%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEU899^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n115.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 13:33:21 EDT 2014

% Result   : Timeout 300.01s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEU899^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n115.star.cs.uiowa.edu
% % Model    : x86_64 x86_64
% % CPU      : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory   : 32286.75MB
% % OS       : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 11:41:26 CDT 2014
% % CPUTime  : 300.01 
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x2a8b440>, <kernel.Type object at 0x2aaab48>) of role type named b_type
% Using role type
% Declaring b:Type
% FOF formula (<kernel.Constant object at 0x2a8b320>, <kernel.Type object at 0x2aaa878>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x2a8b440>, <kernel.DependentProduct object at 0x2aa9908>) of role type named cF
% Using role type
% Declaring cF:(b->a)
% FOF formula (<kernel.Constant object at 0x2a8b440>, <kernel.DependentProduct object at 0x2aa9908>) of role type named cS
% Using role type
% Declaring cS:(b->Prop)
% FOF formula (<kernel.Constant object at 0x2aaab00>, <kernel.DependentProduct object at 0x2aa9b00>) of role type named cR
% Using role type
% Declaring cR:(b->Prop)
% FOF formula (forall (Xx:a), ((iff ((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))))) of role conjecture named cTHM34_pme
% Conjecture to prove = (forall (Xx:a), ((iff ((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))))):Prop
% Parameter b_DUMMY:b.
% Parameter a_DUMMY:a.
% We need to prove ['(forall (Xx:a), ((iff ((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))))']
% Parameter b:Type.
% Parameter a:Type.
% Parameter cF:(b->a).
% Parameter cS:(b->Prop).
% Parameter cR:(b->Prop).
% Trying to prove (forall (Xx:a), ((iff ((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))))
% Found eq_ref000:=(eq_ref00 (ex b)):(((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))))
% Found (eq_ref00 (ex b)) as proof of (P (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found eq_ref000:=(eq_ref00 (ex b)):(((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))
% Found (eq_ref00 (ex b)) as proof of (P (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref0 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found eq_ref000:=(eq_ref00 (ex b)):(((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))))
% Found (eq_ref00 (ex b)) as proof of (P (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found eq_ref000:=(eq_ref00 (ex b)):(((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))
% Found (eq_ref00 (ex b)) as proof of (P (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref0 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found eq_ref000:=(eq_ref00 (ex b)):(((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))))
% Found (eq_ref00 (ex b)) as proof of (P (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found eq_ref000:=(eq_ref00 (ex b)):(((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))
% Found (eq_ref00 (ex b)) as proof of (P (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref0 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found classic0:=(classic ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))):((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) (not ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))))
% Found (classic ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0)
% Found (classic ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0)
% Found (classic ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Found (eq_ref0 (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) b0)
% Found eq_ref000:=(eq_ref00 (ex b)):(((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))
% Found (eq_ref00 (ex b)) as proof of (P (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref0 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found eq_ref000:=(eq_ref00 (ex b)):(((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))))
% Found (eq_ref00 (ex b)) as proof of (P (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% 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 ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and ((or (cR x)) (cS x))) (((eq a) Xx) (cF x)))))
% 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 ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and ((or (cR x)) (cS x))) (((eq a) Xx) (cF x)))))
% Found eq_ref000:=(eq_ref00 (ex b)):(((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))))
% Found (eq_ref00 (ex b)) as proof of (P (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found eq_ref000:=(eq_ref00 (ex b)):(((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))
% Found (eq_ref00 (ex b)) as proof of (P (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref0 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found eq_ref000:=(eq_ref00 (ex b)):(((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))
% Found (eq_ref00 (ex b)) as proof of (P (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref0 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found eq_ref000:=(eq_ref00 (ex b)):(((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))))
% Found (eq_ref00 (ex b)) as proof of (P (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found eq_ref00:=(eq_ref0 a0):(((eq (b->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (b->Prop)) a0) (fun (x:b)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eta_expansion_dep0 (fun (x1:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found classic0:=(classic ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))):((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) (not ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))))
% Found (classic ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0)
% Found (classic ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0)
% Found (classic ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and ((or (cR x)) (cS x))) (((eq a) Xx) (cF x)))))
% Found (eta_expansion_dep00 (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eta_expansion_dep0 (fun (x1:b)=> Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) b0)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P b0)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P b0)
% Found eq_ref00:=(eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found eq_ref00:=(eq_ref0 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (eq_ref0 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found classic0:=(classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))):((or ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) (not ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))))
% Found (classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0)
% Found (classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0)
% Found (classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0)
% Found eta_expansion0000:=(eta_expansion000 (ex b)):(((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->((ex b) (fun (x:b)=> ((and (cR x)) (((eq a) Xx) (cF x))))))
% Found (eta_expansion000 (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->(P b0))
% Found ((eta_expansion00 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->(P b0))
% Found (((eta_expansion0 Prop) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->(P b0))
