%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV220^5 : 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 13:33:53 EDT 2014

% Result   : Timeout 300.04s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV220^5 : 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 08:31:51 CDT 2014
% % CPUTime  : 300.04 
% 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 0x19cba28>, <kernel.Type object at 0x19cb440>) of role type named b_type
% Using role type
% Declaring b:Type
% FOF formula (<kernel.Constant object at 0x1ba9368>, <kernel.Type object at 0x19cbcf8>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x19cb8c0>, <kernel.DependentProduct object at 0x1bc8290>) of role type named f
% Using role type
% Declaring f:(b->a)
% FOF formula (<kernel.Constant object at 0x19cb440>, <kernel.DependentProduct object at 0x1bc8248>) of role type named w
% Using role type
% Declaring w:((b->Prop)->Prop)
% FOF formula (forall (Xx:a), (((ex b) (fun (Xt:b)=> ((and (forall (S:(b->Prop)), ((w S)->(S Xt)))) (((eq a) Xx) (f Xt)))))->(forall (S:(a->Prop)), (((ex (b->Prop)) (fun (Xt:(b->Prop))=> ((and (w Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (Xt Xt0)) (((eq a) Xz) (f Xt0))))))))))->(S Xx))))) of role conjecture named cX5205_pme
% Conjecture to prove = (forall (Xx:a), (((ex b) (fun (Xt:b)=> ((and (forall (S:(b->Prop)), ((w S)->(S Xt)))) (((eq a) Xx) (f Xt)))))->(forall (S:(a->Prop)), (((ex (b->Prop)) (fun (Xt:(b->Prop))=> ((and (w Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (Xt Xt0)) (((eq a) Xz) (f Xt0))))))))))->(S Xx))))):Prop
% Parameter b_DUMMY:b.
% Parameter a_DUMMY:a.
% We need to prove ['(forall (Xx:a), (((ex b) (fun (Xt:b)=> ((and (forall (S:(b->Prop)), ((w S)->(S Xt)))) (((eq a) Xx) (f Xt)))))->(forall (S:(a->Prop)), (((ex (b->Prop)) (fun (Xt:(b->Prop))=> ((and (w Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (Xt Xt0)) (((eq a) Xz) (f Xt0))))))))))->(S Xx)))))']
% Parameter b:Type.
% Parameter a:Type.
% Parameter f:(b->a).
% Parameter w:((b->Prop)->Prop).
% Trying to prove (forall (Xx:a), (((ex b) (fun (Xt:b)=> ((and (forall (S:(b->Prop)), ((w S)->(S Xt)))) (((eq a) Xx) (f Xt)))))->(forall (S:(a->Prop)), (((ex (b->Prop)) (fun (Xt:(b->Prop))=> ((and (w Xt)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (Xt Xt0)) (((eq a) Xz) (f Xt0))))))))))->(S Xx)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found x4:(((eq a) Xx) (f x1))
% Instantiate: b0:=(f x1):a
% Found x4 as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eta_expansion000:=(eta_expansion00 (fun (x50:b)=> (S Xx))):(((eq (b->Prop)) (fun (x50:b)=> (S Xx))) (fun (x:b)=> (S Xx)))
% Found (eta_expansion00 (fun (x50:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x5:b)=> (S Xx))) b0)
% Found ((eta_expansion0 Prop) (fun (x50:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x5:b)=> (S Xx))) b0)
% Found (((eta_expansion b) Prop) (fun (x50:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x5:b)=> (S Xx))) b0)
% Found (((eta_expansion b) Prop) (fun (x50:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x5:b)=> (S Xx))) b0)
% Found (((eta_expansion b) Prop) (fun (x50:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x5:b)=> (S Xx))) b0)
% Found x6:(((eq a) Xx) (f x3))
% Instantiate: b0:=(f x3):a
% Found x6 as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 (f0 x5)):(((eq Prop) (f0 x5)) (f0 x5))
% Found (eq_ref0 (f0 x5)) as proof of (((eq Prop) (f0 x5)) (S Xx))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (S Xx))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (S Xx))
% Found (fun (x5:b)=> ((eq_ref Prop) (f0 x5))) as proof of (((eq Prop) (f0 x5)) (S Xx))
% Found (fun (x5:b)=> ((eq_ref Prop) (f0 x5))) as proof of (forall (x:b), (((eq Prop) (f0 x)) (S Xx)))
% Found eq_ref00:=(eq_ref0 (f0 x5)):(((eq Prop) (f0 x5)) (f0 x5))
% Found (eq_ref0 (f0 x5)) as proof of (((eq Prop) (f0 x5)) (S Xx))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (S Xx))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (S Xx))
% Found (fun (x5:b)=> ((eq_ref Prop) (f0 x5))) as proof of (((eq Prop) (f0 x5)) (S Xx))
% Found (fun (x5:b)=> ((eq_ref Prop) (f0 x5))) as proof of (forall (x:b), (((eq Prop) (f0 x)) (S Xx)))
% Found x6:(((eq a) Xx) (f x1))
% Instantiate: b0:=(f x1):a
% Found x6 as proof of (((eq a) Xx) b0)
% Found x4:(((eq a) Xx) (f x1))
% Instantiate: b0:=(f x1):a
% Found x4 as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (x50:b)=> (forall (x500:(b->Prop)), (((and (w x500)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x500 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx))))):(((eq (b->Prop)) (fun (x50:b)=> (forall (x500:(b->Prop)), (((and (w x500)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x500 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx))))) (fun (x:b)=> (forall (x500:(b->Prop)), (((and (w x500)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x500 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx)))))
% Found (eta_expansion_dep00 (fun (x50:b)=> (forall (x500:(b->Prop)), (((and (w x500)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x500 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx))))) as proof of (((eq (b->Prop)) (fun (x5:b)=> (forall (x50:(b->Prop)), (((and (w x50)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x50 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx))))) b0)
% Found ((eta_expansion_dep0 (fun (x6:b)=> Prop)) (fun (x50:b)=> (forall (x500:(b->Prop)), (((and (w x500)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x500 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx))))) as proof of (((eq (b->Prop)) (fun (x5:b)=> (forall (x50:(b->Prop)), (((and (w x50)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x50 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx))))) b0)
% Found (((eta_expansion_dep b) (fun (x6:b)=> Prop)) (fun (x50:b)=> (forall (x500:(b->Prop)), (((and (w x500)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x500 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx))))) as proof of (((eq (b->Prop)) (fun (x5:b)=> (forall (x50:(b->Prop)), (((and (w x50)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x50 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx))))) b0)
% Found (((eta_expansion_dep b) (fun (x6:b)=> Prop)) (fun (x50:b)=> (forall (x500:(b->Prop)), (((and (w x500)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x500 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx))))) as proof of (((eq (b->Prop)) (fun (x5:b)=> (forall (x50:(b->Prop)), (((and (w x50)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x50 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx))))) b0)
% Found (((eta_expansion_dep b) (fun (x6:b)=> Prop)) (fun (x50:b)=> (forall (x500:(b->Prop)), (((and (w x500)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x500 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx))))) as proof of (((eq (b->Prop)) (fun (x5:b)=> (forall (x50:(b->Prop)), (((and (w x50)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x50 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (x70:b)=> (S Xx))):(((eq (b->Prop)) (fun (x70:b)=> (S Xx))) (fun (x:b)=> (S Xx)))
% Found (eta_expansion_dep00 (fun (x70:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x7:b)=> (S Xx))) b0)
% Found ((eta_expansion_dep0 (fun (x8:b)=> Prop)) (fun (x70:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x7:b)=> (S Xx))) b0)
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) (fun (x70:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x7:b)=> (S Xx))) b0)
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) (fun (x70:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x7:b)=> (S Xx))) b0)
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) (fun (x70:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x7:b)=> (S Xx))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (x60:b)=> (((and (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx)))):(((eq (b->Prop)) (fun (x60:b)=> (((and (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx)))) (fun (x:b)=> (((and (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx))))
