
% Found (fun (x2:((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x1) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x1 Xx)))=> x2) as proof of (x0 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 x:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Instantiate: b:=(fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))):((a->Prop)->Prop)
% Found x as proof of (P b)
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b
% Found eta_expansion000:=(eta_expansion00 x0):(((eq (a->Prop)) x0) (fun (x:a)=> (x0 x)))
% Found (eta_expansion00 x0) as proof of (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (x3 Xx0))))
% Found ((eta_expansion0 Prop) x0) as proof of (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (x3 Xx0))))
% Found (((eta_expansion a) Prop) x0) as proof of (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (x3 Xx0))))
% Found (((eta_expansion a) Prop) x0) as proof of (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (x3 Xx0))))
% Found (((eta_expansion a) Prop) x0) as proof of (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (x3 Xx0))))
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b
% Found x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))
% Instantiate: b:=(fun (S:(a->Prop))=> ((and (cW S)) (S Xx))):((a->Prop)->Prop)
% Found x0 as proof of (P b)
% Found x:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Instantiate: b:=(fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))):((a->Prop)->Prop)
% Found x as proof of (P b)
% Found eq_ref00:=(eq_ref0 (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))
% Found (eq_ref0 (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found eq_ref00:=(eq_ref0 (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Found (eq_ref0 (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S 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)) (x2 Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (x2 Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (x2 Xx)))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (x2 Xx)))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (x Xx))))
% 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)) (x2 Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (x2 Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (x2 Xx)))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (x2 Xx)))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (x Xx))))
% 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)) (x2 Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (x2 Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (x2 Xx)))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (x2 Xx)))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (x Xx))))
% 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)) (x2 Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (x2 Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (x2 Xx)))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (x2 Xx)))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (x Xx))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (a->Prop)) x0) (fun (x:a)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (x3 Xx0))))
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) x0) as proof of (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (x3 Xx0))))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) x0) as proof of (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (x3 Xx0))))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) x0) as proof of (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (x3 Xx0))))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) x0) as proof of (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (x3 Xx0))))
% 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 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 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Found eta_expansion000:=(eta_expansion00 a0):(((eq ((a->Prop)->Prop)) a0) (fun (x:(a->Prop))=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion (a->Prop)) Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion (a->Prop)) Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion (a->Prop)) Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found eta_expansion000:=(eta_expansion00 a0):(((eq ((a->Prop)->Prop)) a0) (fun (x:(a->Prop))=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion (a->Prop)) Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion (a->Prop)) Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion (a->Prop)) Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))
% Found (eq_ref0 (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found eq_ref00:=(eq_ref0 x0):(((eq (a->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (x1 Xx0))))
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (x1 Xx0))))
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (x1 Xx0))))
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (x1 Xx0))))
% Found x:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))):((a->Prop)->Prop)
% Found x as proof of (P f)
% Found x:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))):((a->Prop)->Prop)
% Found x as proof of (P f)
% Found x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and (cW S)) (S Xx))):((a->Prop)->Prop)
% Found x0 as proof of (P f)
% Found x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and (cW S)) (S Xx))):((a->Prop)->Prop)
% Found x0 as proof of (P f)
% 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 x:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))):((a->Prop)->Prop)
% Found x as proof of (P f)
% Found x:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))):((a->Prop)->Prop)