% Found ((((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->(P b0))
% Found ((((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->(P b0))
% Found ((((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->(P b0))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex b)):(((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->((ex b) (fun (x:b)=> ((and (cS x)) (((eq a) Xx) (cF x))))))
% Found (eta_expansion_dep000 (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->(P b0))
% Found ((eta_expansion_dep00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->(P b0))
% Found (((eta_expansion_dep0 (fun (x1:b)=> Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->(P b0))
% Found ((((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->(P b0))
% Found ((((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->(P b0))
% Found ((((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->(P b0))
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: a0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of ((ex b) a0)
% Found (or_intror00 x) as proof of (P ((ex b) a0))
% Found ((or_intror0 ((ex b) a0)) x) as proof of (P ((ex b) a0))
% Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((ex b) a0))
% Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((ex b) a0))
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: a0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of ((ex b) a0)
% Found (or_intror00 x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)))
% Found ((or_intror0 ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)))
% Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)))
% Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)))
% 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 ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and ((or (cR x)) (cS x))) (((eq a) Xx) (cF x)))))
% 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 ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and ((or (cR x)) (cS x))) (((eq a) Xx) (cF x)))))
% 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 (cR x0)) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cR x)) (((eq a) Xx) (cF x)))))
% 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 (cS x0)) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF x)))))
% 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 (cS x0)) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF x)))))
% 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 (cR x0)) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cR x)) (((eq a) Xx) (cF x)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (b->Prop)) a0) (fun (x:b)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eta_expansion_dep0 (fun (x1:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (b->Prop)) a0) (fun (x:b)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eta_expansion_dep0 (fun (x1:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found classic0:=(classic ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))):((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) (not ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))))
% Found (classic ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0)
% Found (classic ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0)
% Found (classic ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0)
% Found eq_ref000:=(eq_ref00 (ex b)):(((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))
% Found (eq_ref00 (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found ((eq_ref0 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found x0:((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found (fun (x0:((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))))=> x0) as proof of (P f)
% Found (fun (x0:((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))))=> x0) as proof of (((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found eta_expansion0000:=(eta_expansion000 (ex b)):(((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->((ex b) (fun (x:b)=> ((and (cS x)) (((eq a) Xx) (cF x))))))
% Found (eta_expansion000 (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found ((eta_expansion00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found (((eta_expansion0 Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found ((((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found ((((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found ((((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex b)):(((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->((ex b) (fun (x:b)=> ((and (cR x)) (((eq a) Xx) (cF x))))))
% Found (eta_expansion_dep000 (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found ((eta_expansion_dep00 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found (((eta_expansion_dep0 (fun (x1:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found ((((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found ((((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found ((((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P b0)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and (cS x)) (((eq a) Xx) (cF x)))))
% Found (eta_expansion00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and (cR x)) (((eq a) Xx) (cF x)))))
% Found (eta_expansion_dep00 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eta_expansion_dep0 (fun (x1:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (b->Prop)) a0) (fun (x:b)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (b->Prop)) a0) (fun (x:b)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found classic0:=(classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))):((or ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) (not ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))))
% Found (classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0)
% Found (classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0)
% Found (classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0)
% Found eta_expansion0000:=(eta_expansion000 (ex b)):(((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->((ex b) (fun (x:b)=> ((and (cS x)) (((eq a) Xx) (cF x))))))
% Found (eta_expansion000 (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->(P b0))
% Found ((eta_expansion00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->(P b0))
% Found (((eta_expansion0 Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->(P b0))
% Found ((((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->(P b0))
% Found ((((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->(P b0))
% Found ((((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->(P b0))
% Found eq_ref000:=(eq_ref00 (ex b)):(((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))))
% Found (eq_ref00 (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->(P b0))
% Found ((eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->(P b0))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->(P b0))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->(P b0))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->(P b0))
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: a0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of ((ex b) a0)
% Found (or_intror00 x) as proof of (P ((ex b) a0))
% Found ((or_intror0 ((ex b) a0)) x) as proof of (P ((ex b) a0))
% Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((ex b) a0))
% Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((ex b) a0))
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: a0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of ((ex b) a0)
% Found (or_intror00 x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)))
% Found ((or_intror0 ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)))
% Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)))
% Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)))
% 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 (cS x0)) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF x)))))
% 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 (cR x0)) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cR x)) (((eq a) Xx) (cF x)))))
% 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 (cS x0)) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF x)))))
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (b->Prop)) a0) (fun (x:b)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (b->Prop)) a0) (fun (x:b)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% 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 (cR x0)) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cR x)) (((eq a) Xx) (cF x)))))
% Found x0:((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Instantiate: b0:=(fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x0 as proof of (P b0)
% Found x0:((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Instantiate: b0:=(fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x0 as proof of (P b0)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P b0)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and (cS x)) (((eq a) Xx) (cF x)))))
% Found (eta_expansion00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found eq_ref00:=(eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found classic0:=(classic ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))):((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) (not ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))))
% Found (classic ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0)
% Found (classic ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0)
% Found (classic ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and ((or (cR x)) (cS x))) (((eq a) Xx) (cF x)))))
% Found (eta_expansion00 (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and ((or (cR x)) (cS x))) (((eq a) Xx) (cF x)))))
% Found (eta_expansion_dep00 (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eta_expansion_dep0 (fun (x2:b)=> Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x2:b)=> Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x2:b)=> Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x2:b)=> Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) b0)
% Found x3:(((eq a) Xx) (cF x0))
% Instantiate: x4:=x0:b
% Found x3 as proof of (((eq a) Xx) (cF x4))
% Found x3:(((eq a) Xx) (cF x0))
% Instantiate: x4:=x0:b
% Found x3 as proof of (((eq a) Xx) (cF x4))
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P b0)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and (cR x)) (((eq a) Xx) (cF x)))))
% Found (eta_expansion_dep00 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eta_expansion_dep0 (fun (x3:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and (cS x)) (((eq a) Xx) (cF x)))))
% Found (eta_expansion00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found eta_expansion0000:=(eta_expansion000 (ex b)):(((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->((ex b) (fun (x:b)=> ((and (cS x)) (((eq a) Xx) (cF x))))))
% Found (eta_expansion000 (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found ((eta_expansion00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found (((eta_expansion0 Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found ((((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found ((((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found ((((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found eta_expansion0000:=(eta_expansion000 (ex b)):(((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->((ex b) (fun (x:b)=> ((and (cR x)) (((eq a) Xx) (cF x))))))
% Found (eta_expansion000 (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found ((eta_expansion00 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found (((eta_expansion0 Prop) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found ((((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found ((((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found ((((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found eta_expansion0000:=(eta_expansion000 (ex b)):(((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->((ex b) (fun (x:b)=> ((and (cR x)) (((eq a) Xx) (cF x))))))
% Found (eta_expansion000 (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found ((eta_expansion00 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found (((eta_expansion0 Prop) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found ((((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found ((((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found ((((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found eq_ref000:=(eq_ref00 (ex b)):(((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))
% Found (eq_ref00 (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found ((eq_ref0 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->(P f))
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found x3:(((eq a) Xx) (cF x0))
% Instantiate: x4:=x0:b
% Found x3 as proof of (((eq a) Xx) (cF x4))
% Found x3:(((eq a) Xx) (cF x0))
% Instantiate: x4:=x0:b
% Found x3 as proof of (((eq a) Xx) (cF x4))
% Found x3:(((eq a) Xx) (cF x0))
% Instantiate: x4:=x0:b
% Found x3 as proof of (((eq a) Xx) (cF x4))
% Found x3:(((eq a) Xx) (cF x0))
% Instantiate: x4:=x0:b
% Found x3 as proof of (((eq a) Xx) (cF x4))
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P b0)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P b0)
% Found x0:((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x0 as proof of (P f)
% Found x0:((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x0 as proof of (P f)
% Found x0:((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x0 as proof of (P f)
% Found x0:((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x0 as proof of (P f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and (cR x)) (((eq a) Xx) (cF x)))))
% Found (eta_expansion_dep00 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eta_expansion_dep0 (fun (x3:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and (cS x)) (((eq a) Xx) (cF x)))))
% Found (eta_expansion00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: a0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of ((ex b) a0)
% Found (or_intror00 x) as proof of (P ((ex b) a0))
% Found ((or_intror0 ((ex b) a0)) x) as proof of (P ((ex b) a0))
% Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((ex b) a0))
% Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((ex b) a0))
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: a0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of ((ex b) a0)
% Found (or_intror00 x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)))
% Found ((or_intror0 ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)))
% Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)))
% Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)))