% Found (eta_expansion_dep00 (fun (x60:b)=> (((and (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx)))) as proof of (((eq (b->Prop)) (fun (x6:b)=> (((and (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx)))) b0)
% Found ((eta_expansion_dep0 (fun (x7:b)=> Prop)) (fun (x60:b)=> (((and (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx)))) as proof of (((eq (b->Prop)) (fun (x6:b)=> (((and (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx)))) b0)
% Found (((eta_expansion_dep b) (fun (x7:b)=> Prop)) (fun (x60:b)=> (((and (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx)))) as proof of (((eq (b->Prop)) (fun (x6:b)=> (((and (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx)))) b0)
% Found (((eta_expansion_dep b) (fun (x7:b)=> Prop)) (fun (x60:b)=> (((and (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx)))) as proof of (((eq (b->Prop)) (fun (x6:b)=> (((and (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx)))) b0)
% Found (((eta_expansion_dep b) (fun (x7:b)=> Prop)) (fun (x60:b)=> (((and (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx)))) as proof of (((eq (b->Prop)) (fun (x6:b)=> (((and (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx)))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (x70:b)=> (S Xx))):(((eq (b->Prop)) (fun (x70:b)=> (S Xx))) (fun (x:b)=> (S Xx)))
% Found (eta_expansion_dep00 (fun (x70:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x7:b)=> (S Xx))) b0)
% Found ((eta_expansion_dep0 (fun (x8:b)=> Prop)) (fun (x70:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x7:b)=> (S Xx))) b0)
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) (fun (x70:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x7:b)=> (S Xx))) b0)
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) (fun (x70:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x7:b)=> (S Xx))) b0)
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) (fun (x70:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x7:b)=> (S Xx))) b0)
% Found eta_expansion000:=(eta_expansion00 (fun (x70:b)=> (S Xx))):(((eq (b->Prop)) (fun (x70:b)=> (S Xx))) (fun (x:b)=> (S Xx)))
% Found (eta_expansion00 (fun (x70:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x7:b)=> (S Xx))) b0)
% Found ((eta_expansion0 Prop) (fun (x70:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x7:b)=> (S Xx))) b0)
% Found (((eta_expansion b) Prop) (fun (x70:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x7:b)=> (S Xx))) b0)
% Found (((eta_expansion b) Prop) (fun (x70:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x7:b)=> (S Xx))) b0)
% Found (((eta_expansion b) Prop) (fun (x70:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x7:b)=> (S Xx))) b0)
% Found eta_expansion000:=(eta_expansion00 (fun (x5:b)=> (S Xx))):(((eq (b->Prop)) (fun (x5:b)=> (S Xx))) (fun (x:b)=> (S Xx)))
% Found (eta_expansion00 (fun (x5:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x5:b)=> (S Xx))) b0)
% Found ((eta_expansion0 Prop) (fun (x5:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x5:b)=> (S Xx))) b0)
% Found (((eta_expansion b) Prop) (fun (x5:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x5:b)=> (S Xx))) b0)
% Found (((eta_expansion b) Prop) (fun (x5:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x5:b)=> (S Xx))) b0)
% Found (((eta_expansion b) Prop) (fun (x5:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x5:b)=> (S Xx))) b0)
% Found eq_ref00:=(eq_ref0 (f0 x5)):(((eq Prop) (f0 x5)) (f0 x5))
% Found (eq_ref0 (f0 x5)) as proof of (((eq Prop) (f0 x5)) (forall (x50:(b->Prop)), (((and (w x50)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x50 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx))))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (forall (x50:(b->Prop)), (((and (w x50)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x50 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx))))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (forall (x50:(b->Prop)), (((and (w x50)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x50 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx))))
% Found (fun (x5:b)=> ((eq_ref Prop) (f0 x5))) as proof of (((eq Prop) (f0 x5)) (forall (x50:(b->Prop)), (((and (w x50)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x50 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx))))
% Found (fun (x5:b)=> ((eq_ref Prop) (f0 x5))) as proof of (forall (x:b), (((eq Prop) (f0 x)) (forall (x50:(b->Prop)), (((and (w x50)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x50 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx)))))
% Found eq_ref00:=(eq_ref0 (f0 x5)):(((eq Prop) (f0 x5)) (f0 x5))
% Found (eq_ref0 (f0 x5)) as proof of (((eq Prop) (f0 x5)) (forall (x50:(b->Prop)), (((and (w x50)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x50 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx))))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (forall (x50:(b->Prop)), (((and (w x50)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x50 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx))))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (forall (x50:(b->Prop)), (((and (w x50)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x50 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx))))
% Found (fun (x5:b)=> ((eq_ref Prop) (f0 x5))) as proof of (((eq Prop) (f0 x5)) (forall (x50:(b->Prop)), (((and (w x50)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x50 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx))))
% Found (fun (x5:b)=> ((eq_ref Prop) (f0 x5))) as proof of (forall (x:b), (((eq Prop) (f0 x)) (forall (x50:(b->Prop)), (((and (w x50)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x50 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx)))))
% Found x8:(((eq a) Xx) (f x5))
% Instantiate: b0:=(f x5):a
% Found x8 as proof of (((eq a) Xx) b0)
% Found x4:(((eq a) Xx) (f x1))
% Instantiate: b0:=(f x1):a
% Found x4 as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 (f0 x7)):(((eq Prop) (f0 x7)) (f0 x7))
% Found (eq_ref0 (f0 x7)) as proof of (((eq Prop) (f0 x7)) (S Xx))
% Found ((eq_ref Prop) (f0 x7)) as proof of (((eq Prop) (f0 x7)) (S Xx))
% Found ((eq_ref Prop) (f0 x7)) as proof of (((eq Prop) (f0 x7)) (S Xx))
% Found (fun (x7:b)=> ((eq_ref Prop) (f0 x7))) as proof of (((eq Prop) (f0 x7)) (S Xx))
% Found (fun (x7:b)=> ((eq_ref Prop) (f0 x7))) as proof of (forall (x:b), (((eq Prop) (f0 x)) (S Xx)))
% Found eq_ref00:=(eq_ref0 (f0 x7)):(((eq Prop) (f0 x7)) (f0 x7))
% Found (eq_ref0 (f0 x7)) as proof of (((eq Prop) (f0 x7)) (S Xx))
% Found ((eq_ref Prop) (f0 x7)) as proof of (((eq Prop) (f0 x7)) (S Xx))
% Found ((eq_ref Prop) (f0 x7)) as proof of (((eq Prop) (f0 x7)) (S Xx))
% Found (fun (x7:b)=> ((eq_ref Prop) (f0 x7))) as proof of (((eq Prop) (f0 x7)) (S Xx))
% Found (fun (x7:b)=> ((eq_ref Prop) (f0 x7))) as proof of (forall (x:b), (((eq Prop) (f0 x)) (S Xx)))
% Found eq_ref00:=(eq_ref0 (f0 x6)):(((eq Prop) (f0 x6)) (f0 x6))
% Found (eq_ref0 (f0 x6)) as proof of (((eq Prop) (f0 x6)) (((and (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx)))
% Found ((eq_ref Prop) (f0 x6)) as proof of (((eq Prop) (f0 x6)) (((and (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx)))
% Found ((eq_ref Prop) (f0 x6)) as proof of (((eq Prop) (f0 x6)) (((and (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx)))
% Found (fun (x6:b)=> ((eq_ref Prop) (f0 x6))) as proof of (((eq Prop) (f0 x6)) (((and (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx)))
% Found (fun (x6:b)=> ((eq_ref Prop) (f0 x6))) as proof of (forall (x:b), (((eq Prop) (f0 x)) (((and (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx))))
% Found eq_ref00:=(eq_ref0 (f0 x6)):(((eq Prop) (f0 x6)) (f0 x6))
% Found (eq_ref0 (f0 x6)) as proof of (((eq Prop) (f0 x6)) (((and (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx)))
% Found ((eq_ref Prop) (f0 x6)) as proof of (((eq Prop) (f0 x6)) (((and (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx)))
