%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV386^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n090.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 13:34:07 EDT 2014

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

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV386^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n090.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 09:03:21 CDT 2014
% % CPUTime  : 300.01 
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x236edd0>, <kernel.Type object at 0x236e518>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x28753b0>, <kernel.DependentProduct object at 0x236e5f0>) of role type named p
% Using role type
% Declaring p:(a->Prop)
% FOF formula ((iff ((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy))))) ((ex a) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))) of role conjecture named cTTTP5306A_pme
% Conjecture to prove = ((iff ((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy))))) ((ex a) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['((iff ((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy))))) ((ex a) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))))']
% Parameter a:Type.
% Parameter p:(a->Prop).
% Trying to prove ((iff ((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy))))) ((ex a) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))))
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))->((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))))
% Found (eq_ref00 (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eq_ref0 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))->((ex a) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))))
% Found (eq_ref00 (ex a)) as proof of (P (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((eq_ref0 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (ex a)) as proof of (P (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (ex a)) as proof of (P (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (ex a)) as proof of (P (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))->((ex a) (fun (x:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eta_expansion00 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eta_expansion0 Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))->((ex a) (fun (x:a)=> ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((eta_expansion00 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (ex a)) as proof of (P (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eta_expansion0 Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (ex a)) as proof of (P (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (ex a)) as proof of (P (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (ex a)) as proof of (P (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))->((ex a) (fun (x:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eta_expansion00 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eta_expansion0 Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))->((ex a) (fun (x:a)=> ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((eta_expansion00 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (ex a)) as proof of (P (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eta_expansion0 Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (ex a)) as proof of (P (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (ex a)) as proof of (P (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (ex a)) as proof of (P (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))->((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))))
% Found (eq_ref00 (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found ((eq_ref0 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))->((ex a) (fun (x:a)=> (((eq (a->Prop)) p) ((eq a) x)))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found ((eta_expansion00 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found (((eta_expansion0 Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))->((ex a) (fun (x:a)=> (((eq (a->Prop)) p) ((eq a) x)))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found ((eta_expansion00 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found (((eta_expansion0 Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found x2:((ex a) ((unique a) p))
% Found (fun (x2:((ex a) ((unique a) p)))=> x2) as proof of ((ex a) ((unique a) p))
% Found (fun (x2:((ex a) ((unique a) p)))=> x2) as proof of (P ((unique a) p))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))->((ex a) (fun (x:a)=> ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((eta_expansion00 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (ex a)) as proof of (P (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eta_expansion0 Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (ex a)) as proof of (P (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (ex a)) as proof of (P (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (ex a)) as proof of (P (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))->((ex a) (fun (x:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eta_expansion00 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eta_expansion0 Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found x2:((ex a) ((unique a) p))
% Found (fun (x2:((ex a) ((unique a) p)))=> x2) as proof of ((ex a) ((unique a) p))
% Found (fun (x2:((ex a) ((unique a) p)))=> x2) as proof of (P ((unique a) p))
% Found x2:((ex a) ((unique a) p))
% Found (fun (x2:((ex a) ((unique a) p)))=> x2) as proof of ((ex a) ((unique a) p))
% Found (fun (x2:((ex a) ((unique a) p)))=> x2) as proof of (P ((unique a) p))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex a)):(((ex a) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))->((ex a) (fun (x:a)=> ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))))
% Found (eta_expansion_dep000 (ex a)) as proof of (P (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((eta_expansion_dep00 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (ex a)) as proof of (P (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (ex a)) as proof of (P (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (ex a)) as proof of (P (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (ex a)) as proof of (P (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))->((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))))
% Found (eq_ref00 (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eq_ref0 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))->((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))))
% Found (eq_ref00 (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eq_ref0 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))->((ex a) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))))
% Found (eq_ref00 (ex a)) as proof of (P (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((eq_ref0 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (ex a)) as proof of (P (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (ex a)) as proof of (P (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (ex a)) as proof of (P (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found eq_ref00:=(eq_ref0 x0):(((eq a) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq a) x0) Xz)
% Found ((eq_ref a) x0) as proof of (((eq a) x0) Xz)
% Found ((eq_ref a) x0) as proof of (((eq a) x0) Xz)
% Found (fun (x00:(p Xz))=> ((eq_ref a) x0)) as proof of (((eq a) x0) Xz)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of ((P x0)->(P Xz))
% Found ((eq_ref0 x0) P) as proof of ((P x0)->(P Xz))
% Found (((eq_ref a) x0) P) as proof of ((P x0)->(P Xz))
% Found (((eq_ref a) x0) P) as proof of ((P x0)->(P Xz))
% Found (fun (P:(a->Prop))=> (((eq_ref a) x0) P)) as proof of ((P x0)->(P Xz))
% Found (fun (x00:(p Xz)) (P:(a->Prop))=> (((eq_ref a) x0) P)) as proof of (((eq a) x0) Xz)
% Found x1:(P x0)
% Instantiate: x0:=Xz:a
% Found (fun (x1:(P x0))=> x1) as proof of (P Xz)
% Found (fun (P:(a->Prop)) (x1:(P x0))=> x1) as proof of ((P x0)->(P Xz))
% Found (fun (x00:(p Xz)) (P:(a->Prop)) (x1:(P x0))=> x1) as proof of (((eq a) x0) Xz)
% Found eq_ref00:=(eq_ref0 x0):(((eq a) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq a) x0) Xz)
% Found ((eq_ref a) x0) as proof of (((eq a) x0) Xz)
% Found ((eq_ref a) x0) as proof of (((eq a) x0) Xz)
% Found (fun (x00:(p Xz))=> ((eq_ref a) x0)) as proof of (((eq a) x0) Xz)
% Found x:((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Instantiate: b:=(fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))):(a->Prop)
% Found x as proof of (P b)
% Found eq_ref00:=(eq_ref0 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))):(((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (eq_ref0 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found x:((ex a) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Instantiate: b:=(fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))):(a->Prop)
% Found x as proof of (P b)
% Found eq_ref00:=(eq_ref0 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))):(((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (eq_ref0 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found x:((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Instantiate: f:=(fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))):(a->Prop)
% Found x as proof of (P f)
% Found x:((ex a) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Instantiate: f:=(fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))):(a->Prop)
% Found x as proof of (P f)
% Found x:((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Instantiate: f:=(fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))):(a->Prop)
% Found x as proof of (P f)
% Found x:((ex a) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Instantiate: f:=(fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))):(a->Prop)
% Found x as proof of (P f)
% Found x:((ex a) ((unique a) p))
% Instantiate: b:=((unique a) p):(a->Prop)
% Found x as proof of (P b)
% Found eq_ref00:=(eq_ref0 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))):(((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found (eq_ref0 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) b)
% Found x:((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Instantiate: b:=(fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))):(a->Prop)
% Found x as proof of (P b)
% Found eq_ref00:=(eq_ref0 ((unique a) p)):(((eq (a->Prop)) ((unique a) p)) ((unique a) p))
% Found (eq_ref0 ((unique a) p)) as proof of (((eq (a->Prop)) ((unique a) p)) b)
% Found ((eq_ref (a->Prop)) ((unique a) p)) as proof of (((eq (a->Prop)) ((unique a) p)) b)
% Found ((eq_ref (a->Prop)) ((unique a) p)) as proof of (((eq (a->Prop)) ((unique a) p)) b)
% Found ((eq_ref (a->Prop)) ((unique a) p)) as proof of (((eq (a->Prop)) ((unique a) p)) b)
% Found x10:=(x1 (fun (x3:(a->Prop))=> (x3 Xz))):((p Xz)->(((eq a) x0) Xz))
% Found (x1 (fun (x3:(a->Prop))=> (x3 Xz))) as proof of ((p Xz)->(((eq a) x2) Xz))
% Found (x1 (fun (x3:(a->Prop))=> (x3 Xz))) as proof of ((p Xz)->(((eq a) x2) Xz))
