%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV195^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n186.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 13:33:52 EDT 2014

% Result   : Timeout 300.03s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV195^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n186.star.cs.uiowa.edu
% % Model    : x86_64 x86_64
% % CPU      : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory   : 32286.75MB
% % OS       : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 08:26:41 CDT 2014
% % CPUTime  : 300.03 
% 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 0x1e5bbd8>, <kernel.Type object at 0x1e5bd40>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x1b42170>, <kernel.DependentProduct object at 0x1e5be60>) of role type named cP
% Using role type
% Declaring cP:(a->(a->a))
% FOF formula (<kernel.Constant object at 0x20b6d88>, <kernel.Constant object at 0x1e5b518>) of role type named cZ
% Using role type
% Declaring cZ:a
% FOF formula (((and ((and (forall (Xx:a) (Xy:a), (not (((eq a) ((cP Xx) Xy)) cZ)))) (forall (Xx:a) (Xy:a) (Xu:a) (Xv:a), ((((eq a) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq a) Xx) Xy)) (((eq a) Xu) Xv)))))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:a), (X Xx)))))->(forall (Xx:a), ((or (((eq a) Xx) cZ)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))))) of role conjecture named cS_LEM1D_pme
% Conjecture to prove = (((and ((and (forall (Xx:a) (Xy:a), (not (((eq a) ((cP Xx) Xy)) cZ)))) (forall (Xx:a) (Xy:a) (Xu:a) (Xv:a), ((((eq a) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq a) Xx) Xy)) (((eq a) Xu) Xv)))))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:a), (X Xx)))))->(forall (Xx:a), ((or (((eq a) Xx) cZ)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))))):Prop
% We need to prove ['(((and ((and (forall (Xx:a) (Xy:a), (not (((eq a) ((cP Xx) Xy)) cZ)))) (forall (Xx:a) (Xy:a) (Xu:a) (Xv:a), ((((eq a) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq a) Xx) Xy)) (((eq a) Xu) Xv)))))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:a), (X Xx)))))->(forall (Xx:a), ((or (((eq a) Xx) cZ)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))))))']
% Parameter a:Type.
% Parameter cP:(a->(a->a)).
% Parameter cZ:a.
% Trying to prove (((and ((and (forall (Xx:a) (Xy:a), (not (((eq a) ((cP Xx) Xy)) cZ)))) (forall (Xx:a) (Xy:a) (Xu:a) (Xv:a), ((((eq a) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq a) Xx) Xy)) (((eq a) Xu) Xv)))))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:a), (X Xx)))))->(forall (Xx:a), ((or (((eq a) Xx) cZ)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))))))
% 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 eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% 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 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 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 eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% 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 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 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 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 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 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 eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found x2:(P cZ)
% Found x2 as proof of (P cZ)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found x2:(P cZ)
% Found x2 as proof of (P cZ)
% Found classic0:=(classic (((eq a) Xx) cZ)):((or (((eq a) Xx) cZ)) (not (((eq a) Xx) cZ)))
% Found (classic (((eq a) Xx) cZ)) as proof of (P b)
% Found (classic (((eq a) Xx) cZ)) as proof of (P b)
% Found (classic (((eq a) Xx) cZ)) as proof of (P b)
% Found eq_ref00:=(eq_ref0 ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))):(((eq Prop) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))))
% Found (eq_ref0 ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) as proof of (((eq Prop) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) b)
% Found ((eq_ref Prop) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) as proof of (((eq Prop) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) b)
% Found ((eq_ref Prop) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) as proof of (((eq Prop) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) b)
% Found ((eq_ref Prop) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) as proof of (((eq Prop) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found x2:(P cZ)
% Found x2 as proof of (P cZ)
% Found x0:(P cZ)
% Found x0 as proof of (P cZ)
% 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 eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (a->Prop)) a0) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (a->Prop)) a0) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))