% Found ((eq_ref Prop) (f0 x6)) as proof of (((eq Prop) (f0 x6)) (((and (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx)))
% Found (fun (x6:b)=> ((eq_ref Prop) (f0 x6))) as proof of (((eq Prop) (f0 x6)) (((and (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx)))
% Found (fun (x6:b)=> ((eq_ref Prop) (f0 x6))) as proof of (forall (x:b), (((eq Prop) (f0 x)) (((and (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx))))
% Found x8:(((eq a) Xx) (f x1))
% Instantiate: b0:=(f x1):a
% Found x8 as proof of (((eq a) Xx) b0)
% Found x8:(((eq a) Xx) (f x3))
% Instantiate: b0:=(f x3):a
% Found x8 as proof of (((eq a) Xx) b0)
% Found x6:(((eq a) Xx) (f x3))
% Instantiate: b0:=(f x3):a
% Found x6 as proof of (((eq a) Xx) b0)
% Found x6:(((eq a) Xx) (f x1))
% Instantiate: b0:=(f x1):a
% Found x6 as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 (f0 x7)):(((eq Prop) (f0 x7)) (f0 x7))
% Found (eq_ref0 (f0 x7)) as proof of (((eq Prop) (f0 x7)) (S Xx))
% Found ((eq_ref Prop) (f0 x7)) as proof of (((eq Prop) (f0 x7)) (S Xx))
% Found ((eq_ref Prop) (f0 x7)) as proof of (((eq Prop) (f0 x7)) (S Xx))
% Found (fun (x7:b)=> ((eq_ref Prop) (f0 x7))) as proof of (((eq Prop) (f0 x7)) (S Xx))
% Found (fun (x7:b)=> ((eq_ref Prop) (f0 x7))) as proof of (forall (x:b), (((eq Prop) (f0 x)) (S Xx)))
% Found eq_ref00:=(eq_ref0 (f0 x7)):(((eq Prop) (f0 x7)) (f0 x7))
% Found (eq_ref0 (f0 x7)) as proof of (((eq Prop) (f0 x7)) (S Xx))
% Found ((eq_ref Prop) (f0 x7)) as proof of (((eq Prop) (f0 x7)) (S Xx))
% Found ((eq_ref Prop) (f0 x7)) as proof of (((eq Prop) (f0 x7)) (S Xx))
% Found (fun (x7:b)=> ((eq_ref Prop) (f0 x7))) as proof of (((eq Prop) (f0 x7)) (S Xx))
% Found (fun (x7:b)=> ((eq_ref Prop) (f0 x7))) as proof of (forall (x:b), (((eq Prop) (f0 x)) (S Xx)))
% Found eq_ref00:=(eq_ref0 (f0 x7)):(((eq Prop) (f0 x7)) (f0 x7))
% Found (eq_ref0 (f0 x7)) as proof of (((eq Prop) (f0 x7)) (S Xx))
% Found ((eq_ref Prop) (f0 x7)) as proof of (((eq Prop) (f0 x7)) (S Xx))
% Found ((eq_ref Prop) (f0 x7)) as proof of (((eq Prop) (f0 x7)) (S Xx))
% Found (fun (x7:b)=> ((eq_ref Prop) (f0 x7))) as proof of (((eq Prop) (f0 x7)) (S Xx))
% Found (fun (x7:b)=> ((eq_ref Prop) (f0 x7))) as proof of (forall (x:b), (((eq Prop) (f0 x)) (S Xx)))
% Found eq_ref00:=(eq_ref0 (f0 x7)):(((eq Prop) (f0 x7)) (f0 x7))
% Found (eq_ref0 (f0 x7)) as proof of (((eq Prop) (f0 x7)) (S Xx))
% Found ((eq_ref Prop) (f0 x7)) as proof of (((eq Prop) (f0 x7)) (S Xx))
% Found ((eq_ref Prop) (f0 x7)) as proof of (((eq Prop) (f0 x7)) (S Xx))
% Found (fun (x7:b)=> ((eq_ref Prop) (f0 x7))) as proof of (((eq Prop) (f0 x7)) (S Xx))
% Found (fun (x7:b)=> ((eq_ref Prop) (f0 x7))) as proof of (forall (x:b), (((eq Prop) (f0 x)) (S Xx)))
% Found x7:(w x5)
% Instantiate: b0:=x5:(b->Prop)
% Found x7 as proof of (P b0)
% Found eta_expansion000:=(eta_expansion00 (fun (x80:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))):(((eq (b->Prop)) (fun (x80:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))) (fun (x:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))
% Found (eta_expansion00 (fun (x80:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))) as proof of (((eq (b->Prop)) (fun (x8:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))) b0)
% Found ((eta_expansion0 Prop) (fun (x80:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))) as proof of (((eq (b->Prop)) (fun (x8:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))) b0)
% Found (((eta_expansion b) Prop) (fun (x80:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))) as proof of (((eq (b->Prop)) (fun (x8:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))) b0)
% Found (((eta_expansion b) Prop) (fun (x80:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))) as proof of (((eq (b->Prop)) (fun (x8:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))) b0)
% Found (((eta_expansion b) Prop) (fun (x80:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))) as proof of (((eq (b->Prop)) (fun (x8:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))) b0)
% Found eq_ref00:=(eq_ref0 (f0 x7)):(((eq Prop) (f0 x7)) (f0 x7))
% Found (eq_ref0 (f0 x7)) as proof of (((eq Prop) (f0 x7)) (S Xx))
% Found ((eq_ref Prop) (f0 x7)) as proof of (((eq Prop) (f0 x7)) (S Xx))
% Found ((eq_ref Prop) (f0 x7)) as proof of (((eq Prop) (f0 x7)) (S Xx))
% Found (fun (x7:b)=> ((eq_ref Prop) (f0 x7))) as proof of (((eq Prop) (f0 x7)) (S Xx))
% Found (fun (x7:b)=> ((eq_ref Prop) (f0 x7))) as proof of (forall (x:b), (((eq Prop) (f0 x)) (S Xx)))
% Found eq_ref00:=(eq_ref0 (f0 x7)):(((eq Prop) (f0 x7)) (f0 x7))
% Found (eq_ref0 (f0 x7)) as proof of (((eq Prop) (f0 x7)) (S Xx))
% Found ((eq_ref Prop) (f0 x7)) as proof of (((eq Prop) (f0 x7)) (S Xx))
% Found ((eq_ref Prop) (f0 x7)) as proof of (((eq Prop) (f0 x7)) (S Xx))
% Found (fun (x7:b)=> ((eq_ref Prop) (f0 x7))) as proof of (((eq Prop) (f0 x7)) (S Xx))
% Found (fun (x7:b)=> ((eq_ref Prop) (f0 x7))) as proof of (forall (x:b), (((eq Prop) (f0 x)) (S Xx)))
% Found x3:(w x1)
% Instantiate: b0:=x1:(b->Prop)
% Found x3 as proof of (P b0)
% Found eq_ref00:=(eq_ref0 (fun (x90:b)=> (S Xx))):(((eq (b->Prop)) (fun (x90:b)=> (S Xx))) (fun (x90:b)=> (S Xx)))
% Found (eq_ref0 (fun (x90:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x9:b)=> (S Xx))) b0)
% Found ((eq_ref (b->Prop)) (fun (x90:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x9:b)=> (S Xx))) b0)
% Found ((eq_ref (b->Prop)) (fun (x90:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x9:b)=> (S Xx))) b0)
% Found ((eq_ref (b->Prop)) (fun (x90:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x9:b)=> (S Xx))) b0)
% Found x7:(w x5)
% Instantiate: b0:=x5:(b->Prop)
% Found x7 as proof of (P b0)
% Found eq_ref00:=(eq_ref0 (fun (x90:b)=> (S Xx))):(((eq (b->Prop)) (fun (x90:b)=> (S Xx))) (fun (x90:b)=> (S Xx)))
% Found (eq_ref0 (fun (x90:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x9:b)=> (S Xx))) b0)
% Found ((eq_ref (b->Prop)) (fun (x90:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x9:b)=> (S Xx))) b0)
% Found ((eq_ref (b->Prop)) (fun (x90:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x9:b)=> (S Xx))) b0)
% Found ((eq_ref (b->Prop)) (fun (x90:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x9:b)=> (S Xx))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (x70:b)=> ((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))):(((eq (b->Prop)) (fun (x70:b)=> ((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))) (fun (x:b)=> ((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))))
% Found (eta_expansion_dep00 (fun (x70:b)=> ((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))) as proof of (((eq (b->Prop)) (fun (x7:b)=> ((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))) b0)
% Found ((eta_expansion_dep0 (fun (x8:b)=> Prop)) (fun (x70:b)=> ((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))) as proof of (((eq (b->Prop)) (fun (x7:b)=> ((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))) b0)
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) (fun (x70:b)=> ((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))) as proof of (((eq (b->Prop)) (fun (x7:b)=> ((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))) b0)
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) (fun (x70:b)=> ((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))) as proof of (((eq (b->Prop)) (fun (x7:b)=> ((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))) b0)
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) (fun (x70:b)=> ((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))) as proof of (((eq (b->Prop)) (fun (x7:b)=> ((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))) b0)
% Found x7:(w x5)
% Instantiate: f0:=x5:(b->Prop)
% Found x7 as proof of (P f0)
% Found x7:(w x5)
% Instantiate: f0:=x5:(b->Prop)
% Found x7 as proof of (P f0)
% Found x7:(w x1)
% Instantiate: b0:=x1:(b->Prop)
% Found x7 as proof of (P b0)
% Found eta_expansion000:=(eta_expansion00 (fun (x80:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))):(((eq (b->Prop)) (fun (x80:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))) (fun (x:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))