% Found (fun (Xz:a)=> (x1 (fun (x3:(a->Prop))=> (x3 Xz)))) as proof of ((p Xz)->(((eq a) x2) Xz))
% Found (fun (Xz:a)=> (x1 (fun (x3:(a->Prop))=> (x3 Xz)))) as proof of (forall (Xz:a), ((p Xz)->(((eq a) x2) Xz)))
% Found x1:(P x0)
% Instantiate: x0:=Xz:a
% Found (fun (x1:(P x0))=> x1) as proof of (P Xz)
% Found (fun (P:(a->Prop)) (x1:(P x0))=> x1) as proof of ((P x0)->(P Xz))
% Found (fun (x00:(p Xz)) (P:(a->Prop)) (x1:(P x0))=> x1) as proof of (((eq a) x0) Xz)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of ((P x0)->(P Xz))
% Found ((eq_ref0 x0) P) as proof of ((P x0)->(P Xz))
% Found (((eq_ref a) x0) P) as proof of ((P x0)->(P Xz))
% Found (((eq_ref a) x0) P) as proof of ((P x0)->(P Xz))
% Found (fun (P:(a->Prop))=> (((eq_ref a) x0) P)) as proof of ((P x0)->(P Xz))
% Found (fun (x00:(p Xz)) (P:(a->Prop))=> (((eq_ref a) x0) P)) as proof of (((eq a) x0) Xz)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))):(((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (fun (x:a)=> ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))))
% Found (eta_expansion_dep00 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))):(((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (fun (x:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))))
% Found (eta_expansion_dep00 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) 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)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x0) Xy0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x0) Xy0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x0) Xy0))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x0) Xy0))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))))
% 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)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x0) Xy0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x0) Xy0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x0) Xy0))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x0) Xy0))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and (p x0)) (forall (Xz:a), ((p Xz)->(((eq a) x0) Xz)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (p x0)) (forall (Xz:a), ((p Xz)->(((eq a) x0) Xz)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (p x0)) (forall (Xz:a), ((p Xz)->(((eq a) x0) Xz)))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (p x0)) (forall (Xz:a), ((p Xz)->(((eq a) x0) Xz)))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) 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)) ((and (p x0)) (forall (Xz:a), ((p Xz)->(((eq a) x0) Xz)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (p x0)) (forall (Xz:a), ((p Xz)->(((eq a) x0) Xz)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (p x0)) (forall (Xz:a), ((p Xz)->(((eq a) x0) Xz)))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (p x0)) (forall (Xz:a), ((p Xz)->(((eq a) x0) Xz)))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))))
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))->((ex a) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))))
% Found (eq_ref00 (ex a)) as proof of (P (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((eq_ref0 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (ex a)) as proof of (P (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (ex a)) as proof of (P (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (ex a)) as proof of (P (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))):(((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (fun (x:a)=> ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))))
% Found (eta_expansion_dep00 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))->((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))))
% Found (eq_ref00 (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eq_ref0 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))):(((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (fun (x:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))))
% Found (eta_expansion_dep00 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found x20:=(x2 (fun (x3:(a->Prop))=> (x3 Xz))):((p Xz)->(((eq a) x1) Xz))
% Found (x2 (fun (x3:(a->Prop))=> (x3 Xz))) as proof of ((p Xz)->(((eq a) x0) Xz))
% Found (x2 (fun (x3:(a->Prop))=> (x3 Xz))) as proof of ((p Xz)->(((eq a) x0) Xz))
% Found (fun (Xz:a)=> (x2 (fun (x3:(a->Prop))=> (x3 Xz)))) as proof of ((p Xz)->(((eq a) x0) Xz))
% Found (fun (Xz:a)=> (x2 (fun (x3:(a->Prop))=> (x3 Xz)))) as proof of (forall (Xz:a), ((p Xz)->(((eq a) x0) Xz)))
% Found x:((ex a) ((unique a) p))
% Instantiate: f:=((unique a) p):(a->Prop)
% Found x as proof of (P f)
% Found x:((ex a) ((unique a) p))
% Instantiate: f:=((unique a) p):(a->Prop)
% Found x as proof of (P f)
% Found x:((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Instantiate: f:=(fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))):(a->Prop)
% Found x as proof of (P f)
% Found x:((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Instantiate: f:=(fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))):(a->Prop)
% Found x as proof of (P f)
% Found x:((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Instantiate: b:=(fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))):(a->Prop)
% Found x as proof of (P b)
% Found x:((ex a) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Instantiate: b:=(fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))):(a->Prop)
% Found x as proof of (P b)
% Found eq_ref00:=(eq_ref0 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))):(((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (eq_ref0 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found eq_ref00:=(eq_ref0 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))):(((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (eq_ref0 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((unique a) p)):(((eq (a->Prop)) ((unique a) p)) (fun (x:a)=> (((unique a) p) x)))
% Found (eta_expansion_dep00 ((unique a) p)) as proof of (((eq (a->Prop)) ((unique a) p)) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) ((unique a) p)) as proof of (((eq (a->Prop)) ((unique a) p)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) ((unique a) p)) as proof of (((eq (a->Prop)) ((unique a) p)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) ((unique a) p)) as proof of (((eq (a->Prop)) ((unique a) p)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) ((unique a) p)) as proof of (((eq (a->Prop)) ((unique a) p)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))):(((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) (fun (x:a)=> (((eq (a->Prop)) p) ((eq a) x))))
% Found (eta_expansion00 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) b)
% Found ((eta_expansion0 Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) ((unique a) p))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) ((unique a) p))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) ((unique a) p))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) ((unique a) p))
% 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)) (((unique a) p) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((unique a) p) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((unique a) p) x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((unique a) p) x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (((unique a) p) x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (((unique a) p) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((unique a) p) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((unique a) p) x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((unique a) p) x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (((unique a) p) x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (((eq (a->Prop)) p) ((eq a) x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (a->Prop)) p) ((eq a) x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (a->Prop)) p) ((eq a) x0)))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((eq (a->Prop)) p) ((eq a) x0)))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (((eq (a->Prop)) p) ((eq a) x))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (((eq (a->Prop)) p) ((eq a) x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (a->Prop)) p) ((eq a) x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (a->Prop)) p) ((eq a) x0)))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((eq (a->Prop)) p) ((eq a) x0)))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (((eq (a->Prop)) p) ((eq a) x))))
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))->((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))))
% Found (eq_ref00 (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found ((eq_ref0 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))):(((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) (fun (x:a)=> (((eq (a->Prop)) p) ((eq a) x))))
% Found (eta_expansion_dep00 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) ((unique a) p))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) ((unique a) p))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) ((unique a) p))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) ((unique a) p))
% Found x:((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Instantiate: f:=(fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))):(a->Prop)
% Found x as proof of (P f)
% Found x:((ex a) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Instantiate: f:=(fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))):(a->Prop)
% Found x as proof of (P f)
% Found x:((ex a) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Instantiate: f:=(fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))):(a->Prop)
% Found x as proof of (P f)
% Found x:((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Instantiate: f:=(fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))):(a->Prop)
% Found x as proof of (P f)
% Found x3:((ex a) ((unique a) p))
% Found (fun (x3:((ex a) ((unique a) p)))=> x3) as proof of ((ex a) ((unique a) p))
% Found (fun (x3:((ex a) ((unique a) p)))=> x3) as proof of (P ((unique a) p))
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((unique a) p)):(((eq (a->Prop)) ((unique a) p)) (fun (x:a)=> (((unique a) p) x)))
% Found (eta_expansion_dep00 ((unique a) p)) as proof of (((eq (a->Prop)) ((unique a) p)) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) ((unique a) p)) as proof of (((eq (a->Prop)) ((unique a) p)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) ((unique a) p)) as proof of (((eq (a->Prop)) ((unique a) p)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) ((unique a) p)) as proof of (((eq (a->Prop)) ((unique a) p)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) ((unique a) p)) as proof of (((eq (a->Prop)) ((unique a) p)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))):(((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (fun (x:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))))
% Found (eta_expansion_dep00 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))):(((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (fun (x:a)=> ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))))