% 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 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 x0:(P Xx)
% Instantiate: b:=Xx:a
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found eq_ref00:=(eq_ref0 ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))):(((eq Prop) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))))
% Found (eq_ref0 ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) as proof of (((eq Prop) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) b)
% Found ((eq_ref Prop) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) as proof of (((eq Prop) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) b)
% Found ((eq_ref Prop) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) as proof of (((eq Prop) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) b)
% Found ((eq_ref Prop) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) as proof of (((eq Prop) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) b)
% Found classic0:=(classic (((eq a) Xx) cZ)):((or (((eq a) Xx) cZ)) (not (((eq a) Xx) cZ)))
% Found (classic (((eq a) Xx) cZ)) as proof of (P b)
% Found (classic (((eq a) Xx) cZ)) as proof of (P b)
% Found (classic (((eq a) Xx) cZ)) as proof of (P b)
% 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 classic0:=(classic ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))):((or ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) (not ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))))
% Found (classic ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) as proof of (P b)
% Found (classic ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) as proof of (P b)
% Found (classic ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (((eq a) Xx) cZ)):(((eq Prop) (((eq a) Xx) cZ)) (((eq a) Xx) cZ))
% Found (eq_ref0 (((eq a) Xx) cZ)) as proof of (((eq Prop) (((eq a) Xx) cZ)) b)
% Found ((eq_ref Prop) (((eq a) Xx) cZ)) as proof of (((eq Prop) (((eq a) Xx) cZ)) b)
% Found ((eq_ref Prop) (((eq a) Xx) cZ)) as proof of (((eq Prop) (((eq a) Xx) cZ)) b)
% Found ((eq_ref Prop) (((eq a) Xx) cZ)) as proof of (((eq Prop) (((eq a) Xx) cZ)) b)
% 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 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 x2:(P1 cZ)
% Found x2 as proof of (P1 cZ)
% Found x2:(P1 cZ)
% Found x2 as proof of (P1 cZ)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))):(((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) (fun (x:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x) Xz))))))
% Found (eta_expansion_dep00 (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found x2:(P cZ)
% Found x2 as proof of (P cZ)
% Found x4:(P cZ)
% Found x4 as proof of (P cZ)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found x2:(P1 cZ)
% Found x2 as proof of (P1 cZ)
% Found x2:(P1 cZ)
% Found x2 as proof of (P1 cZ)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found x2:(P cZ)
% Found x2 as proof of (P cZ)
% Found x4:(P cZ)
% Found x4 as proof of (P cZ)
% 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)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x0) Xz)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x0) Xz)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x0) Xz)))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x0) Xz)))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x) Xz))))))
% 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)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x0) Xz)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x0) Xz)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x0) Xz)))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x0) Xz)))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x) Xz))))))
% Found x4:(P cZ)
% Found x4 as proof of (P cZ)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found x2:(P cZ)
% Found x2 as proof of (P cZ)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found x2:(P1 cZ)
% Found x2 as proof of (P1 cZ)
% Found x2:(P1 cZ)
% Found x2 as proof of (P1 cZ)
% Found x4:(P cZ)
% Found x4 as proof of (P cZ)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found x4:(P cZ)
% Found x4 as proof of (P cZ)
% Found x4:(P cZ)
% Found x4 as proof of (P cZ)
% 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 eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found x0:(P1 cZ)
% Found x0 as proof of (P1 cZ)
% Found x0:(P1 cZ)
% Found x0 as proof of (P1 cZ)
% Found x0:(P cZ)
% Found x0 as proof of (P cZ)
% Found x2:(P Xx)
% Instantiate: b:=Xx:a
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found x2:(P cZ)
% Found x2 as proof of (P cZ)
% Found x2:(P cZ)
% Found x2 as proof of (P cZ)
% Found x2:(P cZ)
% Found x2 as proof of (P cZ)
% Found x0:(P Xx)
% Instantiate: b:=Xx:a
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found x0:(P cZ)
% Instantiate: b:=cZ:a
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))):(((eq Prop) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))))