% Found (eta_expansion00 (fun (x80:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))) as proof of (((eq (b->Prop)) (fun (x8:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))) b0)
% Found ((eta_expansion0 Prop) (fun (x80:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))) as proof of (((eq (b->Prop)) (fun (x8:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))) b0)
% Found (((eta_expansion b) Prop) (fun (x80:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))) as proof of (((eq (b->Prop)) (fun (x8:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))) b0)
% Found (((eta_expansion b) Prop) (fun (x80:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))) as proof of (((eq (b->Prop)) (fun (x8:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))) b0)
% Found (((eta_expansion b) Prop) (fun (x80:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))) as proof of (((eq (b->Prop)) (fun (x8:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))) b0)
% Found x7:(w x3)
% Instantiate: b0:=x3:(b->Prop)
% Found x7 as proof of (P b0)
% Found eta_expansion000:=(eta_expansion00 (fun (x80:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))):(((eq (b->Prop)) (fun (x80:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))) (fun (x:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))
% Found (eta_expansion00 (fun (x80:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))) as proof of (((eq (b->Prop)) (fun (x8:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))) b0)
% Found ((eta_expansion0 Prop) (fun (x80:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))) as proof of (((eq (b->Prop)) (fun (x8:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))) b0)
% Found (((eta_expansion b) Prop) (fun (x80:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))) as proof of (((eq (b->Prop)) (fun (x8:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))) b0)
% Found (((eta_expansion b) Prop) (fun (x80:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))) as proof of (((eq (b->Prop)) (fun (x8:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))) b0)
% Found (((eta_expansion b) Prop) (fun (x80:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))) as proof of (((eq (b->Prop)) (fun (x8:b)=> ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))) b0)
% Found x7:(w x5)
% Instantiate: b0:=x5:(b->Prop)
% Found x7 as proof of (P b0)
% Found eq_ref00:=(eq_ref0 (fun (x5:b)=> (S Xx))):(((eq (b->Prop)) (fun (x5:b)=> (S Xx))) (fun (x5:b)=> (S Xx)))
% Found (eq_ref0 (fun (x5:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x5:b)=> (S Xx))) b0)
% Found ((eq_ref (b->Prop)) (fun (x5:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x5:b)=> (S Xx))) b0)
% Found ((eq_ref (b->Prop)) (fun (x5:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x5:b)=> (S Xx))) b0)
% Found ((eq_ref (b->Prop)) (fun (x5:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x5:b)=> (S Xx))) b0)
% Found x7:(w x5)
% Instantiate: b0:=x5:(b->Prop)
% Found x7 as proof of (P b0)
% Found eq_ref00:=(eq_ref0 (fun (x7:b)=> (S Xx))):(((eq (b->Prop)) (fun (x7:b)=> (S Xx))) (fun (x7:b)=> (S Xx)))
% Found (eq_ref0 (fun (x7:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x7:b)=> (S Xx))) b0)
% Found ((eq_ref (b->Prop)) (fun (x7:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x7:b)=> (S Xx))) b0)
% Found ((eq_ref (b->Prop)) (fun (x7:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x7:b)=> (S Xx))) b0)
% Found ((eq_ref (b->Prop)) (fun (x7:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x7:b)=> (S Xx))) b0)
% Found eq_ref00:=(eq_ref0 (f0 x8)):(((eq Prop) (f0 x8)) (f0 x8))
% Found (eq_ref0 (f0 x8)) as proof of (((eq Prop) (f0 x8)) ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))
% Found ((eq_ref Prop) (f0 x8)) as proof of (((eq Prop) (f0 x8)) ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))
% Found ((eq_ref Prop) (f0 x8)) as proof of (((eq Prop) (f0 x8)) ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))
% Found (fun (x8:b)=> ((eq_ref Prop) (f0 x8))) as proof of (((eq Prop) (f0 x8)) ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))
% Found (fun (x8:b)=> ((eq_ref Prop) (f0 x8))) as proof of (forall (x:b), (((eq Prop) (f0 x)) ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))
% Found eq_ref00:=(eq_ref0 (f0 x8)):(((eq Prop) (f0 x8)) (f0 x8))
% Found (eq_ref0 (f0 x8)) as proof of (((eq Prop) (f0 x8)) ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))
% Found ((eq_ref Prop) (f0 x8)) as proof of (((eq Prop) (f0 x8)) ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))
% Found ((eq_ref Prop) (f0 x8)) as proof of (((eq Prop) (f0 x8)) ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))
% Found (fun (x8:b)=> ((eq_ref Prop) (f0 x8))) as proof of (((eq Prop) (f0 x8)) ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))
% Found (fun (x8:b)=> ((eq_ref Prop) (f0 x8))) as proof of (forall (x:b), (((eq Prop) (f0 x)) ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))
% Found x7:(w x5)
% Instantiate: b0:=x5:(b->Prop)
% Found (fun (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of (P b0)
% Found (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(P b0))
% Found (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of ((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(P b0)))
% Found (and_rect10 (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P b0)
% Found ((and_rect1 (P b0)) (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P b0)
% Found (((fun (P0:Type) (x7:((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->P0)))=> (((((and_rect (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))) P0) x7) x6)) (P b0)) (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P b0)
% Found (((fun (P0:Type) (x7:((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->P0)))=> (((((and_rect (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))) P0) x7) x6)) (P b0)) (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P b0)
% Found x7:(w x5)
% Instantiate: b0:=x5:(b->Prop)
% Found (fun (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of (P b0)
% Found (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(P b0))
% Found (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of ((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(P b0)))
% Found (and_rect10 (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P b0)
% Found ((and_rect1 (P b0)) (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P b0)
% Found (((fun (P0:Type) (x7:((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->P0)))=> (((((and_rect (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))) P0) x7) x6)) (P b0)) (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P b0)
% Found (((fun (P0:Type) (x7:((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->P0)))=> (((((and_rect (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))) P0) x7) x6)) (P b0)) (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P b0)
% Found x3:(w x1)
% Instantiate: f0:=x1:(b->Prop)
% Found x3 as proof of (P f0)
% Found x3:(w x1)
% Instantiate: f0:=x1:(b->Prop)
% Found x3 as proof of (P f0)
% Found x7:(w x5)
% Instantiate: b0:=x5:(b->Prop)
% Found (fun (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of (P b0)
% Found (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(P b0))
% Found (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of ((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(P b0)))
% Found (and_rect10 (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P b0)
% Found ((and_rect1 (P b0)) (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P b0)
% Found (((fun (P0:Type) (x7:((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->P0)))=> (((((and_rect (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))) P0) x7) x6)) (P b0)) (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P b0)