% Found (eta_expansion_dep00 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and (p x0)) (forall (Xz:a), ((p Xz)->(((eq a) x0) Xz)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (p x0)) (forall (Xz:a), ((p Xz)->(((eq a) x0) Xz)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (p x0)) (forall (Xz:a), ((p Xz)->(((eq a) x0) Xz)))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (p x0)) (forall (Xz:a), ((p Xz)->(((eq a) x0) Xz)))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) 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)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x0) Xy0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x0) Xy0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x0) Xy0))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x0) Xy0))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))))
% 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)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x0) Xy0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x0) Xy0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x0) Xy0))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x0) Xy0))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and (p x0)) (forall (Xz:a), ((p Xz)->(((eq a) x0) Xz)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (p x0)) (forall (Xz:a), ((p Xz)->(((eq a) x0) Xz)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (p x0)) (forall (Xz:a), ((p Xz)->(((eq a) x0) Xz)))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (p x0)) (forall (Xz:a), ((p Xz)->(((eq a) x0) Xz)))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))))
% Found x10:=(x1 (fun (x3:(a->Prop))=> (x3 Xz))):((p Xz)->(((eq a) x0) Xz))
% Found (x1 (fun (x3:(a->Prop))=> (x3 Xz))) as proof of ((p Xz)->(((eq a) x2) Xz))
% Found (x1 (fun (x3:(a->Prop))=> (x3 Xz))) as proof of ((p Xz)->(((eq a) x2) Xz))
% Found (fun (Xz:a)=> (x1 (fun (x3:(a->Prop))=> (x3 Xz)))) as proof of ((p Xz)->(((eq a) x2) Xz))
% Found (fun (Xz:a)=> (x1 (fun (x3:(a->Prop))=> (x3 Xz)))) as proof of (forall (Xz:a), ((p Xz)->(((eq a) x2) Xz)))
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))->((ex a) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))))
% Found (eq_ref00 (ex a)) as proof of (P (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((eq_ref0 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (ex a)) as proof of (P (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (ex a)) as proof of (P (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (ex a)) as proof of (P (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))):(((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (fun (x:a)=> ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))))
% Found (eta_expansion_dep00 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))->((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))))
% Found (eq_ref00 (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eq_ref0 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (ex a)) as proof of (P (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))):(((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (fun (x:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))))
% Found (eta_expansion00 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found ((eta_expansion0 Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found x20:=(x2 (fun (x3:(a->Prop))=> (x3 Xz))):((p Xz)->(((eq a) x1) Xz))
% Found (x2 (fun (x3:(a->Prop))=> (x3 Xz))) as proof of ((p Xz)->(((eq a) x0) Xz))
% Found (x2 (fun (x3:(a->Prop))=> (x3 Xz))) as proof of ((p Xz)->(((eq a) x0) Xz))
% Found (fun (Xz:a)=> (x2 (fun (x3:(a->Prop))=> (x3 Xz)))) as proof of ((p Xz)->(((eq a) x0) Xz))
% Found (fun (Xz:a)=> (x2 (fun (x3:(a->Prop))=> (x3 Xz)))) as proof of (forall (Xz:a), ((p Xz)->(((eq a) x0) Xz)))
% Found x:((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Instantiate: b:=(fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))):(a->Prop)
% Found x as proof of (P b)
% Found x:((ex a) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Instantiate: b:=(fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))):(a->Prop)
% Found x as proof of (P b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))):(((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (fun (x:a)=> ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))))
% Found (eta_expansion00 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))):(((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (fun (x:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))))
% Found (eta_expansion_dep00 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) x0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x0)
% Found (eq_sym000 ((eq_ref a) Xz)) as proof of (((eq a) x0) Xz)
% Found ((eq_sym00 x0) ((eq_ref a) Xz)) as proof of (((eq a) x0) Xz)
% Found (((eq_sym0 Xz) x0) ((eq_ref a) Xz)) as proof of (((eq a) x0) Xz)
% Found ((((eq_sym a) Xz) x0) ((eq_ref a) Xz)) as proof of (((eq a) x0) Xz)
% Found (fun (x00:(p Xz))=> ((((eq_sym a) Xz) x0) ((eq_ref a) Xz))) as proof of (((eq a) x0) Xz)
% Found x:((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Instantiate: f:=(fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))):(a->Prop)
% Found x as proof of (P f)
% Found x:((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Instantiate: f:=(fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))):(a->Prop)
% Found x as proof of (P f)
% Found x:((ex a) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Instantiate: f:=(fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))):(a->Prop)
% Found x as proof of (P f)
% Found x:((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Instantiate: b:=(fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))):(a->Prop)
% Found x as proof of (P b)
% Found x:((ex a) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Instantiate: f:=(fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))):(a->Prop)
% Found x as proof of (P f)
% Found x:((ex a) ((unique a) p))
% Instantiate: b:=((unique a) p):(a->Prop)
% Found x as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((unique a) p)):(((eq (a->Prop)) ((unique a) p)) (fun (x:a)=> (((unique a) p) x)))
% Found (eta_expansion_dep00 ((unique a) p)) as proof of (((eq (a->Prop)) ((unique a) p)) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) ((unique a) p)) as proof of (((eq (a->Prop)) ((unique a) p)) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) ((unique a) p)) as proof of (((eq (a->Prop)) ((unique a) p)) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) ((unique a) p)) as proof of (((eq (a->Prop)) ((unique a) p)) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) ((unique a) p)) as proof of (((eq (a->Prop)) ((unique a) p)) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))):(((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) (fun (x:a)=> (((eq (a->Prop)) p) ((eq a) x))))
% Found (eta_expansion00 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) b)
% Found ((eta_expansion0 Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 p):(((eq (a->Prop)) p) (fun (x:a)=> (p x)))
% Found (eta_expansion_dep00 p) as proof of (((eq (a->Prop)) p) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) p) as proof of (((eq (a->Prop)) p) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) p) as proof of (((eq (a->Prop)) p) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) p) as proof of (((eq (a->Prop)) p) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) p) as proof of (((eq (a->Prop)) p) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xy0:a)=> (((eq a) x0) Xy0)))
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy0:a)=> (((eq a) x0) Xy0)))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy0:a)=> (((eq a) x0) Xy0)))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy0:a)=> (((eq a) x0) Xy0)))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy0:a)=> (((eq a) x0) Xy0)))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))->(P0 (fun (x:a)=> ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))))
% Found (eta_expansion_dep000 P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((eta_expansion_dep00 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))->(P0 (fun (x:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))))
% Found (eta_expansion_dep000 P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eta_expansion_dep00 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))->(P0 (fun (x:a)=> ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))))
% Found (eta_expansion_dep000 P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((eta_expansion_dep00 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))->(P0 (fun (x:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))))
% Found (eta_expansion_dep000 P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eta_expansion_dep00 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found eq_ref000:=(eq_ref00 P):((P p)->(P p))
% Found (eq_ref00 P) as proof of (P0 p)
% Found ((eq_ref0 p) P) as proof of (P0 p)
% Found (((eq_ref (a->Prop)) p) P) as proof of (P0 p)
% Found (((eq_ref (a->Prop)) p) P) as proof of (P0 p)
% 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)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x2) Xy0))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x2) Xy0))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x2) Xy0))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x2) Xy0))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (p x2)) (forall (Xz:a), ((p Xz)->(((eq a) x2) Xz)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (p x2)) (forall (Xz:a), ((p Xz)->(((eq a) x2) Xz)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (p x2)) (forall (Xz:a), ((p Xz)->(((eq a) x2) Xz)))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (p x2)) (forall (Xz:a), ((p Xz)->(((eq a) x2) Xz)))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) 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)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x2) Xy0))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x2) Xy0))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x2) Xy0))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x2) Xy0))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (p x2)) (forall (Xz:a), ((p Xz)->(((eq a) x2) Xz)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (p x2)) (forall (Xz:a), ((p Xz)->(((eq a) x2) Xz)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (p x2)) (forall (Xz:a), ((p Xz)->(((eq a) x2) Xz)))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (p x2)) (forall (Xz:a), ((p Xz)->(((eq a) x2) Xz)))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))))
% Found x:((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Instantiate: f:=(fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))):(a->Prop)
% Found x as proof of (P f)
% Found x:((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Instantiate: f:=(fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))):(a->Prop)
% Found x as proof of (P f)
% Found x:((ex a) ((unique a) p))
% Instantiate: f:=((unique a) p):(a->Prop)
% Found x as proof of (P f)