% Found (eq_ref0 ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) as proof of (((eq Prop) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) b)
% Found ((eq_ref Prop) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) as proof of (((eq Prop) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) b)
% Found ((eq_ref Prop) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) as proof of (((eq Prop) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) b)
% Found ((eq_ref Prop) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) as proof of (((eq Prop) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) b)
% Found classic0:=(classic (((eq a) Xx) cZ)):((or (((eq a) Xx) cZ)) (not (((eq a) Xx) cZ)))
% Found (classic (((eq a) Xx) cZ)) as proof of (P b)
% Found (classic (((eq a) Xx) cZ)) as proof of (P b)
% Found (classic (((eq a) Xx) cZ)) as proof of (P b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found classic0:=(classic ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))):((or ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) (not ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))))
% Found (classic ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) as proof of (P b)
% Found (classic ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) as proof of (P b)
% Found (classic ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (((eq a) Xx) cZ)):(((eq Prop) (((eq a) Xx) cZ)) (((eq a) Xx) cZ))
% Found (eq_ref0 (((eq a) Xx) cZ)) as proof of (((eq Prop) (((eq a) Xx) cZ)) b)
% Found ((eq_ref Prop) (((eq a) Xx) cZ)) as proof of (((eq Prop) (((eq a) Xx) cZ)) b)
% Found ((eq_ref Prop) (((eq a) Xx) cZ)) as proof of (((eq Prop) (((eq a) Xx) cZ)) b)
% Found ((eq_ref Prop) (((eq a) Xx) cZ)) as proof of (((eq Prop) (((eq a) Xx) cZ)) b)
% Found x2:(P cZ)
% Found x2 as proof of (P cZ)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))):(((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) (fun (x:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x) Xz))))))
% Found (eta_expansion_dep00 (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))):(((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) (fun (x:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x) Xz))))))
% Found (eta_expansion00 (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found x0:(P cZ)
% Found x0 as proof of (P cZ)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found x2:(P Xx)
% Instantiate: b:=Xx:a
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found x0:(P cZ)
% Found x0 as proof of (P cZ)
% Found x2:(P cZ)
% Found x2 as proof of (P cZ)
% Found x2:(P cZ)
% Found x2 as proof of (P cZ)
% Found x2:(P cZ)
% Found x2 as proof of (P cZ)
% Found eq_ref00:=(eq_ref0 (((eq a) Xx) cZ)):(((eq Prop) (((eq a) Xx) cZ)) (((eq a) Xx) cZ))
% Found (eq_ref0 (((eq a) Xx) cZ)) as proof of (((eq Prop) (((eq a) Xx) cZ)) b)
% Found ((eq_ref Prop) (((eq a) Xx) cZ)) as proof of (((eq Prop) (((eq a) Xx) cZ)) b)
% Found ((eq_ref Prop) (((eq a) Xx) cZ)) as proof of (((eq Prop) (((eq a) Xx) cZ)) b)
% Found ((eq_ref Prop) (((eq a) Xx) cZ)) as proof of (((eq Prop) (((eq a) Xx) cZ)) b)
% Found classic0:=(classic ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))):((or ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) (not ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))))
% Found (classic ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) as proof of (P b)
% Found (classic ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) as proof of (P b)
% Found (classic ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) as proof of (P b)
% Found x0:(P Xx)
% Instantiate: b:=Xx:a
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))):(((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) (fun (x:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x) Xz))))))
% Found (eta_expansion00 (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found x2:(P1 cZ)
% Found x2 as proof of (P1 cZ)
% Found x2:(P1 cZ)
% Found x2 as proof of (P1 cZ)
% 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)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x2) Xz)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x2) Xz)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x2) Xz)))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x2) Xz)))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x) Xz))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x2) Xz)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x2) Xz)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x2) Xz)))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x2) Xz)))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x) Xz))))))
% 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)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x0) Xz)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x0) Xz)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x0) Xz)))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x0) Xz)))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x) Xz))))))