% Found (fun (x6:((and (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))))=> (((fun (P0:Type) (x7:((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->P0)))=> (((((and_rect (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))) P0) x7) x6)) (P b0)) (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7))) as proof of (P b0)
% Found x7:(w x5)
% Instantiate: f0:=x5:(b->Prop)
% Found x7 as proof of (P f0)
% Found x7:(w x5)
% Instantiate: f0:=x5:(b->Prop)
% Found x7 as proof of (P f0)
% Found x5:(w x1)
% Instantiate: b0:=x1:(b->Prop)
% Found x5 as proof of (P b0)
% Found eq_ref00:=(eq_ref0 (fun (x90:b)=> (S Xx))):(((eq (b->Prop)) (fun (x90:b)=> (S Xx))) (fun (x90:b)=> (S Xx)))
% Found (eq_ref0 (fun (x90:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x9:b)=> (S Xx))) b0)
% Found ((eq_ref (b->Prop)) (fun (x90:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x9:b)=> (S Xx))) b0)
% Found ((eq_ref (b->Prop)) (fun (x90:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x9:b)=> (S Xx))) b0)
% Found ((eq_ref (b->Prop)) (fun (x90:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x9:b)=> (S Xx))) b0)
% Found x5:(w x3)
% Instantiate: b0:=x3:(b->Prop)
% Found x5 as proof of (P b0)
% Found eq_ref00:=(eq_ref0 (fun (x90:b)=> (S Xx))):(((eq (b->Prop)) (fun (x90:b)=> (S Xx))) (fun (x90:b)=> (S Xx)))
% Found (eq_ref0 (fun (x90:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x9:b)=> (S Xx))) b0)
% Found ((eq_ref (b->Prop)) (fun (x90:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x9:b)=> (S Xx))) b0)
% Found ((eq_ref (b->Prop)) (fun (x90:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x9:b)=> (S Xx))) b0)
% Found ((eq_ref (b->Prop)) (fun (x90:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x9:b)=> (S Xx))) b0)
% Found x7:(w x1)
% Instantiate: b0:=x1:(b->Prop)
% Found x7 as proof of (P b0)
% Found eq_ref00:=(eq_ref0 (fun (x90:b)=> (S Xx))):(((eq (b->Prop)) (fun (x90:b)=> (S Xx))) (fun (x90:b)=> (S Xx)))
% Found (eq_ref0 (fun (x90:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x9:b)=> (S Xx))) b0)
% Found ((eq_ref (b->Prop)) (fun (x90:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x9:b)=> (S Xx))) b0)
% Found ((eq_ref (b->Prop)) (fun (x90:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x9:b)=> (S Xx))) b0)
% Found ((eq_ref (b->Prop)) (fun (x90:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x9:b)=> (S Xx))) b0)
% Found x7:(w x3)
% Instantiate: b0:=x3:(b->Prop)
% Found x7 as proof of (P b0)
% Found eq_ref00:=(eq_ref0 (fun (x90:b)=> (S Xx))):(((eq (b->Prop)) (fun (x90:b)=> (S Xx))) (fun (x90:b)=> (S Xx)))
% Found (eq_ref0 (fun (x90:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x9:b)=> (S Xx))) b0)
% Found ((eq_ref (b->Prop)) (fun (x90:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x9:b)=> (S Xx))) b0)
% Found ((eq_ref (b->Prop)) (fun (x90:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x9:b)=> (S Xx))) b0)
% Found ((eq_ref (b->Prop)) (fun (x90:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x9:b)=> (S Xx))) b0)
% Found eta_expansion000:=(eta_expansion00 (fun (x70:b)=> ((w x3)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))):(((eq (b->Prop)) (fun (x70:b)=> ((w x3)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))) (fun (x:b)=> ((w x3)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))))
% Found (eta_expansion00 (fun (x70:b)=> ((w x3)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))) as proof of (((eq (b->Prop)) (fun (x7:b)=> ((w x3)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))) b0)
% Found ((eta_expansion0 Prop) (fun (x70:b)=> ((w x3)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))) as proof of (((eq (b->Prop)) (fun (x7:b)=> ((w x3)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))) b0)
% Found (((eta_expansion b) Prop) (fun (x70:b)=> ((w x3)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))) as proof of (((eq (b->Prop)) (fun (x7:b)=> ((w x3)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))) b0)
% Found (((eta_expansion b) Prop) (fun (x70:b)=> ((w x3)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))) as proof of (((eq (b->Prop)) (fun (x7:b)=> ((w x3)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))) b0)
% Found (((eta_expansion b) Prop) (fun (x70:b)=> ((w x3)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))) as proof of (((eq (b->Prop)) (fun (x7:b)=> ((w x3)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (x70:b)=> ((w x1)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))):(((eq (b->Prop)) (fun (x70:b)=> ((w x1)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))) (fun (x:b)=> ((w x1)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))))
% Found (eta_expansion_dep00 (fun (x70:b)=> ((w x1)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))) as proof of (((eq (b->Prop)) (fun (x7:b)=> ((w x1)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))) b0)
% Found ((eta_expansion_dep0 (fun (x8:b)=> Prop)) (fun (x70:b)=> ((w x1)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))) as proof of (((eq (b->Prop)) (fun (x7:b)=> ((w x1)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))) b0)
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) (fun (x70:b)=> ((w x1)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))) as proof of (((eq (b->Prop)) (fun (x7:b)=> ((w x1)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))) b0)
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) (fun (x70:b)=> ((w x1)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))) as proof of (((eq (b->Prop)) (fun (x7:b)=> ((w x1)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))) b0)
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) (fun (x70:b)=> ((w x1)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))) as proof of (((eq (b->Prop)) (fun (x7:b)=> ((w x1)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))) b0)
% Found x7:(w x3)
% Instantiate: f0:=x3:(b->Prop)
% Found x7 as proof of (P f0)
% Found x7:(w x3)
% Instantiate: f0:=x3:(b->Prop)
% Found x7 as proof of (P f0)
% Found x7:(w x1)
% Instantiate: f0:=x1:(b->Prop)
% Found x7 as proof of (P f0)
% Found x7:(w x1)
% Instantiate: f0:=x1:(b->Prop)
% Found x7 as proof of (P f0)
% Found eq_ref00:=(eq_ref0 (f0 x9)):(((eq Prop) (f0 x9)) (f0 x9))
% Found (eq_ref0 (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (forall (x:b), (((eq Prop) (f0 x)) (S Xx)))
% Found eq_ref00:=(eq_ref0 (f0 x9)):(((eq Prop) (f0 x9)) (f0 x9))
% Found (eq_ref0 (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (forall (x:b), (((eq Prop) (f0 x)) (S Xx)))
% Found x7:(w x5)
% Instantiate: f0:=x5:(b->Prop)
% Found x7 as proof of (P f0)
% Found x7:(w x5)
% Instantiate: f0:=x5:(b->Prop)
% Found x7 as proof of (P f0)
% Found x7:(w x5)
% Instantiate: f0:=x5:(b->Prop)
% Found x7 as proof of (P f0)
% Found x7:(w x5)
% Instantiate: f0:=x5:(b->Prop)
% Found x7 as proof of (P f0)
% Found eq_ref00:=(eq_ref0 (f0 x9)):(((eq Prop) (f0 x9)) (f0 x9))
% Found (eq_ref0 (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (forall (x:b), (((eq Prop) (f0 x)) (S Xx)))
% Found eq_ref00:=(eq_ref0 (f0 x9)):(((eq Prop) (f0 x9)) (f0 x9))
% Found (eq_ref0 (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (forall (x:b), (((eq Prop) (f0 x)) (S Xx)))
% Found x7:(w x1)
% Instantiate: b0:=x1:(b->Prop)
% Found x7 as proof of (P b0)
% Found eq_ref00:=(eq_ref0 (fun (x7:b)=> (S Xx))):(((eq (b->Prop)) (fun (x7:b)=> (S Xx))) (fun (x7:b)=> (S Xx)))
% Found (eq_ref0 (fun (x7:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x7:b)=> (S Xx))) b0)
% Found ((eq_ref (b->Prop)) (fun (x7:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x7:b)=> (S Xx))) b0)
% Found ((eq_ref (b->Prop)) (fun (x7:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x7:b)=> (S Xx))) b0)
% Found ((eq_ref (b->Prop)) (fun (x7:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x7:b)=> (S Xx))) b0)
% Found x7:(w x3)
% Instantiate: b0:=x3:(b->Prop)
% Found x7 as proof of (P b0)
% Found eq_ref00:=(eq_ref0 (fun (x7:b)=> (S Xx))):(((eq (b->Prop)) (fun (x7:b)=> (S Xx))) (fun (x7:b)=> (S Xx)))
% Found (eq_ref0 (fun (x7:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x7:b)=> (S Xx))) b0)