% Found x as proof of (P f)
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: b:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of 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 x4:(x0 Xx)
% Instantiate: x2:=x0:(a->Prop)
% Found (fun (x4:(x0 Xx))=> x4) as proof of (x2 Xx)
% Found (fun (x3:((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4:(x0 Xx))=> x4) as proof of ((x0 Xx)->(x2 Xx))
% Found (fun (x3:((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4:(x0 Xx))=> x4) as proof of (((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))->((x0 Xx)->(x2 Xx)))
% Found (and_rect00 (fun (x3:((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4:(x0 Xx))=> x4)) as proof of (x2 Xx)
% Found ((and_rect0 (x2 Xx)) (fun (x3:((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4:(x0 Xx))=> x4)) as proof of (x2 Xx)
% Found (((fun (P:Type) (x3:(((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))->((x0 Xx)->P)))=> (((((and_rect ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x0 Xx)) P) x3) x1)) (x2 Xx)) (fun (x3:((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4:(x0 Xx))=> x4)) as proof of (x2 Xx)
% Found (((fun (P:Type) (x3:(((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))->((x0 Xx)->P)))=> (((((and_rect ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x0 Xx)) P) x3) x1)) (x2 Xx)) (fun (x3:((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4:(x0 Xx))=> x4)) as proof of (x2 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 iff_refl:=(fun (A:Prop)=> ((((conj (A->A)) (A->A)) (fun (H:A)=> H)) (fun (H:A)=> H))):(forall (P:Prop), ((iff P) P))
% Instantiate: a0:=(forall (P:Prop), ((iff P) P)):Prop
% Found iff_refl as proof of a0
% Found eq_ref00:=(eq_ref0 (f x6)):(((eq Prop) (f x6)) (f x6))
% Found (eq_ref0 (f x6)) as proof of (((eq Prop) (f x6)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x6) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x6 Xx)))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x6) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x6 Xx)))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x6) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x6 Xx)))
% Found (fun (x6:(a->Prop))=> ((eq_ref Prop) (f x6))) as proof of (((eq Prop) (f x6)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x6) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x6 Xx)))
% Found (fun (x6:(a->Prop))=> ((eq_ref Prop) (f x6))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x Xx))))
% Found eq_ref00:=(eq_ref0 (f x6)):(((eq Prop) (f x6)) (f x6))
% Found (eq_ref0 (f x6)) as proof of (((eq Prop) (f x6)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x6) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x6 Xx)))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x6) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x6 Xx)))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x6) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x6 Xx)))
% Found (fun (x6:(a->Prop))=> ((eq_ref Prop) (f x6))) as proof of (((eq Prop) (f x6)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x6) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x6 Xx)))
% Found (fun (x6:(a->Prop))=> ((eq_ref Prop) (f x6))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x 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 x2:((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))
% Instantiate: b:=(fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))):((a->Prop)->Prop)
% Found x2 as proof of (P b)
% Found x:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Instantiate: b:=(fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))):((a->Prop)->Prop)
% Found x as proof of (P b)
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: b:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of b
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: b:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of b
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: b:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of b
% Found x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))
% Instantiate: a0:=(fun (S:(a->Prop))=> ((and (cW S)) (S Xx))):((a->Prop)->Prop)
% Found (fun (x1:(cZ Xx))=> x0) as proof of (P a0)
% Found (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0) as proof of ((cZ Xx)->(P a0))
% Found (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0) as proof of (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))->((cZ Xx)->(P a0)))
% Found (and_rect00 (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0)) as proof of (P a0)
% Found ((and_rect0 (P a0)) (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0)) as proof of (P a0)
% Found (((fun (P0:Type) (x0:(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))->((cZ Xx)->P0)))=> (((((and_rect ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (cZ Xx)) P0) x0) x)) (P a0)) (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0)) as proof of (P a0)
% Found (((fun (P0:Type) (x0:(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))->((cZ Xx)->P0)))=> (((((and_rect ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (cZ Xx)) P0) x0) x)) (P a0)) (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0)) as proof of (P a0)
% 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)) ((and (cW x4)) (x4 Xx)))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (cW x4)) (x4 Xx)))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (cW x4)) (x4 Xx)))
% Found (fun (x4:(a->Prop))=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and (cW x4)) (x4 Xx)))
% Found (fun (x4:(a->Prop))=> ((eq_ref Prop) (f x4))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (x Xx))))