% 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)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x0) Xz)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x0) Xz)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x0) Xz)))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x0) Xz)))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x) Xz))))))
% Found x4:(P1 cZ)
% Found x4 as proof of (P1 cZ)
% Found x4:(P1 cZ)
% Found x4 as proof of (P1 cZ)
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (a->Prop)) a0) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (a->Prop)) a0) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (a->Prop)) a0) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (a->Prop)) a0) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))
% Found x2:(P1 cZ)
% Found x2 as proof of (P1 cZ)
% Found x2:(P1 cZ)
% Found x2 as proof of (P1 cZ)
% Found x4:(P cZ)
% Found x4 as proof of (P cZ)
% 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 eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) cZ)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) cZ)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) cZ)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) cZ)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x2) Xz)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x2) Xz)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x2) Xz)))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x2) Xz)))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x) Xz))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x2) Xz)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x2) Xz)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x2) Xz)))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x2) Xz)))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x) Xz))))))
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% 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 eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found x4:(P1 cZ)
% Found x4 as proof of (P1 cZ)
% Found x4:(P1 cZ)
% Found x4 as proof of (P1 cZ)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% 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 eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found x4:(P1 cZ)
% Found x4 as proof of (P1 cZ)
% Found x4:(P1 cZ)
% Found x4 as proof of (P1 cZ)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found x4:(P cZ)
% Found x4 as proof of (P cZ)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found x2:(P b)
% Found x2 as proof of (P cZ)
% Found x4:(P cZ)
% Found x4 as proof of (P cZ)
% Found x2:(P cZ)
% Found x2 as proof of (P cZ)
% Found x2:(P1 cZ)
% Found x2 as proof of (P1 cZ)
% Found x2:(P1 cZ)
% Found x2 as proof of (P1 cZ)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found x0:(P cZ)
% Instantiate: b:=cZ:a
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref000:=(eq_ref00 P):((P 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 x2:(P Xx)
% Instantiate: b:=Xx:a
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found x2:(P cZ)
% Instantiate: b:=cZ:a
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found x4:(P Xx)
% Instantiate: b:=Xx:a
% Found x4 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found eq_ref000:=(eq_ref00 P):((P cZ)->(P cZ))
% Found (eq_ref00 P) as proof of (P0 cZ)
% Found ((eq_ref0 cZ) P) as proof of (P0 cZ)
% Found (((eq_ref a) cZ) P) as proof of (P0 cZ)
% Found (((eq_ref a) cZ) P) as proof of (P0 cZ)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found classic0:=(classic ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))):((or ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) (not ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))))
% Found (classic ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) as proof of (P b)
% Found (classic ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) as proof of (P b)
% Found (classic ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (((eq a) Xx) cZ)):(((eq Prop) (((eq a) Xx) cZ)) (((eq a) Xx) cZ))
% Found (eq_ref0 (((eq a) Xx) cZ)) as proof of (((eq Prop) (((eq a) Xx) cZ)) b)
% Found ((eq_ref Prop) (((eq a) Xx) cZ)) as proof of (((eq Prop) (((eq a) Xx) cZ)) b)
% Found ((eq_ref Prop) (((eq a) Xx) cZ)) as proof of (((eq Prop) (((eq a) Xx) cZ)) b)
% Found ((eq_ref Prop) (((eq a) Xx) cZ)) as proof of (((eq Prop) (((eq a) Xx) cZ)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))):(((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) (fun (x:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x) Xz))))))
% Found (eta_expansion_dep00 (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))):(((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) (fun (x:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x) Xz))))))