% Found ((eq_ref (b->Prop)) (fun (x7:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x7:b)=> (S Xx))) b0)
% Found ((eq_ref (b->Prop)) (fun (x7:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x7:b)=> (S Xx))) b0)
% Found ((eq_ref (b->Prop)) (fun (x7:b)=> (S Xx))) as proof of (((eq (b->Prop)) (fun (x7:b)=> (S Xx))) b0)
% Found eq_ref00:=(eq_ref0 (f0 x8)):(((eq Prop) (f0 x8)) (f0 x8))
% Found (eq_ref0 (f0 x8)) as proof of (((eq Prop) (f0 x8)) ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))
% Found ((eq_ref Prop) (f0 x8)) as proof of (((eq Prop) (f0 x8)) ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))
% Found ((eq_ref Prop) (f0 x8)) as proof of (((eq Prop) (f0 x8)) ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))
% Found (fun (x8:b)=> ((eq_ref Prop) (f0 x8))) as proof of (((eq Prop) (f0 x8)) ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))
% Found (fun (x8:b)=> ((eq_ref Prop) (f0 x8))) as proof of (forall (x:b), (((eq Prop) (f0 x)) ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))
% Found eq_ref00:=(eq_ref0 (f0 x8)):(((eq Prop) (f0 x8)) (f0 x8))
% Found (eq_ref0 (f0 x8)) as proof of (((eq Prop) (f0 x8)) ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))
% Found ((eq_ref Prop) (f0 x8)) as proof of (((eq Prop) (f0 x8)) ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))
% Found ((eq_ref Prop) (f0 x8)) as proof of (((eq Prop) (f0 x8)) ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))
% Found (fun (x8:b)=> ((eq_ref Prop) (f0 x8))) as proof of (((eq Prop) (f0 x8)) ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))
% Found (fun (x8:b)=> ((eq_ref Prop) (f0 x8))) as proof of (forall (x:b), (((eq Prop) (f0 x)) ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))
% Found eq_ref00:=(eq_ref0 (f0 x8)):(((eq Prop) (f0 x8)) (f0 x8))
% Found (eq_ref0 (f0 x8)) as proof of (((eq Prop) (f0 x8)) ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))
% Found ((eq_ref Prop) (f0 x8)) as proof of (((eq Prop) (f0 x8)) ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))
% Found ((eq_ref Prop) (f0 x8)) as proof of (((eq Prop) (f0 x8)) ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))
% Found (fun (x8:b)=> ((eq_ref Prop) (f0 x8))) as proof of (((eq Prop) (f0 x8)) ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))
% Found (fun (x8:b)=> ((eq_ref Prop) (f0 x8))) as proof of (forall (x:b), (((eq Prop) (f0 x)) ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))
% Found eq_ref00:=(eq_ref0 (f0 x8)):(((eq Prop) (f0 x8)) (f0 x8))
% Found (eq_ref0 (f0 x8)) as proof of (((eq Prop) (f0 x8)) ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))
% Found ((eq_ref Prop) (f0 x8)) as proof of (((eq Prop) (f0 x8)) ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))
% Found ((eq_ref Prop) (f0 x8)) as proof of (((eq Prop) (f0 x8)) ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))
% Found (fun (x8:b)=> ((eq_ref Prop) (f0 x8))) as proof of (((eq Prop) (f0 x8)) ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))
% Found (fun (x8:b)=> ((eq_ref Prop) (f0 x8))) as proof of (forall (x:b), (((eq Prop) (f0 x)) ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))
% Found x7:(w x3)
% Instantiate: b0:=x3:(b->Prop)
% Found (fun (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of (P b0)
% Found (fun (x7:(w x3)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(P b0))
% Found (fun (x7:(w x3)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of ((w x3)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(P b0)))
% Found (and_rect10 (fun (x7:(w x3)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P b0)
% Found ((and_rect1 (P b0)) (fun (x7:(w x3)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P b0)
% Found (((fun (P0:Type) (x7:((w x3)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->P0)))=> (((((and_rect (w x3)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))) P0) x7) x4)) (P b0)) (fun (x7:(w x3)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P b0)
% Found (((fun (P0:Type) (x7:((w x3)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->P0)))=> (((((and_rect (w x3)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))) P0) x7) x4)) (P b0)) (fun (x7:(w x3)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P b0)
% Found x7:(w x1)
% Instantiate: b0:=x1:(b->Prop)
% Found (fun (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of (P b0)
% Found (fun (x7:(w x1)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(P b0))
% Found (fun (x7:(w x1)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of ((w x1)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(P b0)))
% Found (and_rect10 (fun (x7:(w x1)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P b0)
% Found ((and_rect1 (P b0)) (fun (x7:(w x1)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P b0)
% Found (((fun (P0:Type) (x7:((w x1)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->P0)))=> (((((and_rect (w x1)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))) P0) x7) x2)) (P b0)) (fun (x7:(w x1)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P b0)
% Found (((fun (P0:Type) (x7:((w x1)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->P0)))=> (((((and_rect (w x1)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))) P0) x7) x2)) (P b0)) (fun (x7:(w x1)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P b0)
% Found x7:(w x5)
% Instantiate: f0:=x5:(b->Prop)
% Found (fun (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of (P f0)
% Found (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(P f0))
% Found (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of ((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(P f0)))
% Found (and_rect10 (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P f0)
% Found ((and_rect1 (P f0)) (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P f0)
% Found (((fun (P0:Type) (x7:((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->P0)))=> (((((and_rect (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))) P0) x7) x6)) (P f0)) (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P f0)
% Found (((fun (P0:Type) (x7:((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->P0)))=> (((((and_rect (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))) P0) x7) x6)) (P f0)) (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P f0)
% Found x7:(w x5)
% Instantiate: f0:=x5:(b->Prop)
% Found (fun (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of (P f0)
% Found (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(P f0))
% Found (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of ((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(P f0)))
% Found (and_rect10 (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P f0)
% Found ((and_rect1 (P f0)) (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P f0)
% Found (((fun (P0:Type) (x7:((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->P0)))=> (((((and_rect (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))) P0) x7) x6)) (P f0)) (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P f0)
% Found (((fun (P0:Type) (x7:((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->P0)))=> (((((and_rect (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))) P0) x7) x6)) (P f0)) (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P f0)
% Found x7:(w x5)
% Instantiate: f0:=x5:(b->Prop)
% Found (fun (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of (P f0)
% Found (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(P f0))
% Found (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of ((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(P f0)))
% Found (and_rect10 (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P f0)
% Found ((and_rect1 (P f0)) (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P f0)
% Found (((fun (P0:Type) (x7:((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->P0)))=> (((((and_rect (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))) P0) x7) x6)) (P f0)) (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P f0)