% Found x:((ex a) ((unique a) p))
% Instantiate: f:=((unique a) p):(a->Prop)
% Found x as proof of (P f)
% Found eta_expansion000:=(eta_expansion00 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))):(((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (fun (x:a)=> ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))))
% Found (eta_expansion00 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))):(((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (fun (x:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))))
% Found (eta_expansion_dep00 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))):(((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (fun (x:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))))
% Found (eta_expansion_dep00 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))):(((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (fun (x:a)=> ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))))
% Found (eta_expansion_dep00 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found eta_expansion000:=(eta_expansion00 p):(((eq (a->Prop)) p) (fun (x:a)=> (p x)))
% Found (eta_expansion00 p) as proof of (((eq (a->Prop)) p) b)
% Found ((eta_expansion0 Prop) p) as proof of (((eq (a->Prop)) p) b)
% Found (((eta_expansion a) Prop) p) as proof of (((eq (a->Prop)) p) b)
% Found (((eta_expansion a) Prop) p) as proof of (((eq (a->Prop)) p) b)
% Found (((eta_expansion a) Prop) p) as proof of (((eq (a->Prop)) p) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) ((eq a) x0))
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) ((eq a) x0))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) ((eq a) x0))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) ((eq a) x0))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) ((eq a) x0))
% Found eq_ref00:=(eq_ref0 x0):(((eq a) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq a) x0) Xz)
% Found ((eq_ref a) x0) as proof of (((eq a) x0) Xz)
% Found ((eq_ref a) x0) as proof of (((eq a) x0) Xz)
% Found (fun (x2:(((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x1) Xy0))))=> ((eq_ref a) x0)) as proof of (((eq a) x0) Xz)
% Found (fun (x1:a) (x2:(((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x1) Xy0))))=> ((eq_ref a) x0)) as proof of ((((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x1) Xy0)))->(((eq a) x0) Xz))
% Found (fun (x1:a) (x2:(((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x1) Xy0))))=> ((eq_ref a) x0)) as proof of (forall (x:a), ((((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))->(((eq a) x0) Xz)))
% Found (ex_ind00 (fun (x1:a) (x2:(((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x1) Xy0))))=> ((eq_ref a) x0))) as proof of (((eq a) x0) Xz)
% Found ((ex_ind0 (((eq a) x0) Xz)) (fun (x1:a) (x2:(((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x1) Xy0))))=> ((eq_ref a) x0))) as proof of (((eq a) x0) Xz)
% Found (((fun (P:Prop) (x1:(forall (x:a), ((((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))->P)))=> (((((ex_ind a) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P) x1) x)) (((eq a) x0) Xz)) (fun (x1:a) (x2:(((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x1) Xy0))))=> ((eq_ref a) x0))) as proof of (((eq a) x0) Xz)
% Found (fun (x00:(p Xz))=> (((fun (P:Prop) (x1:(forall (x:a), ((((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))->P)))=> (((((ex_ind a) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P) x1) x)) (((eq a) x0) Xz)) (fun (x1:a) (x2:(((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x1) Xy0))))=> ((eq_ref a) x0)))) as proof of (((eq a) x0) Xz)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))->(P0 (fun (x:a)=> (((eq (a->Prop)) p) ((eq a) x)))))
% Found (eta_expansion_dep000 P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found ((eta_expansion_dep00 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found (((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))->(P0 (fun (x:a)=> (((eq (a->Prop)) p) ((eq a) x)))))
% Found (eta_expansion_dep000 P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found ((eta_expansion_dep00 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found (((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found eq_ref000:=(eq_ref00 P):((P p)->(P p))
% Found (eq_ref00 P) as proof of (P0 p)
% Found ((eq_ref0 p) P) as proof of (P0 p)
% Found (((eq_ref (a->Prop)) p) P) as proof of (P0 p)
% Found (((eq_ref (a->Prop)) p) P) as proof of (P0 p)
% Found eq_ref000:=(eq_ref00 P):((P p)->(P p))
% Found (eq_ref00 P) as proof of (P0 p)
% Found ((eq_ref0 p) P) as proof of (P0 p)
% Found (((eq_ref (a->Prop)) p) P) as proof of (P0 p)
% Found (((eq_ref (a->Prop)) p) P) as proof of (P0 p)
% Found x:((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Instantiate: b:=(fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))):(a->Prop)
% Found x as proof of (P b)
% Found x:((ex a) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Instantiate: b:=(fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))):(a->Prop)
% Found x as proof of (P b)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (((eq (a->Prop)) p) ((eq a) x2)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((eq (a->Prop)) p) ((eq a) x2)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((eq (a->Prop)) p) ((eq a) x2)))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (((eq (a->Prop)) p) ((eq a) x2)))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (((eq (a->Prop)) p) ((eq a) x))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (((unique a) p) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((unique a) p) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((unique a) p) x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (((unique a) p) x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (((unique a) p) x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (((eq (a->Prop)) p) ((eq a) x2)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((eq (a->Prop)) p) ((eq a) x2)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((eq (a->Prop)) p) ((eq a) x2)))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (((eq (a->Prop)) p) ((eq a) x2)))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (((eq (a->Prop)) p) ((eq a) x))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (((unique a) p) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((unique a) p) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((unique a) p) x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (((unique a) p) x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (((unique a) p) x)))
% Found eq_ref00:=(eq_ref0 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))):(((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (eq_ref0 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))):(((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (fun (x:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))))
% Found (eta_expansion_dep00 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) ((unique a) p))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) ((unique a) p))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) ((unique a) p))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) ((unique a) p))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) ((unique a) p))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))):(((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) (fun (x:a)=> (((eq (a->Prop)) p) ((eq a) x))))
% Found (eta_expansion_dep00 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) ((unique a) p))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) ((unique a) p))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) ((unique a) p))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) ((unique a) p))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))):(((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) (fun (x:a)=> (((eq (a->Prop)) p) ((eq a) x))))
% Found (eta_expansion_dep00 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((unique a) p)):(((eq (a->Prop)) ((unique a) p)) (fun (x:a)=> (((unique a) p) x)))
% Found (eta_expansion_dep00 ((unique a) p)) as proof of (((eq (a->Prop)) ((unique a) p)) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) ((unique a) p)) as proof of (((eq (a->Prop)) ((unique a) p)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) ((unique a) p)) as proof of (((eq (a->Prop)) ((unique a) p)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) ((unique a) p)) as proof of (((eq (a->Prop)) ((unique a) p)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) ((unique a) p)) as proof of (((eq (a->Prop)) ((unique a) p)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((unique a) p)):(((eq (a->Prop)) ((unique a) p)) (fun (x:a)=> (((unique a) p) x)))
% Found (eta_expansion_dep00 ((unique a) p)) as proof of (((eq (a->Prop)) ((unique a) p)) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) ((unique a) p)) as proof of (((eq (a->Prop)) ((unique a) p)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) ((unique a) p)) as proof of (((eq (a->Prop)) ((unique a) p)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) ((unique a) p)) as proof of (((eq (a->Prop)) ((unique a) p)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) ((unique a) p)) as proof of (((eq (a->Prop)) ((unique a) p)) b)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))->(P0 (fun (x:a)=> ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((eta_expansion00 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eta_expansion0 Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))->(P0 (fun (x:a)=> ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((eta_expansion00 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eta_expansion0 Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))->(P0 (fun (x:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eta_expansion00 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eta_expansion0 Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))->(P0 (fun (x:a)=> ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((eta_expansion00 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eta_expansion0 Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))->(P0 (fun (x:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eta_expansion00 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eta_expansion0 Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))->(P0 (fun (x:a)=> ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((eta_expansion00 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eta_expansion0 Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))->(P0 (fun (x:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eta_expansion00 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eta_expansion0 Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))->(P0 (fun (x:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eta_expansion00 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eta_expansion0 Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) x0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x0)
% Found (eq_sym000 ((eq_ref a) Xz)) as proof of (((eq a) x0) Xz)