% Found (eta_expansion_dep00 (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found x2:(P b)
% Found x2 as proof of (P cZ)
% Found x4:(P cZ)
% Found x4 as proof of (P cZ)
% Found x4:(P cZ)
% Found x4 as proof of (P cZ)
% Found x2:(P cZ)
% Found x2 as proof of (P cZ)
% Found x4:(P cZ)
% Found x4 as proof of (P cZ)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found x2:(P Xx)
% Instantiate: b:=Xx:a
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found x2:(P cZ)
% Instantiate: b:=cZ:a
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found x4:(P Xx)
% Instantiate: b:=Xx:a
% Found x4 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found x4:(P Xx)
% Instantiate: b:=Xx:a
% Found x4 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found eq_ref00:=(eq_ref0 (((eq a) Xx) cZ)):(((eq Prop) (((eq a) Xx) cZ)) (((eq a) Xx) cZ))
% Found (eq_ref0 (((eq a) Xx) cZ)) as proof of (((eq Prop) (((eq a) Xx) cZ)) b)
% Found ((eq_ref Prop) (((eq a) Xx) cZ)) as proof of (((eq Prop) (((eq a) Xx) cZ)) b)
% Found ((eq_ref Prop) (((eq a) Xx) cZ)) as proof of (((eq Prop) (((eq a) Xx) cZ)) b)
% Found ((eq_ref Prop) (((eq a) Xx) cZ)) as proof of (((eq Prop) (((eq a) Xx) cZ)) b)
% Found classic0:=(classic ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))):((or ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) (not ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))))
% Found (classic ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) as proof of (P b)
% Found (classic ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) as proof of (P b)
% Found (classic ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) as proof of (P b)
% Found x4:(P1 cZ)
% Found x4 as proof of (P1 cZ)
% Found x4:(P1 cZ)
% Found x4 as proof of (P1 cZ)
% Found eq_ref00:=(eq_ref0 (((eq a) Xx) cZ)):(((eq Prop) (((eq a) Xx) cZ)) (((eq a) Xx) cZ))
% Found (eq_ref0 (((eq a) Xx) cZ)) as proof of (((eq Prop) (((eq a) Xx) cZ)) b)
% Found ((eq_ref Prop) (((eq a) Xx) cZ)) as proof of (((eq Prop) (((eq a) Xx) cZ)) b)
% Found ((eq_ref Prop) (((eq a) Xx) cZ)) as proof of (((eq Prop) (((eq a) Xx) cZ)) b)
% Found ((eq_ref Prop) (((eq a) Xx) cZ)) as proof of (((eq Prop) (((eq a) Xx) cZ)) b)
% Found classic0:=(classic ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))):((or ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) (not ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))))
% Found (classic ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) as proof of (P b)
% Found (classic ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) as proof of (P b)
% Found (classic ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz))))))) as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))):(((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) (fun (x:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x) Xz))))))
% Found (eta_expansion_dep00 (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found x4:(P1 cZ)
% Found x4 as proof of (P1 cZ)
% Found x4:(P1 cZ)
% Found x4 as proof of (P1 cZ)
% Found x4:(P1 cZ)
% Found x4 as proof of (P1 cZ)
% Found x4:(P1 cZ)
% Found x4 as proof of (P1 cZ)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))):(((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) (fun (x:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x) Xz))))))
% Found (eta_expansion_dep00 (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))):(((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) (fun (x:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x) Xz))))))
% Found (eta_expansion00 (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found x0:(P Xx)
% Instantiate: b:=Xx:a
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found x0:(P1 cZ)
% Found x0 as proof of (P1 cZ)
% Found x0:(P1 cZ)
% Found x0 as proof of (P1 cZ)
% Found x2:(P Xx)
% Instantiate: b:=Xx:a
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found x2:(P Xx)
% Instantiate: b:=Xx:a
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found x2:(P1 cZ)
% Found x2 as proof of (P1 cZ)
% Found x2:(P1 cZ)
% Found x2 as proof of (P1 cZ)
% Found x2:(P1 cZ)
% Found x2 as proof of (P1 cZ)
% Found x2:(P1 cZ)
% Found x2 as proof of (P1 cZ)
% Found x2:(P1 cZ)
% Found x2 as proof of (P1 cZ)
% Found x2:(P1 cZ)
% Found x2 as proof of (P1 cZ)
% Found x2:(P1 cZ)
% Found x2 as proof of (P1 cZ)
% Found x2:(P1 cZ)
% Found x2 as proof of (P1 cZ)
% 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 eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found x2:(P b)
% Found x2 as proof of (P cZ)
% Found x4:(P cZ)
% Found x4 as proof of (P cZ)
% Found x4:(P cZ)
% Found x4 as proof of (P cZ)
% Found x0:(P1 cZ)
% Found x0 as proof of (P1 cZ)
% Found x0:(P1 cZ)
% Found x0 as proof of (P1 cZ)
% Found x4:(P cZ)
% Found x4 as proof of (P cZ)
% Found x4:(P cZ)
% Found x4 as proof of (P cZ)
% Found x2:(P Xx)
% Instantiate: b:=Xx:a