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found x4:(((eq a) Xx) (cF x1))
% Instantiate: x0:=x1:b
% Found x4 as proof of (((eq a) Xx) (cF x0))
% Found x4:(((eq a) Xx) (cF x1))
% Instantiate: x0:=x1:b
% Found x4 as proof of (((eq a) Xx) (cF x0))
% Found x4:(((eq a) Xx) (cF x0))
% Instantiate: x2:=x0:b
% Found x4 as proof of (((eq a) Xx) (cF x2))
% 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 (cS x0)) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF x)))))
% 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 (cR x0)) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cR x)) (((eq a) Xx) (cF x)))))
% 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 (cS x0)) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF x)))))
% 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 (cR x0)) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cR x)) (((eq a) Xx) (cF x)))))
% Found x4:(((eq a) Xx) (cF x0))
% Instantiate: x2:=x0:b
% Found x4 as proof of (((eq a) Xx) (cF x2))
% Found x4:(((eq a) Xx) (cF x0))
% Instantiate: x2:=x0:b
% Found x4 as proof of (((eq a) Xx) (cF x2))
% Found x4:(((eq a) Xx) (cF x0))
% Instantiate: x2:=x0:b
% Found x4 as proof of (((eq a) Xx) (cF x2))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (b->Prop)) a0) (fun (x:b)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eta_expansion_dep0 (fun (x1:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found eq_ref00:=(eq_ref0 a0):(((eq (b->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found classic0:=(classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))):((or ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) (not ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))))
% Found (classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0)
% Found (classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0)
% Found (classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (eq_ref0 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (b->Prop)) b0) (fun (x:b)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (b->Prop)) b0) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion b) Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion b) Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion b) Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Found eq_ref00:=(eq_ref0 b0):(((eq (b->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (b->Prop)) b0) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Found eq_ref00:=(eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: a0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of ((ex b) a0)
% Found (or_intror00 x) as proof of (P ((ex b) a0))
% Found ((or_intror0 ((ex b) a0)) x) as proof of (P ((ex b) a0))
% Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((ex b) a0))
% Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((ex b) a0))
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: a0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of ((ex b) a0)
% Found (or_intror00 x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)))
% Found ((or_intror0 ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)))
% Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)))
% Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)))
% Found x0:((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Instantiate: b0:=(fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x0 as proof of (P b0)
% Found x0:((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Instantiate: b0:=(fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x0 as proof of (P b0)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF x)))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cR x)) (((eq a) Xx) (cF x)))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cR x)) (((eq a) Xx) (cF x)))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF x)))))
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found classic0:=(classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))):((or ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) (not ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))))
% Found (classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0)
% Found (classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0)
% Found (classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P b0)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq (b->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (b->Prop)) a0) (fun (x:b)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found classic0:=(classic ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))):((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) (not ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))))
% Found (classic ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0)
% Found (classic ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0)
% Found (classic ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and (cR x)) (((eq a) Xx) (cF x)))))
% Found (eta_expansion00 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and (cS x)) (((eq a) Xx) (cF x)))))
% Found (eta_expansion00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))))
% Found (fun (x1:b)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))))
% Found (fun (x1:b)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and ((or (cR x)) (cS x))) (((eq a) Xx) (cF x)))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))))
% Found (fun (x1:b)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))))
% Found (fun (x1:b)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and ((or (cR x)) (cS x))) (((eq a) Xx) (cF x)))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))))
% Found (fun (x1:b)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))))
% Found (fun (x1:b)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and ((or (cR x)) (cS x))) (((eq a) Xx) (cF x)))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))))
% Found (fun (x1:b)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))))
% Found (fun (x1:b)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and ((or (cR x)) (cS x))) (((eq a) Xx) (cF x)))))
% Found eq_ref000:=(eq_ref00 (ex b)):(((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))))
% Found (eq_ref00 (ex b)) as proof of (P (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found eq_ref00:=(eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (b->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (b->Prop)) b0) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Found eq_ref000:=(eq_ref00 (ex b)):(((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))
% Found (eq_ref00 (ex b)) as proof of (P (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref0 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found eq_ref00:=(eq_ref0 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (eq_ref0 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (b->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (b->Prop)) b0) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and ((or (cR x)) (cS x))) (((eq a) Xx) (cF x)))))
% Found (eta_expansion00 (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and ((or (cR x)) (cS x))) (((eq a) Xx) (cF x)))))
% Found (eta_expansion00 (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) b0)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF x)))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF x)))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cR x)) (((eq a) Xx) (cF x)))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cR x)) (((eq a) Xx) (cF x)))))