% Found ((eq_sym00 x0) ((eq_ref a) Xz)) as proof of (((eq a) x0) Xz)
% Found (((eq_sym0 Xz) x0) ((eq_ref a) Xz)) as proof of (((eq a) x0) Xz)
% Found ((((eq_sym a) Xz) x0) ((eq_ref a) Xz)) as proof of (((eq a) x0) Xz)
% Found (fun (x00:(p Xz))=> ((((eq_sym a) Xz) x0) ((eq_ref a) Xz))) as proof of (((eq a) x0) Xz)
% Found x2:(P0 ((unique a) p))
% Found (fun (x2:(P0 ((unique a) p)))=> x2) as proof of (P0 ((unique a) p))
% Found (fun (x2:(P0 ((unique a) p)))=> x2) as proof of (P1 ((unique a) p))
% Found x2:(P0 ((unique a) p))
% Found (fun (x2:(P0 ((unique a) p)))=> x2) as proof of (P0 ((unique a) p))
% Found (fun (x2:(P0 ((unique a) p)))=> x2) as proof of (P1 ((unique a) p))
% Found x10:(P p)
% Found (fun (x10:(P p))=> x10) as proof of (P p)
% Found (fun (x10:(P p))=> x10) as proof of (P0 p)
% Found eq_ref00:=(eq_ref0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))):(((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found (eq_ref0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) as proof of (((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) b)
% Found ((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) as proof of (((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) b)
% Found ((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) as proof of (((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) b)
% Found ((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) as proof of (((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found eq_ref00:=(eq_ref0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))):(((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found (eq_ref0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) as proof of (((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) as proof of (((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) as proof of (((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) as proof of (((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found eq_ref00:=(eq_ref0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))):(((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found (eq_ref0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) as proof of (((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) b)
% Found ((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) as proof of (((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) b)
% Found ((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) as proof of (((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) b)
% Found ((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) as proof of (((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found eq_ref00:=(eq_ref0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))):(((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found (eq_ref0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) as proof of (((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) as proof of (((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) as proof of (((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) as proof of (((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found eq_ref00:=(eq_ref0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))):(((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found (eq_ref0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) as proof of (((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) as proof of (((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) as proof of (((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) as proof of (((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found eq_ref00:=(eq_ref0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))):(((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found (eq_ref0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) as proof of (((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) as proof of (((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) as proof of (((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) as proof of (((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found eq_ref00:=(eq_ref0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))):(((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found (eq_ref0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) as proof of (((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) b)
% Found ((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) as proof of (((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) b)
% Found ((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) as proof of (((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) b)
% Found ((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) as proof of (((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found eq_ref00:=(eq_ref0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))):(((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found (eq_ref0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) as proof of (((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) b)
% Found ((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) as proof of (((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) b)
% Found ((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) as proof of (((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) b)
% Found ((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) as proof of (((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 p):(((eq (a->Prop)) p) (fun (x:a)=> (p x)))
% Found (eta_expansion_dep00 p) as proof of (((eq (a->Prop)) p) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) p) as proof of (((eq (a->Prop)) p) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) p) as proof of (((eq (a->Prop)) p) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) p) as proof of (((eq (a->Prop)) p) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) p) as proof of (((eq (a->Prop)) p) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xy0:a)=> (((eq a) x0) Xy0)))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy0:a)=> (((eq a) x0) Xy0)))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy0:a)=> (((eq a) x0) Xy0)))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy0:a)=> (((eq a) x0) Xy0)))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy0:a)=> (((eq a) x0) Xy0)))
% Found x:((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Instantiate: f:=(fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))):(a->Prop)
% Found x as proof of (P f)
% Found x:((ex a) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Instantiate: f:=(fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))):(a->Prop)
% Found x as proof of (P f)
% Found x:((ex a) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Instantiate: f:=(fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))):(a->Prop)
% Found x as proof of (P f)
% Found x:((ex a) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Instantiate: f:=(fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))):(a->Prop)
% Found x as proof of (P f)
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))->(P0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))))
% Found (eq_ref00 P0) as proof of (P1 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found ((eq_ref0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) P0) as proof of (P1 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found (((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) P0) as proof of (P1 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found (((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) P0) as proof of (P1 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))->(P0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))))
% Found (eq_ref00 P0) as proof of (P1 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found ((eq_ref0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) P0) as proof of (P1 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found (((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) P0) as proof of (P1 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found (((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) P0) as proof of (P1 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))->(P0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))))
% Found (eq_ref00 P0) as proof of (P1 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found ((eq_ref0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) P0) as proof of (P1 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found (((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) P0) as proof of (P1 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found (((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) P0) as proof of (P1 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))->(P0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))))
% Found (eq_ref00 P0) as proof of (P1 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found ((eq_ref0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) P0) as proof of (P1 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found (((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) P0) as proof of (P1 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found (((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) P0) as proof of (P1 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))->(P0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))))
% Found (eq_ref00 P0) as proof of (P1 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found ((eq_ref0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) P0) as proof of (P1 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found (((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) P0) as proof of (P1 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found (((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) P0) as proof of (P1 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))->(P0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))))
% Found (eq_ref00 P0) as proof of (P1 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found ((eq_ref0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) P0) as proof of (P1 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found (((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) P0) as proof of (P1 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found (((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) P0) as proof of (P1 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))->(P0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))))
% Found (eq_ref00 P0) as proof of (P1 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found ((eq_ref0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) P0) as proof of (P1 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found (((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) P0) as proof of (P1 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found (((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) P0) as proof of (P1 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))->(P0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))))