% 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)) ((and (cW x4)) (x4 Xx)))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (cW x4)) (x4 Xx)))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (cW x4)) (x4 Xx)))
% Found (fun (x4:(a->Prop))=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and (cW x4)) (x4 Xx)))
% Found (fun (x4:(a->Prop))=> ((eq_ref Prop) (f x4))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (x 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 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 x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))
% Instantiate: a0:=(fun (S:(a->Prop))=> ((and (cW S)) (S Xx))):((a->Prop)->Prop)
% Found (fun (x1:(cZ Xx))=> x0) as proof of (P a0)
% Found (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0) as proof of ((cZ Xx)->(P a0))
% Found (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0) as proof of (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))->((cZ Xx)->(P a0)))
% Found (and_rect00 (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0)) as proof of (P a0)
% Found ((and_rect0 (P a0)) (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0)) as proof of (P a0)
% Found (((fun (P0:Type) (x0:(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))->((cZ Xx)->P0)))=> (((((and_rect ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (cZ Xx)) P0) x0) x)) (P a0)) (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0)) as proof of (P a0)
% Found (((fun (P0:Type) (x0:(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))->((cZ Xx)->P0)))=> (((((and_rect ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (cZ Xx)) P0) x0) x)) (P a0)) (fun (x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))) (x1:(cZ Xx))=> x0)) as proof of (P a0)
% 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 (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Found eta_expansion000:=(eta_expansion00 a0):(((eq ((a->Prop)->Prop)) a0) (fun (x:(a->Prop))=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion (a->Prop)) Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion (a->Prop)) Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion (a->Prop)) Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found eta_expansion000:=(eta_expansion00 a0):(((eq ((a->Prop)->Prop)) a0) (fun (x:(a->Prop))=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion (a->Prop)) Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion (a->Prop)) Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion (a->Prop)) Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 (x0 Xx)):(((eq Prop) (x0 Xx)) (x0 Xx))
% Found (eq_ref0 (x0 Xx)) as proof of (((eq Prop) (x0 Xx)) b)
% Found ((eq_ref Prop) (x0 Xx)) as proof of (((eq Prop) (x0 Xx)) b)
% Found ((eq_ref Prop) (x0 Xx)) as proof of (((eq Prop) (x0 Xx)) b)
% Found ((eq_ref Prop) (x0 Xx)) as proof of (((eq Prop) (x0 Xx)) b)
% Found eta_expansion000:=(eta_expansion00 (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) (fun (x:(a->Prop))=> ((and (cW x)) (x Xx))))
% Found (eta_expansion00 (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) (fun (x:(a->Prop))=> ((and (cW x)) (x Xx))))
% Found (eta_expansion_dep00 (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x5:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S 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)) (x2 Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (x2 Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (x2 Xx)))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (x2 Xx)))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (x Xx))))
% 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)) (x2 Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (x2 Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (x2 Xx)))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cW x2)) (x2 Xx)))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and (cW x)) (x Xx))))
% Found x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cW S)) (S Xx))))
% Instantiate: a0:=(fun (S:(a->Prop))=> ((and (cW S)) (S Xx))):((a->Prop)->Prop)
% Found x0 as proof of (P a0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (a->Prop)) x0) (fun (x:a)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (x3 Xx0))))
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) x0) as proof of (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (x3 Xx0))))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) x0) as proof of (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (x3 Xx0))))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) x0) as proof of (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (x3 Xx0))))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) x0) as proof of (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (x3 Xx0))))
% 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 (x3:(a->Prop))=> Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) 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 (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq ((a->Prop)->Prop)) b) (fun (x:(a->Prop))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Found ((eta_expansion_dep0 (fun (x3:(a->Prop))=> Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (a->Prop)) x0) (fun (x:a)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (x1 Xx0))))
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) x0) as proof of (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (x1 Xx0))))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) x0) as proof of (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (x1 Xx0))))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) x0) as proof of (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (x1 Xx0))))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) x0) as proof of (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (x1 Xx0))))