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P b0)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P b0)
% Found eq_ref00:=(eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and (cS x)) (((eq a) Xx) (cF x)))))
% Found (eta_expansion_dep00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eta_expansion_dep0 (fun (x3:b)=> Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found x3:(((eq a) Xx) (cF x0))
% Instantiate: x4:=x0:b
% Found x3 as proof of (((eq a) Xx) (cF x4))
% Found x3:(((eq a) Xx) (cF x0))
% Instantiate: x4:=x0:b
% Found x3 as proof of (((eq a) Xx) (cF x4))
% Found x0:((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x0 as proof of (P f)
% Found x0:((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x0 as proof of (P f)
% Found x0:((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x0 as proof of (P f)
% Found x0:((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x0 as proof of (P f)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P b0)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and (cS x)) (((eq a) Xx) (cF x)))))
% Found (eta_expansion_dep00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eta_expansion_dep0 (fun (x3:b)=> Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and (cR x)) (((eq a) Xx) (cF x)))))
% Found (eta_expansion_dep00 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eta_expansion_dep0 (fun (x3:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found x3:(((eq a) Xx) (cF x0))
% Instantiate: x4:=x0:b
% Found x3 as proof of (((eq a) Xx) (cF x4))
% Found x3:(((eq a) Xx) (cF x0))
% Instantiate: x4:=x0:b
% Found x3 as proof of (((eq a) Xx) (cF x4))
% Found x3:(((eq a) Xx) (cF x0))
% Instantiate: x4:=x0:b
% Found x3 as proof of (((eq a) Xx) (cF x4))
% Found x3:(((eq a) Xx) (cF x0))
% Instantiate: x4:=x0:b
% Found x3 as proof of (((eq a) Xx) (cF x4))
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: a0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of ((ex b) a0)
% Found (or_intror00 x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)))
% Found ((or_intror0 ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)))
% Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)))
% Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)))
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: a0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of ((ex b) a0)
% Found (or_intror00 x) as proof of (P ((ex b) a0))
% Found ((or_intror0 ((ex b) a0)) x) as proof of (P ((ex b) a0))
% Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((ex b) a0))
% Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((ex b) a0))
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (b->Prop)) a0) (fun (x:b)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found eq_ref00:=(eq_ref0 a0):(((eq (b->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x 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 (cR x0)) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cR x)) (((eq a) Xx) (cF x)))))
% 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 (cR x0)) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cR x)) (((eq a) Xx) (cF x)))))
% 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 (cS x0)) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF x)))))
% 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 (cS x0)) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF x)))))
% Found eq_ref00:=(eq_ref0 b0):(((eq (b->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (b->Prop)) b0) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and (cS x)) (((eq a) Xx) (cF x)))))
% Found (eta_expansion_dep00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eta_expansion_dep0 (fun (x1:b)=> Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (b->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (b->Prop)) b0) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and (cR x)) (((eq a) Xx) (cF x)))))
% Found (eta_expansion_dep00 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eta_expansion_dep0 (fun (x1:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found classic0:=(classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))):((or ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) (not ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))))
% Found (classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0)
% Found (classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0)
% Found (classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0)
% Found eq_substitution00000:=(eq_substitution0000 (fun (x6:a)=> x6)):((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x1)))
% Found (eq_substitution0000 (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0)))
% Found ((eq_substitution000 (cF x1)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0)))
% Found (((eq_substitution00 Xx) (cF x1)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0)))
% Found ((((eq_substitution0 a) Xx) (cF x1)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0)))
% Found (((((eq_substitution a) a) Xx) (cF x1)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0)))
% Found (((((eq_substitution a) a) Xx) (cF x1)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0)))
% Found (fun (x3:((or (cR x1)) (cS x1)))=> (((((eq_substitution a) a) Xx) (cF x1)) (fun (x6:a)=> x6))) as proof of ((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0)))
% Found (fun (x3:((or (cR x1)) (cS x1)))=> (((((eq_substitution a) a) Xx) (cF x1)) (fun (x6:a)=> x6))) as proof of (((or (cR x1)) (cS x1))->((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0))))
% Found (and_rect00 (fun (x3:((or (cR x1)) (cS x1)))=> (((((eq_substitution a) a) Xx) (cF x1)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x0))
% Found ((and_rect0 (((eq a) Xx) (cF x0))) (fun (x3:((or (cR x1)) (cS x1)))=> (((((eq_substitution a) a) Xx) (cF x1)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x0))
% Found (((fun (P:Type) (x3:(((or (cR x1)) (cS x1))->((((eq a) Xx) (cF x1))->P)))=> (((((and_rect ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))) P) x3) x2)) (((eq a) Xx) (cF x0))) (fun (x3:((or (cR x1)) (cS x1)))=> (((((eq_substitution a) a) Xx) (cF x1)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x0))
% Found (((fun (P:Type) (x3:(((or (cR x1)) (cS x1))->((((eq a) Xx) (cF x1))->P)))=> (((((and_rect ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))) P) x3) x2)) (((eq a) Xx) (cF x0))) (fun (x3:((or (cR x1)) (cS x1)))=> (((((eq_substitution a) a) Xx) (cF x1)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x0))
% Found x4:(((eq a) Xx) (cF x1))
% Instantiate: x0:=x1:b
% Found (fun (x4:(((eq a) Xx) (cF x1)))=> x4) as proof of (((eq a) Xx) (cF x0))
% Found (fun (x3:((or (cR x1)) (cS x1))) (x4:(((eq a) Xx) (cF x1)))=> x4) as proof of ((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0)))
% Found (fun (x3:((or (cR x1)) (cS x1))) (x4:(((eq a) Xx) (cF x1)))=> x4) as proof of (((or (cR x1)) (cS x1))->((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0))))
% Found (and_rect00 (fun (x3:((or (cR x1)) (cS x1))) (x4:(((eq a) Xx) (cF x1)))=> x4)) as proof of (((eq a) Xx) (cF x0))
% Found ((and_rect0 (((eq a) Xx) (cF x0))) (fun (x3:((or (cR x1)) (cS x1))) (x4:(((eq a) Xx) (cF x1)))=> x4)) as proof of (((eq a) Xx) (cF x0))