% Found (eq_ref00 P0) as proof of (P1 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found ((eq_ref0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) P0) as proof of (P1 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found (((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) P0) as proof of (P1 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found (((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) P0) as proof of (P1 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))->(P0 (fun (x:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eta_expansion00 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eta_expansion0 Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))->(P0 (fun (x:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eta_expansion00 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eta_expansion0 Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))->(P0 (fun (x:a)=> ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((eta_expansion00 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eta_expansion0 Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))->(P0 (fun (x:a)=> ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((eta_expansion00 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eta_expansion0 Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found eq_ref000:=(eq_ref00 P):((P p)->(P p))
% Found (eq_ref00 P) as proof of (P0 p)
% Found ((eq_ref0 p) P) as proof of (P0 p)
% Found (((eq_ref (a->Prop)) p) P) as proof of (P0 p)
% Found (((eq_ref (a->Prop)) p) P) as proof of (P0 p)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))->(P0 (fun (x:a)=> (((eq (a->Prop)) p) ((eq a) x)))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found ((eta_expansion00 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found (((eta_expansion0 Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))->(P0 (fun (x:a)=> (((eq (a->Prop)) p) ((eq a) x)))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found ((eta_expansion00 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found (((eta_expansion0 Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))->(P0 (fun (x:a)=> (((eq (a->Prop)) p) ((eq a) x)))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found ((eta_expansion00 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found (((eta_expansion0 Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))->(P0 (fun (x:a)=> (((eq (a->Prop)) p) ((eq a) x)))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found ((eta_expansion00 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found (((eta_expansion0 Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy))))
% Found x10:(P p)
% Found (fun (x10:(P p))=> x10) as proof of (P p)
% Found (fun (x10:(P p))=> x10) as proof of (P0 p)
% Found x10:(P p)
% Found (fun (x10:(P p))=> x10) as proof of (P p)
% Found (fun (x10:(P p))=> x10) as proof of (P0 p)
% 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)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x2) Xy0))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x2) Xy0))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x2) Xy0))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x2) Xy0))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (p x2)) (forall (Xz:a), ((p Xz)->(((eq a) x2) Xz)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (p x2)) (forall (Xz:a), ((p Xz)->(((eq a) x2) Xz)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (p x2)) (forall (Xz:a), ((p Xz)->(((eq a) x2) Xz)))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (p x2)) (forall (Xz:a), ((p Xz)->(((eq a) x2) Xz)))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))))
% Found eq_ref00:=(eq_ref0 (((eq (a->Prop)) p) ((eq a) x))):(((eq Prop) (((eq (a->Prop)) p) ((eq a) x))) (((eq (a->Prop)) p) ((eq a) x)))
% Found (eq_ref0 (((eq (a->Prop)) p) ((eq a) x))) as proof of (((eq Prop) (((eq (a->Prop)) p) ((eq a) x))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) p) ((eq a) x))) as proof of (((eq Prop) (((eq (a->Prop)) p) ((eq a) x))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) p) ((eq a) x))) as proof of (((eq Prop) (((eq (a->Prop)) p) ((eq a) x))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) p) ((eq a) x))) as proof of (((eq Prop) (((eq (a->Prop)) p) ((eq a) x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((unique a) p) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((unique a) p) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((unique a) p) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((unique a) p) x))
% Found eq_ref00:=(eq_ref0 (((unique a) p) x)):(((eq Prop) (((unique a) p) x)) (((unique a) p) x))
% Found (eq_ref0 (((unique a) p) x)) as proof of (((eq Prop) (((unique a) p) x)) b)
% Found ((eq_ref Prop) (((unique a) p) x)) as proof of (((eq Prop) (((unique a) p) x)) b)
% Found ((eq_ref Prop) (((unique a) p) x)) as proof of (((eq Prop) (((unique a) p) x)) b)
% Found ((eq_ref Prop) (((unique a) p) x)) as proof of (((eq Prop) (((unique a) p) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) ((eq a) x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) ((eq a) x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) ((eq a) x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) ((eq a) x)))
% Found eq_ref00:=(eq_ref0 (((unique a) p) x)):(((eq Prop) (((unique a) p) x)) (((unique a) p) x))
% Found (eq_ref0 (((unique a) p) x)) as proof of (((eq Prop) (((unique a) p) x)) b)
% Found ((eq_ref Prop) (((unique a) p) x)) as proof of (((eq Prop) (((unique a) p) x)) b)
% Found ((eq_ref Prop) (((unique a) p) x)) as proof of (((eq Prop) (((unique a) p) x)) b)
% Found ((eq_ref Prop) (((unique a) p) x)) as proof of (((eq Prop) (((unique a) p) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) ((eq a) x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) ((eq a) x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) ((eq a) x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) ((eq a) x)))
% Found eq_ref00:=(eq_ref0 (((unique a) p) x)):(((eq Prop) (((unique a) p) x)) (((unique a) p) x))
% Found (eq_ref0 (((unique a) p) x)) as proof of (((eq Prop) (((unique a) p) x)) b)
% Found ((eq_ref Prop) (((unique a) p) x)) as proof of (((eq Prop) (((unique a) p) x)) b)
% Found ((eq_ref Prop) (((unique a) p) x)) as proof of (((eq Prop) (((unique a) p) x)) b)
% Found ((eq_ref Prop) (((unique a) p) x)) as proof of (((eq Prop) (((unique a) p) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) ((eq a) x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) ((eq a) x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) ((eq a) x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) ((eq a) x)))
% Found eq_ref00:=(eq_ref0 (((eq (a->Prop)) p) ((eq a) x))):(((eq Prop) (((eq (a->Prop)) p) ((eq a) x))) (((eq (a->Prop)) p) ((eq a) x)))
% Found (eq_ref0 (((eq (a->Prop)) p) ((eq a) x))) as proof of (((eq Prop) (((eq (a->Prop)) p) ((eq a) x))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) p) ((eq a) x))) as proof of (((eq Prop) (((eq (a->Prop)) p) ((eq a) x))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) p) ((eq a) x))) as proof of (((eq Prop) (((eq (a->Prop)) p) ((eq a) x))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) p) ((eq a) x))) as proof of (((eq Prop) (((eq (a->Prop)) p) ((eq a) x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((unique a) p) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((unique a) p) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((unique a) p) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((unique a) p) x))
% Found eq_ref00:=(eq_ref0 (((unique a) p) x)):(((eq Prop) (((unique a) p) x)) (((unique a) p) x))
% Found (eq_ref0 (((unique a) p) x)) as proof of (((eq Prop) (((unique a) p) x)) b)
% Found ((eq_ref Prop) (((unique a) p) x)) as proof of (((eq Prop) (((unique a) p) x)) b)
% Found ((eq_ref Prop) (((unique a) p) x)) as proof of (((eq Prop) (((unique a) p) x)) b)
% Found ((eq_ref Prop) (((unique a) p) x)) as proof of (((eq Prop) (((unique a) p) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) ((eq a) x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) ((eq a) x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) ((eq a) x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) ((eq a) x)))
% Found eq_ref00:=(eq_ref0 (((eq (a->Prop)) p) ((eq a) x))):(((eq Prop) (((eq (a->Prop)) p) ((eq a) x))) (((eq (a->Prop)) p) ((eq a) x)))
% Found (eq_ref0 (((eq (a->Prop)) p) ((eq a) x))) as proof of (((eq Prop) (((eq (a->Prop)) p) ((eq a) x))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) p) ((eq a) x))) as proof of (((eq Prop) (((eq (a->Prop)) p) ((eq a) x))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) p) ((eq a) x))) as proof of (((eq Prop) (((eq (a->Prop)) p) ((eq a) x))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) p) ((eq a) x))) as proof of (((eq Prop) (((eq (a->Prop)) p) ((eq a) x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((unique a) p) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((unique a) p) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((unique a) p) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((unique a) p) x))
% Found eq_ref00:=(eq_ref0 (((eq (a->Prop)) p) ((eq a) x))):(((eq Prop) (((eq (a->Prop)) p) ((eq a) x))) (((eq (a->Prop)) p) ((eq a) x)))
% Found (eq_ref0 (((eq (a->Prop)) p) ((eq a) x))) as proof of (((eq Prop) (((eq (a->Prop)) p) ((eq a) x))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) p) ((eq a) x))) as proof of (((eq Prop) (((eq (a->Prop)) p) ((eq a) x))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) p) ((eq a) x))) as proof of (((eq Prop) (((eq (a->Prop)) p) ((eq a) x))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) p) ((eq a) x))) as proof of (((eq Prop) (((eq (a->Prop)) p) ((eq a) x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((unique a) p) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((unique a) p) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((unique a) p) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((unique a) p) x))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (p x2)) (forall (Xz:a), ((p Xz)->(((eq a) x2) Xz)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (p x2)) (forall (Xz:a), ((p Xz)->(((eq a) x2) Xz)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (p x2)) (forall (Xz:a), ((p Xz)->(((eq a) x2) Xz)))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (p x2)) (forall (Xz:a), ((p Xz)->(((eq a) x2) Xz)))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) 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)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x2) Xy0))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x2) Xy0))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x2) Xy0))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x2) Xy0))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))))
% Found eq_ref00:=(eq_ref0 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))):(((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (eq_ref0 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))):(((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (fun (x:a)=> ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))))