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found x2:(P cZ)
% Found x2 as proof of (P cZ)
% Found eq_ref00:=(eq_ref0 (forall (Xx0:a) (Xy:a), (((and (P Xx0)) (P Xy))->(P ((cP Xx0) Xy))))):(((eq Prop) (forall (Xx0:a) (Xy:a), (((and (P Xx0)) (P Xy))->(P ((cP Xx0) Xy))))) (forall (Xx0:a) (Xy:a), (((and (P Xx0)) (P Xy))->(P ((cP Xx0) Xy)))))
% Found (eq_ref0 (forall (Xx0:a) (Xy:a), (((and (P Xx0)) (P Xy))->(P ((cP Xx0) Xy))))) as proof of (((eq Prop) (forall (Xx0:a) (Xy:a), (((and (P Xx0)) (P Xy))->(P ((cP Xx0) Xy))))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy:a), (((and (P Xx0)) (P Xy))->(P ((cP Xx0) Xy))))) as proof of (((eq Prop) (forall (Xx0:a) (Xy:a), (((and (P Xx0)) (P Xy))->(P ((cP Xx0) Xy))))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy:a), (((and (P Xx0)) (P Xy))->(P ((cP Xx0) Xy))))) as proof of (((eq Prop) (forall (Xx0:a) (Xy:a), (((and (P Xx0)) (P Xy))->(P ((cP Xx0) Xy))))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy:a), (((and (P Xx0)) (P Xy))->(P ((cP Xx0) Xy))))) as proof of (((eq Prop) (forall (Xx0:a) (Xy:a), (((and (P Xx0)) (P Xy))->(P ((cP Xx0) Xy))))) b)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found x0:(P cZ)
% Found x0 as proof of (P cZ)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found x2:(P cZ)
% Found x2 as proof of (P cZ)
% Found x2:(P cZ)
% Found x2 as proof of (P cZ)
% Found x2:(P cZ)
% Instantiate: b:=cZ:a
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))):(((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) (fun (x:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x) Xz))))))
% Found (eta_expansion_dep00 (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP Xy) Xz)))))) b)
% Found x2:(P cZ)
% Found x2 as proof of (P cZ)
% Found x2:(P cZ)
% Found x2 as proof of (P cZ)
% Found x0:(P Xx)
% Instantiate: b:=Xx:a
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b)
% Found x0:(P1 cZ)
% Found x0 as proof of (P1 cZ)
% Found x0:(P1 cZ)
% Found x0 as proof of (P1 cZ)
% Found x2:(P1 cZ)
% Found x2 as proof of (P1 cZ)
% Found x2:(P1 cZ)
% Found x2 as proof of (P1 cZ)
% 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)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x2) Xz)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x2) Xz)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x2) Xz)))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x2) Xz)))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x) Xz))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x2) Xz)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x2) Xz)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x2) Xz)))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x2) Xz)))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x) Xz))))))
% Found x2:(P1 cZ)
% Found x2 as proof of (P1 cZ)
% Found x2:(P1 cZ)
% Found x2 as proof of (P1 cZ)
% Found x2:(P1 cZ)
% Found x2 as proof of (P1 cZ)
% Found x2:(P1 cZ)
% Found x2 as proof of (P1 cZ)
% Found x0:(P cZ)
% Instantiate: b:=cZ:a
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x4) Xz)))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x4) Xz)))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x4) Xz)))))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x4) Xz)))))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x) Xz))))))
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x4) Xz)))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x4) Xz)))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x4) Xz)))))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x4) Xz)))))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x) Xz))))))
% 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 eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) cZ)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) cZ)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) cZ)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) cZ)
% 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 eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) cZ)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) cZ)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) cZ)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) cZ)
% Found eq_ref00:=(eq_ref0 (forall (Xx0:a) (Xy:a), (((and (P Xx0)) (P Xy))->(P ((cP Xx0) Xy))))):(((eq Prop) (forall (Xx0:a) (Xy:a), (((and (P Xx0)) (P Xy))->(P ((cP Xx0) Xy))))) (forall (Xx0:a) (Xy:a), (((and (P Xx0)) (P Xy))->(P ((cP Xx0) Xy)))))
% Found (eq_ref0 (forall (Xx0:a) (Xy:a), (((and (P Xx0)) (P Xy))->(P ((cP Xx0) Xy))))) as proof of (((eq Prop) (forall (Xx0:a) (Xy:a), (((and (P Xx0)) (P Xy))->(P ((cP Xx0) Xy))))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy:a), (((and (P Xx0)) (P Xy))->(P ((cP Xx0) Xy))))) as proof of (((eq Prop) (forall (Xx0:a) (Xy:a), (((and (P Xx0)) (P Xy))->(P ((cP Xx0) Xy))))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy:a), (((and (P Xx0)) (P Xy))->(P ((cP Xx0) Xy))))) as proof of (((eq Prop) (forall (Xx0:a) (Xy:a), (((and (P Xx0)) (P Xy))->(P ((cP Xx0) Xy))))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy:a), (((and (P Xx0)) (P Xy))->(P ((cP Xx0) Xy))))) as proof of (((eq Prop) (forall (Xx0:a) (Xy:a), (((and (P Xx0)) (P Xy))->(P ((cP Xx0) Xy))))) b)