% Found (((fun (P:Type) (x3:(((or (cR x1)) (cS x1))->((((eq a) Xx) (cF x1))->P)))=> (((((and_rect ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))) P) x3) x2)) (((eq a) Xx) (cF x0))) (fun (x3:((or (cR x1)) (cS x1))) (x4:(((eq a) Xx) (cF x1)))=> x4)) as proof of (((eq a) Xx) (cF x0))
% Found (((fun (P:Type) (x3:(((or (cR x1)) (cS x1))->((((eq a) Xx) (cF x1))->P)))=> (((((and_rect ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))) P) x3) x2)) (((eq a) Xx) (cF x0))) (fun (x3:((or (cR x1)) (cS x1))) (x4:(((eq a) Xx) (cF x1)))=> x4)) as proof of (((eq a) Xx) (cF x0))
% Found x4:(((eq a) Xx) (cF x0))
% Instantiate: x2:=x0:b
% Found (fun (x4:(((eq a) Xx) (cF x0)))=> x4) as proof of (((eq a) Xx) (cF x2))
% Found (fun (x3:((or (cR x0)) (cS x0))) (x4:(((eq a) Xx) (cF x0)))=> x4) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2)))
% Found (fun (x3:((or (cR x0)) (cS x0))) (x4:(((eq a) Xx) (cF x0)))=> x4) as proof of (((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))))
% Found (and_rect00 (fun (x3:((or (cR x0)) (cS x0))) (x4:(((eq a) Xx) (cF x0)))=> x4)) as proof of (((eq a) Xx) (cF x2))
% Found ((and_rect0 (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0))) (x4:(((eq a) Xx) (cF x0)))=> x4)) as proof of (((eq a) Xx) (cF x2))
% Found (((fun (P:Type) (x3:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x3) x1)) (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0))) (x4:(((eq a) Xx) (cF x0)))=> x4)) as proof of (((eq a) Xx) (cF x2))
% Found (((fun (P:Type) (x3:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x3) x1)) (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0))) (x4:(((eq a) Xx) (cF x0)))=> x4)) as proof of (((eq a) Xx) (cF x2))
% Found eq_substitution00000:=(eq_substitution0000 (fun (x6:a)=> x6)):((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x0)))
% Found (eq_substitution0000 (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2)))
% Found ((eq_substitution000 (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2)))
% Found (((eq_substitution00 Xx) (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2)))
% Found ((((eq_substitution0 a) Xx) (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2)))
% Found (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2)))
% Found (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2)))
% Found (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6))) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2)))
% Found (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6))) as proof of (((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))))
% Found (and_rect00 (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2))
% Found ((and_rect0 (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2))
% Found (((fun (P:Type) (x3:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x3) x1)) (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2))
% Found (((fun (P:Type) (x3:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x3) x1)) (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2))
% Found x4:(((eq a) Xx) (cF x1))
% Instantiate: x0:=x1:b
% Found x4 as proof of (((eq a) Xx) (cF x0))
% Found x4:(((eq a) Xx) (cF x0))
% Instantiate: x2:=x0:b
% Found (fun (x4:(((eq a) Xx) (cF x0)))=> x4) as proof of (((eq a) Xx) (cF x2))
% Found (fun (x3:((or (cR x0)) (cS x0))) (x4:(((eq a) Xx) (cF x0)))=> x4) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2)))
% Found (fun (x3:((or (cR x0)) (cS x0))) (x4:(((eq a) Xx) (cF x0)))=> x4) as proof of (((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))))
% Found (and_rect00 (fun (x3:((or (cR x0)) (cS x0))) (x4:(((eq a) Xx) (cF x0)))=> x4)) as proof of (((eq a) Xx) (cF x2))
% Found ((and_rect0 (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0))) (x4:(((eq a) Xx) (cF x0)))=> x4)) as proof of (((eq a) Xx) (cF x2))
% Found (((fun (P:Type) (x3:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x3) x1)) (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0))) (x4:(((eq a) Xx) (cF x0)))=> x4)) as proof of (((eq a) Xx) (cF x2))
% Found (((fun (P:Type) (x3:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x3) x1)) (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0))) (x4:(((eq a) Xx) (cF x0)))=> x4)) as proof of (((eq a) Xx) (cF x2))
% Found eq_substitution00000:=(eq_substitution0000 (fun (x6:a)=> x6)):((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x0)))
% Found (eq_substitution0000 (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2)))
% Found ((eq_substitution000 (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2)))
% Found (((eq_substitution00 Xx) (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2)))
% Found ((((eq_substitution0 a) Xx) (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2)))
% Found (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2)))
% Found (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2)))
% Found (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6))) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2)))
% Found (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6))) as proof of (((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))))
% Found (and_rect00 (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2))
% Found ((and_rect0 (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2))
% Found (((fun (P:Type) (x3:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x3) x1)) (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2))
% Found (((fun (P:Type) (x3:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x3) x1)) (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2))
% Found x4:(((eq a) Xx) (cF x1))
% Instantiate: x0:=x1:b
% Found x4 as proof of (((eq a) Xx) (cF x0))
% Found x4:(((eq a) Xx) (cF x0))
% Instantiate: x2:=x0:b
% Found x4 as proof of (((eq a) Xx) (cF x2))
% Found x4:(((eq a) Xx) (cF x0))
% Instantiate: x2:=x0:b
% Found x4 as proof of (((eq a) Xx) (cF x2))
% Found x4:(((eq a) Xx) (cF x0))
% Instantiate: x2:=x0:b
% Found x4 as proof of (((eq a) Xx) (cF x2))
% Found x4:(((eq a) Xx) (cF x0))
% Instantiate: x2:=x0:b
% Found x4 as proof of (((eq a) Xx) (cF x2))
% Found eq_ref00:=(eq_ref0 a0):(((eq (b->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (b->Prop)) a0) (fun (x:b)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: a0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of ((ex b) a0)
% Found (or_intror00 x) as proof of (P ((ex b) a0))
% Found ((or_intror0 ((ex b) a0)) x) as proof of (P ((ex b) a0))
% Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((ex b) a0))
% Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((ex b) a0))
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: a0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of ((ex b) a0)
% Found (or_intror00 x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)))
% Found ((or_intror0 ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)))
% Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)))
% Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)))
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P b0)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P b0)
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (b->Prop)) a0) (fun (x:b)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (b->Prop)) a0) (fun (x:b)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found eq_ref00:=(eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and (cS x)) (((eq a) Xx) (cF x)))))