% Found (eta_expansion00 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))):(((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (fun (x:a)=> ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))))
% Found (eta_expansion00 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))):(((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (fun (x:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))))
% Found (eta_expansion_dep00 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found eq_ref000:=(eq_ref00 P0):((P0 (((eq (a->Prop)) p) ((eq a) x)))->(P0 (((eq (a->Prop)) p) ((eq a) x))))
% Found (eq_ref00 P0) as proof of (P1 (((eq (a->Prop)) p) ((eq a) x)))
% Found ((eq_ref0 (((eq (a->Prop)) p) ((eq a) x))) P0) as proof of (P1 (((eq (a->Prop)) p) ((eq a) x)))
% Found (((eq_ref Prop) (((eq (a->Prop)) p) ((eq a) x))) P0) as proof of (P1 (((eq (a->Prop)) p) ((eq a) x)))
% Found (((eq_ref Prop) (((eq (a->Prop)) p) ((eq a) x))) P0) as proof of (P1 (((eq (a->Prop)) p) ((eq a) x)))
% Found eq_ref000:=(eq_ref00 P0):((P0 (((eq (a->Prop)) p) ((eq a) x)))->(P0 (((eq (a->Prop)) p) ((eq a) x))))
% Found (eq_ref00 P0) as proof of (P1 (((eq (a->Prop)) p) ((eq a) x)))
% Found ((eq_ref0 (((eq (a->Prop)) p) ((eq a) x))) P0) as proof of (P1 (((eq (a->Prop)) p) ((eq a) x)))
% Found (((eq_ref Prop) (((eq (a->Prop)) p) ((eq a) x))) P0) as proof of (P1 (((eq (a->Prop)) p) ((eq a) x)))
% Found (((eq_ref Prop) (((eq (a->Prop)) p) ((eq a) x))) P0) as proof of (P1 (((eq (a->Prop)) p) ((eq a) x)))
% Found eq_ref000:=(eq_ref00 P0):((P0 (((eq (a->Prop)) p) ((eq a) x)))->(P0 (((eq (a->Prop)) p) ((eq a) x))))
% Found (eq_ref00 P0) as proof of (P1 (((eq (a->Prop)) p) ((eq a) x)))
% Found ((eq_ref0 (((eq (a->Prop)) p) ((eq a) x))) P0) as proof of (P1 (((eq (a->Prop)) p) ((eq a) x)))
% Found (((eq_ref Prop) (((eq (a->Prop)) p) ((eq a) x))) P0) as proof of (P1 (((eq (a->Prop)) p) ((eq a) x)))
% Found (((eq_ref Prop) (((eq (a->Prop)) p) ((eq a) x))) P0) as proof of (P1 (((eq (a->Prop)) p) ((eq a) x)))
% Found eq_ref000:=(eq_ref00 P0):((P0 (((eq (a->Prop)) p) ((eq a) x)))->(P0 (((eq (a->Prop)) p) ((eq a) x))))
% Found (eq_ref00 P0) as proof of (P1 (((eq (a->Prop)) p) ((eq a) x)))
% Found ((eq_ref0 (((eq (a->Prop)) p) ((eq a) x))) P0) as proof of (P1 (((eq (a->Prop)) p) ((eq a) x)))
% Found (((eq_ref Prop) (((eq (a->Prop)) p) ((eq a) x))) P0) as proof of (P1 (((eq (a->Prop)) p) ((eq a) x)))
% Found (((eq_ref Prop) (((eq (a->Prop)) p) ((eq a) x))) P0) as proof of (P1 (((eq (a->Prop)) p) ((eq a) x)))
% Found eq_ref000:=(eq_ref00 P):((P p)->(P p))
% Found (eq_ref00 P) as proof of (P0 p)
% Found ((eq_ref0 p) P) as proof of (P0 p)
% Found (((eq_ref (a->Prop)) p) P) as proof of (P0 p)
% Found (((eq_ref (a->Prop)) p) P) as proof of (P0 p)
% Found eq_ref000:=(eq_ref00 P):((P p)->(P p))
% Found (eq_ref00 P) as proof of (P0 p)
% Found ((eq_ref0 p) P) as proof of (P0 p)
% Found (((eq_ref (a->Prop)) p) P) as proof of (P0 p)
% Found (((eq_ref (a->Prop)) p) P) as proof of (P0 p)
% Found eq_ref00:=(eq_ref0 x0):(((eq a) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq a) x0) Xz)
% Found ((eq_ref a) x0) as proof of (((eq a) x0) Xz)
% Found ((eq_ref a) x0) as proof of (((eq a) x0) Xz)
% Found (fun (x2:(((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x1) Xy0))))=> ((eq_ref a) x0)) as proof of (((eq a) x0) Xz)
% Found (fun (x1:a) (x2:(((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x1) Xy0))))=> ((eq_ref a) x0)) as proof of ((((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x1) Xy0)))->(((eq a) x0) Xz))
% Found (fun (x1:a) (x2:(((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x1) Xy0))))=> ((eq_ref a) x0)) as proof of (forall (x:a), ((((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))->(((eq a) x0) Xz)))
% Found (ex_ind00 (fun (x1:a) (x2:(((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x1) Xy0))))=> ((eq_ref a) x0))) as proof of (((eq a) x0) Xz)
% Found ((ex_ind0 (((eq a) x0) Xz)) (fun (x1:a) (x2:(((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x1) Xy0))))=> ((eq_ref a) x0))) as proof of (((eq a) x0) Xz)
% Found (((fun (P:Prop) (x1:(forall (x:a), ((((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))->P)))=> (((((ex_ind a) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P) x1) x)) (((eq a) x0) Xz)) (fun (x1:a) (x2:(((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x1) Xy0))))=> ((eq_ref a) x0))) as proof of (((eq a) x0) Xz)
% Found (fun (x00:(p Xz))=> (((fun (P:Prop) (x1:(forall (x:a), ((((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))->P)))=> (((((ex_ind a) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P) x1) x)) (((eq a) x0) Xz)) (fun (x1:a) (x2:(((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x1) Xy0))))=> ((eq_ref a) x0)))) as proof of (((eq a) x0) Xz)
% Found x:((ex a) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Instantiate: b:=(fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))):(a->Prop)
% Found x as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))):(((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (fun (x:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))))
% Found (eta_expansion_dep00 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found x2:(P0 ((unique a) p))
% Found (fun (x2:(P0 ((unique a) p)))=> x2) as proof of (P0 ((unique a) p))
% Found (fun (x2:(P0 ((unique a) p)))=> x2) as proof of (P1 ((unique a) p))
% Found x2:(P0 ((unique a) p))
% Found (fun (x2:(P0 ((unique a) p)))=> x2) as proof of (P0 ((unique a) p))
% Found (fun (x2:(P0 ((unique a) p)))=> x2) as proof of (P1 ((unique a) p))
% Found x2:(P0 ((unique a) p))
% Found (fun (x2:(P0 ((unique a) p)))=> x2) as proof of (P0 ((unique a) p))
% Found (fun (x2:(P0 ((unique a) p)))=> x2) as proof of (P1 ((unique a) p))
% Found x2:(P0 ((unique a) p))
% Found (fun (x2:(P0 ((unique a) p)))=> x2) as proof of (P0 ((unique a) p))
% Found (fun (x2:(P0 ((unique a) p)))=> x2) as proof of (P1 ((unique a) p))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))->(P0 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((eq_ref0 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))->(P0 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((eq_ref0 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))->(P0 (fun (x:a)=> ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((eta_expansion00 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eta_expansion0 Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))->(P0 (fun (x:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eta_expansion00 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eta_expansion0 Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))->(P0 (fun (x:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eta_expansion00 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eta_expansion0 Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))->(P0 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eq_ref0 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))->(P0 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eq_ref0 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) P0) as proof of (P1 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))->(P0 (fun (x:a)=> ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((eta_expansion00 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eta_expansion0 Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found x01:(P0 (((unique a) p) x))
% Found (fun (x01:(P0 (((unique a) p) x)))=> x01) as proof of (P0 (((unique a) p) x))
% Found (fun (x01:(P0 (((unique a) p) x)))=> x01) as proof of (P1 (((unique a) p) x))
% Found x01:(P0 (((unique a) p) x))
% Found (fun (x01:(P0 (((unique a) p) x)))=> x01) as proof of (P0 (((unique a) p) x))
% Found (fun (x01:(P0 (((unique a) p) x)))=> x01) as proof of (P1 (((unique a) p) x))
% Found x01:(P0 (((unique a) p) x))
% Found (fun (x01:(P0 (((unique a) p) x)))=> x01) as proof of (P0 (((unique a) p) x))
% Found (fun (x01:(P0 (((unique a) p) x)))=> x01) as proof of (P1 (((unique a) p) x))
% Found x01:(P0 (((unique a) p) x))
% Found (fun (x01:(P0 (((unique a) p) x)))=> x01) as proof of (P0 (((unique a) p) x))
% Found (fun (x01:(P0 (((unique a) p) x)))=> x01) as proof of (P1 (((unique a) p) x))
% Found eta_expansion000:=(eta_expansion00 p):(((eq (a->Prop)) p) (fun (x:a)=> (p x)))
% Found (eta_expansion00 p) as proof of (((eq (a->Prop)) p) b)
% Found ((eta_expansion0 Prop) p) as proof of (((eq (a->Prop)) p) b)
% Found (((eta_expansion a) Prop) p) as proof of (((eq (a->Prop)) p) b)
% Found (((eta_expansion a) Prop) p) as proof of (((eq (a->Prop)) p) b)
% Found (((eta_expansion a) Prop) p) as proof of (((eq (a->Prop)) p) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xy0:a)=> (((eq a) x2) Xy0)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy0:a)=> (((eq a) x2) Xy0)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy0:a)=> (((eq a) x2) Xy0)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy0:a)=> (((eq a) x2) Xy0)))
% Found x:((ex a) ((unique a) p))
% Instantiate: b:=((unique a) p):(a->Prop)
% Found x as proof of (P b)
% Found x:((ex a) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Instantiate: f:=(fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))):(a->Prop)
% Found x as proof of (P f)
% Found x:((ex a) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Instantiate: f:=(fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))):(a->Prop)
% Found x as proof of (P f)
% Found eq_ref00:=(eq_ref0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))):(((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found (eq_ref0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) as proof of (((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) b)
% Found ((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) as proof of (((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) b)
% Found ((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) as proof of (((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) b)
% Found ((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) as proof of (((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found eq_ref00:=(eq_ref0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))):(((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found (eq_ref0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) as proof of (((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) b)
% Found ((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) as proof of (((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) b)
% Found ((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) as proof of (((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) b)