% Found x:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))):((a->Prop)->Prop)
% Found x as proof of (P f)
% Found x:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))):((a->Prop)->Prop)
% Found x as proof of (P f)
% Found x:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))):((a->Prop)->Prop)
% Found x as proof of (P f)
% Found x:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) S) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (S Xx))):((a->Prop)->Prop)
% Found x as proof of (P f)
% 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 x4:(x1 Xx)
% Instantiate: x0:=x1:(a->Prop)
% Found (fun (x4:(x1 Xx))=> x4) as proof of (x0 Xx)
% Found (fun (x3:((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x1) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4:(x1 Xx))=> x4) as proof of ((x1 Xx)->(x0 Xx))
% Found (fun (x3:((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x1) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4:(x1 Xx))=> x4) as proof of (((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x1) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))->((x1 Xx)->(x0 Xx)))
% Found (and_rect00 (fun (x3:((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x1) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4:(x1 Xx))=> x4)) as proof of (x0 Xx)
% Found ((and_rect0 (x0 Xx)) (fun (x3:((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x1) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4:(x1 Xx))=> x4)) as proof of (x0 Xx)
% Found (((fun (P:Type) (x3:(((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x1) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))->((x1 Xx)->P)))=> (((((and_rect ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x1) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x1 Xx)) P) x3) x2)) (x0 Xx)) (fun (x3:((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x1) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4:(x1 Xx))=> x4)) as proof of (x0 Xx)
% Found (((fun (P:Type) (x3:(((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x1) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))->((x1 Xx)->P)))=> (((((and_rect ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x1) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x1 Xx)) P) x3) x2)) (x0 Xx)) (fun (x3:((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x1) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4:(x1 Xx))=> x4)) as proof of (x0 Xx)
% Found eq_sym:=(fun (T:Type) (a:T) (b:T) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq T) x) a))) ((eq_ref T) a))):(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a)))
% Instantiate: a0:=(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a))):Prop
% Found eq_sym as proof of a0
% Found x4:(x0 Xx)
% Instantiate: x2:=x0:(a->Prop)
% Found (fun (x4:(x0 Xx))=> x4) as proof of (x2 Xx)
% Found (fun (x3:((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4:(x0 Xx))=> x4) as proof of ((x0 Xx)->(x2 Xx))
% Found (fun (x3:((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4:(x0 Xx))=> x4) as proof of (((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))->((x0 Xx)->(x2 Xx)))
% Found (and_rect00 (fun (x3:((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4:(x0 Xx))=> x4)) as proof of (x2 Xx)
% Found ((and_rect0 (x2 Xx)) (fun (x3:((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4:(x0 Xx))=> x4)) as proof of (x2 Xx)
% Found (((fun (P:Type) (x3:(((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))->((x0 Xx)->P)))=> (((((and_rect ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x0 Xx)) P) x3) x1)) (x2 Xx)) (fun (x3:((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4:(x0 Xx))=> x4)) as proof of (x2 Xx)
% Found (((fun (P:Type) (x3:(((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))->((x0 Xx)->P)))=> (((((and_rect ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x0 Xx)) P) x3) x1)) (x2 Xx)) (fun (x3:((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4:(x0 Xx))=> x4)) as proof of (x2 Xx)
% Found x4:(x0 Xx)
% Instantiate: x2:=x0:(a->Prop)
% Found (fun (x4:(x0 Xx))=> x4) as proof of (x2 Xx)
% Found (fun (x3:((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4:(x0 Xx))=> x4) as proof of ((x0 Xx)->(x2 Xx))
% Found (fun (x3:((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4:(x0 Xx))=> x4) as proof of (((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))->((x0 Xx)->(x2 Xx)))
% Found (and_rect00 (fun (x3:((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4:(x0 Xx))=> x4)) as proof of (x2 Xx)
% Found ((and_rect0 (x2 Xx)) (fun (x3:((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4:(x0 Xx))=> x4)) as proof of (x2 Xx)
% Found (((fun (P:Type) (x3:(((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))->((x0 Xx)->P)))=> (((((and_rect ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x0 Xx)) P) x3) x1)) (x2 Xx)) (fun (x3:((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4:(x0 Xx))=> x4)) as proof of (x2 Xx)
% Found (((fun (P:Type) (x3:(((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))->((x0 Xx)->P)))=> (((((and_rect ((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x0 Xx)) P) x3) x1)) (x2 Xx)) (fun (x3:((ex (a->Prop)) (fun (Xt:(a->Prop))=> ((and (cW Xt)) (((eq (a->Prop)) x0) (fun (Xx0:a)=> ((and (cZ Xx0)) (Xt Xx0)))))))) (x4:(x0 Xx))=> x4)) as proof of (x2 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 iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((i
% EOF
%------------------------------------------------------------------------------