% Found (eta_expansion_dep00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eta_expansion_dep0 (fun (x3:b)=> Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found classic0:=(classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))):((or ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) (not ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))))
% Found (classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0)
% Found (classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0)
% Found (classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cR x)) (((eq a) Xx) (cF x)))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cR x)) (((eq a) Xx) (cF x)))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF x)))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF x)))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))))
% Found (fun (x1:b)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))))
% Found (fun (x1:b)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and ((or (cR x)) (cS x))) (((eq a) Xx) (cF x)))))
% Found x3:(((eq a) Xx) (cF x0))
% Instantiate: x4:=x0:b
% Found x3 as proof of (((eq a) Xx) (cF x4))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))))
% Found (fun (x1:b)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))))
% Found (fun (x1:b)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and ((or (cR x)) (cS x))) (((eq a) Xx) (cF x)))))
% Found x3:(((eq a) Xx) (cF x0))
% Instantiate: x4:=x0:b
% Found x3 as proof of (((eq a) Xx) (cF x4))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))))
% Found (fun (x1:b)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))))
% Found (fun (x1:b)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and ((or (cR x)) (cS x))) (((eq a) Xx) (cF x)))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))))
% Found (fun (x1:b)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))))
% Found (fun (x1:b)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and ((or (cR x)) (cS x))) (((eq a) Xx) (cF x)))))
% Found eq_ref000:=(eq_ref00 (ex b)):(((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))->((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))
% Found (eq_ref00 (ex b)) as proof of (P (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref0 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found eq_ref00:=(eq_ref0 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (eq_ref0 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (b->Prop)) b0) (fun (x:b)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (b->Prop)) b0) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion b) Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion b) Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion b) Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Found eq_ref000:=(eq_ref00 (ex b)):(((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))->((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))))
% Found (eq_ref00 (ex b)) as proof of (P (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found ((eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (ex b)) as proof of (P (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found eq_ref00:=(eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (b->Prop)) b0) (fun (x:b)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (b->Prop)) b0) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion b) Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion b) Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Found (((eta_expansion b) Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) b0)
% Found classic0:=(classic ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))):((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) (not ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))))
% Found (classic ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0)
% Found (classic ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0)
% Found (classic ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P b0)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P b0)
% Found eq_ref00:=(eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and (cS x)) (((eq a) Xx) (cF x)))))
% Found (eta_expansion_dep00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eta_expansion_dep0 (fun (x3:b)=> Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cR x)) (((eq a) Xx) (cF x)))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF x)))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF x)))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cR x)) (((eq a) Xx) (cF x)))))
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P b0)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P b0)
% Found x3:(((eq a) Xx) (cF x0))
% Instantiate: x4:=x0:b
% Found x3 as proof of (((eq a) Xx) (cF x4))
% Found x3:(((eq a) Xx) (cF x0))
% Instantiate: x4:=x0:b
% Found x3 as proof of (((eq a) Xx) (cF x4))
% Found x3:(((eq a) Xx) (cF x0))
% Instantiate: x4:=x0:b
% Found x3 as proof of (((eq a) Xx) (cF x4))
% Found x3:(((eq a) Xx) (cF x0))
% Instantiate: x4:=x0:b
% Found x3 as proof of (((eq a) Xx) (cF x4))
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and (cR x)) (((eq a) Xx) (cF x)))))
% Found (eta_expansion_dep00 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eta_expansion_dep0 (fun (x5:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x5:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x5:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x5:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and (cS x)) (((eq a) Xx) (cF x)))))
% Found (eta_expansion00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P b0)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P b0)
% Found eq_ref000:=(eq_ref00 cS):((cS x1)->(cS x1))
% Found (eq_ref00 cS) as proof of ((cS x1)->(cS x0))
% Found ((eq_ref0 x1) cS) as proof of ((cS x1)->(cS x0))
% Found (((eq_ref b) x1) cS) as proof of ((cS x1)->(cS x0))
% Found (((eq_ref b) x1) cS) as proof of ((cS x1)->(cS x0))
% Found eq_ref000:=(eq_ref00 cR):((cR x1)->(cR x1))
% Found (eq_ref00 cR) as proof of ((cR x1)->(cR x0))
% Found ((eq_ref0 x1) cR) as proof of ((cR x1)->(cR x0))
% Found (((eq_ref b) x1) cR) as proof of ((cR x1)->(cR x0))
% Found (((eq_ref b) x1) cR) as proof of ((cR x1)->(cR x0))
% Found eq_ref000:=(eq_ref00 cR):((cR x0)->(cR x0))
% Found (eq_ref00 cR) as proof of ((cR x0)->(cR x2))
% Found ((eq_ref0 x0) cR) as proof of ((cR x0)->(cR x2))
% Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x2))
% Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x2))
% Found eq_ref000:=(eq_ref00 cS):((cS x0)->(cS x0))
% Found (eq_ref00 cS) as proof of ((cS x0)->(cS x2))
% Found ((eq_ref0 x0) cS) as proof of ((cS x0)->(cS x2))
% Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x2))
% Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x2))
% Found eq_ref000:=(eq_ref00 cS):((cS x0)->(cS x0))
% Found (eq_ref00 cS) as proof of ((cS x0)->(cS x2))
% Found ((eq_ref0 x0) cS) as proof of ((cS x0)->(cS x2))
% Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x2))
% Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x2))
% Found eq_ref000:=(eq_ref00 cR):((cR x0)->(cR x0))
% Found (eq_ref00 cR) as proof of ((cR x0)->(cR x2))
% Found ((eq_ref0 x0) cR) as proof of ((cR x0)->(cR x2))
% Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x2))
% Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x2))