% Found ((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) as proof of (((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found eq_ref00:=(eq_ref0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))):(((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found (eq_ref0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) as proof of (((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) as proof of (((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) as proof of (((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) as proof of (((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found eq_ref00:=(eq_ref0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))):(((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found (eq_ref0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) as proof of (((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) b)
% Found ((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) as proof of (((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) b)
% Found ((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) as proof of (((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) b)
% Found ((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) as proof of (((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found eq_ref00:=(eq_ref0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))):(((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found (eq_ref0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) as proof of (((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) as proof of (((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) as proof of (((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) as proof of (((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found eq_ref00:=(eq_ref0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))):(((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found (eq_ref0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) as proof of (((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) b)
% Found ((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) as proof of (((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) b)
% Found ((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) as proof of (((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) b)
% Found ((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) as proof of (((eq Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found eq_ref00:=(eq_ref0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))):(((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found (eq_ref0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) as proof of (((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) as proof of (((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) as proof of (((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) as proof of (((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found eq_ref00:=(eq_ref0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))):(((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found (eq_ref0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) as proof of (((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) as proof of (((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) as proof of (((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) as proof of (((eq Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))):(((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) (fun (x:a)=> (((eq (a->Prop)) p) ((eq a) x))))
% Found (eta_expansion_dep00 (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) b)
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) ((eq a) Xy)))) b)
% Found eq_ref000:=(eq_ref00 P):((P p)->(P p))
% Found (eq_ref00 P) as proof of (P0 p)
% Found ((eq_ref0 p) P) as proof of (P0 p)
% Found (((eq_ref (a->Prop)) p) P) as proof of (P0 p)
% Found (((eq_ref (a->Prop)) p) P) as proof of (P0 p)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) p)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) p)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) p)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) p)
% Found eq_ref00:=(eq_ref0 (fun (Xy0:a)=> (((eq a) x0) Xy0))):(((eq (a->Prop)) (fun (Xy0:a)=> (((eq a) x0) Xy0))) (fun (Xy0:a)=> (((eq a) x0) Xy0)))
% Found (eq_ref0 (fun (Xy0:a)=> (((eq a) x0) Xy0))) as proof of (((eq (a->Prop)) (fun (Xy0:a)=> (((eq a) x0) Xy0))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy0:a)=> (((eq a) x0) Xy0))) as proof of (((eq (a->Prop)) (fun (Xy0:a)=> (((eq a) x0) Xy0))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy0:a)=> (((eq a) x0) Xy0))) as proof of (((eq (a->Prop)) (fun (Xy0:a)=> (((eq a) x0) Xy0))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy0:a)=> (((eq a) x0) Xy0))) as proof of (((eq (a->Prop)) (fun (Xy0:a)=> (((eq a) x0) Xy0))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))):(((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (fun (x:a)=> ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))))
% Found (eta_expansion00 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0)))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))):(((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) (fun (x:a)=> ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))))
% Found (eta_expansion00 (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz)))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))):(((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (fun (x:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))))
% Found (eta_expansion00 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found ((eta_expansion0 Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((and (p Xy)) (forall (Xz:a), ((p Xz)->(((eq a) Xy) Xz))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))):(((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) (fun (x:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))))
% Found (eta_expansion00 (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found ((eta_expansion0 Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) Xy) Xy0))))) b)
% Found eq_ref000:=(eq_ref00 P0):((P0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))->(P0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))))
% Found (eq_ref00 P0) as proof of (P1 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found ((eq_ref0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) P0) as proof of (P1 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found (((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) P0) as proof of (P1 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found (((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) P0) as proof of (P1 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))->(P0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))))
% Found (eq_ref00 P0) as proof of (P1 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found ((eq_ref0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) P0) as proof of (P1 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found (((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) P0) as proof of (P1 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found (((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) P0) as proof of (P1 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))->(P0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))))
% Found (eq_ref00 P0) as proof of (P1 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found ((eq_ref0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) P0) as proof of (P1 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found (((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) P0) as proof of (P1 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found (((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) P0) as proof of (P1 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))->(P0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))))
% Found (eq_ref00 P0) as proof of (P1 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found ((eq_ref0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) P0) as proof of (P1 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found (((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) P0) as proof of (P1 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found (((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) P0) as proof of (P1 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))->(P0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))))
% Found (eq_ref00 P0) as proof of (P1 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found ((eq_ref0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) P0) as proof of (P1 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found (((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) P0) as proof of (P1 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found (((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) P0) as proof of (P1 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))->(P0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))))
% Found (eq_ref00 P0) as proof of (P1 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found ((eq_ref0 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) P0) as proof of (P1 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found (((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) P0) as proof of (P1 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found (((eq_ref Prop) (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0)))) P0) as proof of (P1 (((eq (a->Prop)) p) (fun (Xy0:a)=> (((eq a) x) Xy0))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))->(P0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))))
% Found (eq_ref00 P0) as proof of (P1 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found ((eq_ref0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) P0) as proof of (P1 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found (((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) P0) as proof of (P1 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found (((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) P0) as proof of (P1 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))->(P0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))))
% Found (eq_ref00 P0) as proof of (P1 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found ((eq_ref0 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) P0) as proof of (P1 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found (((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) P0) as proof of (P1 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found (((eq_ref Prop) ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz))))) P0) as proof of (P1 ((and (p x)) (forall (Xz:a), ((p Xz)->(((eq a) x) Xz)))))
% Found eq_ref00:=(eq_ref0 p):(((eq (a->Prop)) p) p)
% Found (eq_ref0 p) as proof of (((eq (a->Prop)) p) b)
% Found ((eq_ref (a->Prop)) p) as proof of (((eq (a->Prop)) p) b)
% Found ((eq_ref (a->Prop)) p) as proof of (((eq (a->Prop)) p) b)
% Found ((eq_ref (a->Prop)) p) as proof of (((eq (a->Prop)) p) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xy0:a)=> (((eq a) x0) Xy0)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy0:a)=> (((eq a) x0) Xy0)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy0:a)=> (((eq a) x0) Xy0)))
% Found ((eq_re
% EOF
%------------------------------------------------------------------------------