% Found x0:(P cZ)
% Found x0 as proof of (P cZ)
% Found x2:(P cZ)
% Found x2 as proof of (P cZ)
% Found x2:(P cZ)
% Found x2 as proof of (P cZ)
% Found x2:(P cZ)
% Found x2 as proof of (P cZ)
% Found x2:(P cZ)
% Found x2 as proof of (P cZ)
% Found x2:(P cZ)
% Found x2 as proof of (P cZ)
% Found x2:(P cZ)
% Found x2 as proof of (P cZ)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy:a), (((and (P Xx)) (P Xy))->(P ((cP Xx) Xy))))):(((eq Prop) (forall (Xx:a) (Xy:a), (((and (P Xx)) (P Xy))->(P ((cP Xx) Xy))))) (forall (Xx:a) (Xy:a), (((and (P Xx)) (P Xy))->(P ((cP Xx) Xy)))))
% Found (eq_ref0 (forall (Xx:a) (Xy:a), (((and (P Xx)) (P Xy))->(P ((cP Xx) Xy))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (((and (P Xx)) (P Xy))->(P ((cP Xx) Xy))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (((and (P Xx)) (P Xy))->(P ((cP Xx) Xy))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (((and (P Xx)) (P Xy))->(P ((cP Xx) Xy))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (((and (P Xx)) (P Xy))->(P ((cP Xx) Xy))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (((and (P Xx)) (P Xy))->(P ((cP Xx) Xy))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (((and (P Xx)) (P Xy))->(P ((cP Xx) Xy))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (((and (P Xx)) (P Xy))->(P ((cP Xx) Xy))))) b)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x2) Xz)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x2) Xz)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x2) Xz)))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x2) Xz)))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x) Xz))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x2) Xz)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x2) Xz)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x2) Xz)))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x2) Xz)))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x) Xz))))))
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found x0:(P cZ)
% Found x0 as proof of (P cZ)
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x4) Xz)))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x4) Xz)))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x4) Xz)))))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x4) Xz)))))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x) Xz))))))
% Found x0:(P cZ)
% Found x0 as proof of (P cZ)
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x4) Xz)))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x4) Xz)))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x4) Xz)))))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x4) Xz)))))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x) Xz))))))
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x4) Xz)))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x4) Xz)))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x4) Xz)))))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x4) Xz)))))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x) Xz))))))
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x4) Xz)))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x4) Xz)))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x4) Xz)))))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x4) Xz)))))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex a) (fun (Xz:a)=> (((eq a) Xx) ((cP x) Xz))))))
% Found eq_ref000:=(eq_ref00 P):((P cZ)->(P cZ))
% Found (eq_ref00 P) as proof of (P0 cZ)
% Found ((eq_ref0 cZ) P) as proof of (P0 cZ)
% Found (((eq_ref a) cZ) P) as proof of (P0 cZ)
% Found (((eq_ref a) cZ) P) as proof of (P0 cZ)
% 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 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 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 eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) cZ)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) cZ)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) cZ)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) cZ)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found eq_ref00:=(eq_ref0 cZ):(((eq a) cZ) cZ)
% Found (eq_ref0 cZ) as proof of (((eq a) cZ) b0)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b0)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b0)
% Found ((eq_ref a) cZ) as proof of (((eq a) cZ) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found ((eq_ref a) b) as proof of (((eq a) b) cZ)
% Found x4:(P1 cZ)
% Found x4 as proof of (P1 cZ)
% Found x4:(P1 cZ)
% Found x4 as pr
% EOF
%------------------------------------------------------------------------------