% Found eq_ref00:=(eq_ref0 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (eq_ref0 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and (cR x)) (((eq a) Xx) (cF x)))))
% Found (eta_expansion00 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P b0)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P b0)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P b0)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P b0)
% Found x3:(((eq a) Xx) (cF x0))
% Instantiate: x4:=x0:b
% Found x3 as proof of (((eq a) Xx) (cF x4))
% Found x3:(((eq a) Xx) (cF x0))
% Instantiate: x4:=x0:b
% Found x3 as proof of (((eq a) Xx) (cF x4))
% Found x3:(((eq a) Xx) (cF x0))
% Instantiate: x4:=x0:b
% Found x3 as proof of (((eq a) Xx) (cF x4))
% Found x3:(((eq a) Xx) (cF x0))
% Instantiate: x4:=x0:b
% Found x3 as proof of (((eq a) Xx) (cF x4))
% Found x3:(((eq a) Xx) (cF x0))
% Instantiate: x4:=x0:b
% Found x3 as proof of (((eq a) Xx) (cF x4))
% Found x3:(((eq a) Xx) (cF x0))
% Instantiate: x4:=x0:b
% Found x3 as proof of (((eq a) Xx) (cF x4))
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found x40:=(x4 (fun (x5:a)=> (cR x1))):((cR x1)->(cR x1))
% Found (x4 (fun (x5:a)=> (cR x1))) as proof of ((cR x1)->(cR x0))
% Found (x4 (fun (x5:a)=> (cR x1))) as proof of ((cR x1)->(cR x0))
% Found x40:=(x4 (fun (x5:a)=> (cS x1))):((cS x1)->(cS x1))
% Found (x4 (fun (x5:a)=> (cS x1))) as proof of ((cS x1)->(cS x0))
% Found (x4 (fun (x5:a)=> (cS x1))) as proof of ((cS x1)->(cS x0))
% Found eq_ref00:=(eq_ref0 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))
% Found (eq_ref0 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found eq_ref00:=(eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))
% Found (eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and (cR x)) (((eq a) Xx) (cF x)))))
% Found (eta_expansion_dep00 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eta_expansion_dep0 (fun (x5:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x5:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x5:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x5:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and (cS x)) (((eq a) Xx) (cF x)))))
% Found (eta_expansion00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0)
% Found eq_ref000:=(eq_ref00 P):((P Xx)->(P Xx))
% Found (eq_ref00 P) as proof of (P0 Xx)
% Found ((eq_ref0 Xx) P) as proof of (P0 Xx)
% Found (((eq_ref a) Xx) P) as proof of (P0 Xx)
% Found (((eq_ref a) Xx) P) as proof of (P0 Xx)
% Found x40:=(x4 (fun (x5:a)=> (cS x0))):((cS x0)->(cS x0))
% Found (x4 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x2))
% Found (x4 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x2))
% Found x40:=(x4 (fun (x5:a)=> (cR x0))):((cR x0)->(cR x0))
% Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2))
% Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2))
% Found x40:=(x4 (fun (x5:a)=> (cR x1))):((cR x1)->(cR x1))
% Found (x4 (fun (x5:a)=> (cR x1))) as proof of ((cR x1)->(cR x0))
% Found (x4 (fun (x5:a)=> (cR x1))) as proof of ((cR x1)->(cR x0))
% Found x40:=(x4 (fun (x5:a)=> (cR x0))):((cR x0)->(cR x0))
% Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2))
% Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2))
% Found x40:=(x4 (fun (x5:a)=> (cS x0))):((cS x0)->(cS x0))
% Found (x4 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x2))
% Found (x4 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x2))
% Found eq_ref000:=(eq_ref00 cS):((cS x1)->(cS x1))
% Found (eq_ref00 cS) as proof of ((cS x1)->(cS x0))
% Found ((eq_ref0 x1) cS) as proof of ((cS x1)->(cS x0))
% Found (((eq_ref b) x1) cS) as proof of ((cS x1)->(cS x0))
% Found (((eq_ref b) x1) cS) as proof of ((cS x1)->(cS x0))
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) (cF x0))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) (cF x0))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) (cF x0))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) (cF x0))
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found x40:=(x4 (fun (x5:a)=> (cR x0))):((cR x0)->(cR x0))
% Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2))
% Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2))
% Found x40:=(x4 (fun (x5:a)=> (cS x0))):((cS x0)->(cS x0))
% Found (x4 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x2))
% Found (x4 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x2))
% Found eq_ref000:=(eq_ref00 cR):((cR x0)->(cR x0))
% Found (eq_ref00 cR) as proof of ((cR x0)->(cR x4))
% Found ((eq_ref0 x0) cR) as proof of ((cR x0)->(cR x4))
% Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x4))
% Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x4))
% Found eq_ref000:=(eq_ref00 cS):((cS x0)->(cS x0))
% Found (eq_ref00 cS) as proof of ((cS x0)->(cS x4))
% Found ((eq_ref0 x0) cS) as proof of ((cS x0)->(cS x4))
% Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x4))
% Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x4))
% Found x40:=(x4 (fun (x5:a)=> (cR x0))):((cR x0)->(cR x0))
% Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2))
% Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2))
% Found x40:=(x4 (fun (x5:a)=> (cS x0))):((cS x0)->(cS x0))
% Found (x4 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x2))
% Found (x4 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x2))
% Found x30:=(x3 (fun (x5:a)=> (cS x0))):((cS x0)->(cS x0))
% Found (x3 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x4))
% Found (x3 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x4))
% Found x30:=(x3 (fun (x5:a)=> (cR x0))):((cR x0)->(cR x0))
% Found (x3 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x4))
% Found (x3 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x4))
% Found x30:=(x3 (fun (x5:a)=> (cR x0))):((cR x0)->(cR x0))
% Found (x3 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x4))
% Found (x3 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x4))
% Found x30:=(x3 (fun (x5:a)=> (cS x0))):((cS x0)->(cS x0))
% Found (x3 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x4))
% Found (x3 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x4))
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: a0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of ((ex b) a0)
% Found (or_intror00 x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)))
% Found ((or_intror0 ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)))
% Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)))
% Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)))
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: a0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of ((ex b) a0)
% Found (or_intror00 x) as proof of (P ((ex b) a0))
% Found ((or_intror0 ((ex b) a0)) x) as proof of (P ((ex b) a0))
% Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((ex b) a0))
% Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((ex b) a0))
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))
% Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop)
% Found x as proof of (P f)
% 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 (cR x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cR x)) (((eq a) Xx) (cF x)))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF x)))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF x)))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2))))
% Found ((eq_ref P
% EOF
%------------------------------------------------------------------------------