% Found (((fun (P0:Type) (x7:((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->P0)))=> (((((and_rect (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))) P0) x7) x6)) (P f0)) (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P f0)
% Found x7:(w x5)
% Instantiate: f0:=x5:(b->Prop)
% Found (fun (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of (P f0)
% Found (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(P f0))
% Found (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of ((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(P f0)))
% Found (and_rect10 (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P f0)
% Found ((and_rect1 (P f0)) (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P f0)
% Found (((fun (P0:Type) (x7:((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->P0)))=> (((((and_rect (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))) P0) x7) x6)) (P f0)) (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P f0)
% Found (((fun (P0:Type) (x7:((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->P0)))=> (((((and_rect (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))) P0) x7) x6)) (P f0)) (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P f0)
% Found x5:(w x1)
% Instantiate: f0:=x1:(b->Prop)
% Found x5 as proof of (P f0)
% Found x5:(w x3)
% Instantiate: f0:=x3:(b->Prop)
% Found x5 as proof of (P f0)
% Found x5:(w x3)
% Instantiate: f0:=x3:(b->Prop)
% Found x5 as proof of (P f0)
% Found x5:(w x1)
% Instantiate: f0:=x1:(b->Prop)
% Found x5 as proof of (P f0)
% Found x7:(w x5)
% Instantiate: f0:=x5:(b->Prop)
% Found (fun (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of (P f0)
% Found (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(P f0))
% Found (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of ((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(P f0)))
% Found (and_rect10 (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P f0)
% Found ((and_rect1 (P f0)) (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P f0)
% Found (((fun (P0:Type) (x7:((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->P0)))=> (((((and_rect (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))) P0) x7) x6)) (P f0)) (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P f0)
% Found (fun (x6:((and (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))))=> (((fun (P0:Type) (x7:((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->P0)))=> (((((and_rect (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))) P0) x7) x6)) (P f0)) (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7))) as proof of (P f0)
% Found x7:(w x5)
% Instantiate: f0:=x5:(b->Prop)
% Found (fun (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of (P f0)
% Found (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(P f0))
% Found (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of ((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(P f0)))
% Found (and_rect10 (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P f0)
% Found ((and_rect1 (P f0)) (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P f0)
% Found (((fun (P0:Type) (x7:((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->P0)))=> (((((and_rect (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))) P0) x7) x6)) (P f0)) (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P f0)
% Found (fun (x6:((and (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))))=> (((fun (P0:Type) (x7:((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->P0)))=> (((((and_rect (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))) P0) x7) x6)) (P f0)) (fun (x7:(w x5)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7))) as proof of (P f0)
% Found x7:(w x1)
% Instantiate: f0:=x1:(b->Prop)
% Found x7 as proof of (P f0)
% Found x7:(w x1)
% Instantiate: f0:=x1:(b->Prop)
% Found x7 as proof of (P f0)
% Found x7:(w x3)
% Instantiate: f0:=x3:(b->Prop)
% Found x7 as proof of (P f0)
% Found x7:(w x3)
% Instantiate: f0:=x3:(b->Prop)
% Found x7 as proof of (P f0)
% Found eq_ref00:=(eq_ref0 (f0 x9)):(((eq Prop) (f0 x9)) (f0 x9))
% Found (eq_ref0 (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (forall (x:b), (((eq Prop) (f0 x)) (S Xx)))
% Found eq_ref00:=(eq_ref0 (f0 x9)):(((eq Prop) (f0 x9)) (f0 x9))
% Found (eq_ref0 (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (forall (x:b), (((eq Prop) (f0 x)) (S Xx)))
% Found eq_ref00:=(eq_ref0 (f0 x9)):(((eq Prop) (f0 x9)) (f0 x9))
% Found (eq_ref0 (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (forall (x:b), (((eq Prop) (f0 x)) (S Xx)))
% Found eq_ref00:=(eq_ref0 (f0 x9)):(((eq Prop) (f0 x9)) (f0 x9))
% Found (eq_ref0 (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (forall (x:b), (((eq Prop) (f0 x)) (S Xx)))
% Found eq_ref00:=(eq_ref0 (f0 x7)):(((eq Prop) (f0 x7)) (f0 x7))
% Found (eq_ref0 (f0 x7)) as proof of (((eq Prop) (f0 x7)) ((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))
% Found ((eq_ref Prop) (f0 x7)) as proof of (((eq Prop) (f0 x7)) ((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))
% Found ((eq_ref Prop) (f0 x7)) as proof of (((eq Prop) (f0 x7)) ((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))
% Found (fun (x7:b)=> ((eq_ref Prop) (f0 x7))) as proof of (((eq Prop) (f0 x7)) ((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))
% Found (fun (x7:b)=> ((eq_ref Prop) (f0 x7))) as proof of (forall (x:b), (((eq Prop) (f0 x)) ((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))))
% Found eq_ref00:=(eq_ref0 (f0 x7)):(((eq Prop) (f0 x7)) (f0 x7))
% Found (eq_ref0 (f0 x7)) as proof of (((eq Prop) (f0 x7)) ((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))
% Found ((eq_ref Prop) (f0 x7)) as proof of (((eq Prop) (f0 x7)) ((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))
% Found ((eq_ref Prop) (f0 x7)) as proof of (((eq Prop) (f0 x7)) ((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))
% Found (fun (x7:b)=> ((eq_ref Prop) (f0 x7))) as proof of (((eq Prop) (f0 x7)) ((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))
% Found (fun (x7:b)=> ((eq_ref Prop) (f0 x7))) as proof of (forall (x:b), (((eq Prop) (f0 x)) ((w x5)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))))
% Found eq_ref00:=(eq_ref0 (f0 x9)):(((eq Prop) (f0 x9)) (f0 x9))
% Found (eq_ref0 (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (forall (x:b), (((eq Prop) (f0 x)) (S Xx)))
% Found eq_ref00:=(eq_ref0 (f0 x9)):(((eq Prop) (f0 x9)) (f0 x9))
% Found (eq_ref0 (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (forall (x:b), (((eq Prop) (f0 x)) (S Xx)))
% Found eq_ref00:=(eq_ref0 (f0 x9)):(((eq Prop) (f0 x9)) (f0 x9))
% Found (eq_ref0 (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (forall (x:b), (((eq Prop) (f0 x)) (S Xx)))
% Found eq_ref00:=(eq_ref0 (f0 x9)):(((eq Prop) (f0 x9)) (f0 x9))
% Found (eq_ref0 (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (forall (x:b), (((eq Prop) (f0 x)) (S Xx)))
% Found x7:(w x3)
% Instantiate: f0:=x3:(b->Prop)
% Found x7 as proof of (P f0)
% Found x7:(w x1)
% Instantiate: f0:=x1:(b->Prop)
% Found x7 as proof of (P f0)
% Found x7:(w x3)
% Instantiate: f0:=x3:(b->Prop)
% Found x7 as proof of (P f0)
% Found x7:(w x1)
% Instantiate: f0:=x1:(b->Prop)
% Found x7 as proof of (P f0)
% Found eq_ref00:=(eq_ref0 (f0 x9)):(((eq Prop) (f0 x9)) (f0 x9))
% Found (eq_ref0 (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (forall (x:b), (((eq Prop) (f0 x)) (S Xx)))
% Found eq_ref00:=(eq_ref0 (f0 x9)):(((eq Prop) (f0 x9)) (f0 x9))
% Found (eq_ref0 (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (forall (x:b), (((eq Prop) (f0 x)) (S Xx)))
% Found eq_ref00:=(eq_ref0 (f0 x9)):(((eq Prop) (f0 x9)) (f0 x9))
% Found (eq_ref0 (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (forall (x:b), (((eq Prop) (f0 x)) (S Xx)))
% Found eq_ref00:=(eq_ref0 (f0 x9)):(((eq Prop) (f0 x9)) (f0 x9))
% Found (eq_ref0 (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (forall (x:b), (((eq Prop) (f0 x)) (S Xx)))
% Found x7:(w x1)
% Instantiate: f0:=x1:(b->Prop)
% Found (fun (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of (P f0)
% Found (fun (x7:(w x1)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(P f0))
% Found (fun (x7:(w x1)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of ((w x1)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(P f0)))
% Found (and_rect10 (fun (x7:(w x1)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P f0)
% Found ((and_rect1 (P f0)) (fun (x7:(w x1)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P f0)
% Found (((fun (P0:Type) (x7:((w x1)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->P0)))=> (((((and_rect (w x1)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))) P0) x7) x2)) (P f0)) (fun (x7:(w x1)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P f0)
% Found (((fun (P0:Type) (x7:((w x1)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->P0)))=> (((((and_rect (w x1)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))) P0) x7) x2)) (P f0)) (fun (x7:(w x1)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P f0)
% Found x7:(w x3)
% Instantiate: f0:=x3:(b->Prop)
% Found (fun (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of (P f0)
% Found (fun (x7:(w x3)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(P f0))
% Found (fun (x7:(w x3)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of ((w x3)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(P f0)))
% Found (and_rect10 (fun (x7:(w x3)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P f0)
% Found ((and_rect1 (P f0)) (fun (x7:(w x3)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P f0)
% Found (((fun (P0:Type) (x7:((w x3)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->P0)))=> (((((and_rect (w x3)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))) P0) x7) x4)) (P f0)) (fun (x7:(w x3)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P f0)
% Found (((fun (P0:Type) (x7:((w x3)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->P0)))=> (((((and_rect (w x3)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))) P0) x7) x4)) (P f0)) (fun (x7:(w x3)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P f0)
% Found x7:(w x3)
% Instantiate: f0:=x3:(b->Prop)
% Found (fun (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of (P f0)
% Found (fun (x7:(w x3)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(P f0))
% Found (fun (x7:(w x3)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of ((w x3)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(P f0)))
% Found (and_rect10 (fun (x7:(w x3)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P f0)
% Found ((and_rect1 (P f0)) (fun (x7:(w x3)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P f0)
% Found (((fun (P0:Type) (x7:((w x3)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->P0)))=> (((((and_rect (w x3)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))) P0) x7) x4)) (P f0)) (fun (x7:(w x3)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P f0)
% Found (((fun (P0:Type) (x7:((w x3)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->P0)))=> (((((and_rect (w x3)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))) P0) x7) x4)) (P f0)) (fun (x7:(w x3)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P f0)
% Found x7:(w x1)
% Instantiate: f0:=x1:(b->Prop)
% Found (fun (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of (P f0)
% Found (fun (x7:(w x1)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of ((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(P f0))
% Found (fun (x7:(w x1)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7) as proof of ((w x1)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(P f0)))
% Found (and_rect10 (fun (x7:(w x1)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P f0)
% Found ((and_rect1 (P f0)) (fun (x7:(w x1)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P f0)
% Found (((fun (P0:Type) (x7:((w x1)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->P0)))=> (((((and_rect (w x1)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))) P0) x7) x2)) (P f0)) (fun (x7:(w x1)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P f0)
% Found (((fun (P0:Type) (x7:((w x1)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->P0)))=> (((((and_rect (w x1)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))) P0) x7) x2)) (P f0)) (fun (x7:(w x1)) (x8:(((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0))))))))=> x7)) as proof of (P f0)
% Found eq_ref00:=(eq_ref0 (f0 x9)):(((eq Prop) (f0 x9)) (f0 x9))
% Found (eq_ref0 (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (forall (x:b), (((eq Prop) (f0 x)) (S Xx)))
% Found eq_ref00:=(eq_ref0 (f0 x9)):(((eq Prop) (f0 x9)) (f0 x9))
% Found (eq_ref0 (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (forall (x:b), (((eq Prop) (f0 x)) (S Xx)))
% Found eq_ref00:=(eq_ref0 (f0 x9)):(((eq Prop) (f0 x9)) (f0 x9))
% Found (eq_ref0 (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (forall (x:b), (((eq Prop) (f0 x)) (S Xx)))
% Found eq_ref00:=(eq_ref0 (f0 x9)):(((eq Prop) (f0 x9)) (f0 x9))
% Found (eq_ref0 (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found ((eq_ref Prop) (f0 x9)) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (((eq Prop) (f0 x9)) (S Xx))
% Found (fun (x9:b)=> ((eq_ref Prop) (f0 x9))) as proof of (forall (x:b), (((eq Prop) (f0 x)) (S Xx)))
% Found eq_ref00:=(eq_ref0 (f0 x7)):(((eq Prop) (f0 x7)) (f0 x7))
% Found (eq_ref0 (f0 x7)) as proof of (((eq Prop) (f0 x7)) ((w x1)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))
% Found ((eq_ref Prop) (f0 x7)) as proof of (((eq Prop) (f0 x7)) ((w x1)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))
% Found ((eq_ref Prop) (f0 x7)) as proof of (((eq Prop) (f0 x7)) ((w x1)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))
% Found (fun (x7:b)=> ((eq_ref Prop) (f0 x7))) as proof of (((eq Prop) (f0 x7)) ((w x1)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))
% Found (fun (x7:b)=> ((eq_ref Prop) (f0 x7))) as proof of (forall (x:b), (((eq Prop) (f0 x)) ((w x1)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))))
% Found eq_ref00:=(eq_ref0 (f0 x7)):(((eq Prop) (f0 x7)) (f0 x7))
% Found (eq_ref0 (f0 x7)) as proof of (((eq Prop) (f0 x7)) ((w x3)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))
% Found ((eq_ref Prop) (f0 x7)) as proof of (((eq Prop) (f0 x7)) ((w x3)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))
% Found ((eq_ref Prop) (f0 x7)) as proof of (((eq Prop) (f0 x7)) ((w x3)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))
% Found (fun (x7:b)=> ((eq_ref Prop) (f0 x7))) as proof of (((eq Prop) (f0 x7)) ((w x3)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))
% Found (fun (x7:b)=> ((eq_ref Prop) (f0 x7))) as proof of (forall (x:b), (((eq Prop) (f0 x)) ((w x3)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))))
% Found eq_ref00:=(eq_ref0 (f0 x7)):(((eq Prop) (f0 x7)) (f0 x7))
% Found (eq_ref0 (f0 x7)) as proof of (((eq Prop) (f0 x7)) ((w x1)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))
% Found ((eq_ref Prop) (f0 x7)) as proof of (((eq Prop) (f0 x7)) ((w x1)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))
% Found ((eq_ref Prop) (f0 x7)) as proof of (((eq Prop) (f0 x7)) ((w x1)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))
% Found (fun (x7:b)=> ((eq_ref Prop) (f0 x7))) as proof of (((eq Prop) (f0 x7)) ((w x1)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))
% Found (fun (x7:b)=> ((eq_ref Prop) (f0 x7))) as proof of (forall (x:b), (((eq Prop) (f0 x)) ((w x1)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x1 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))))
% Found eq_ref00:=(eq_ref0 (f0 x7)):(((eq Prop) (f0 x7)) (f0 x7))
% Found (eq_ref0 (f0 x7)) as proof of (((eq Prop) (f0 x7)) ((w x3)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))
% Found ((eq_ref Prop) (f0 x7)) as proof of (((eq Prop) (f0 x7)) ((w x3)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))
% Found ((eq_ref Prop) (f0 x7)) as proof of (((eq Prop) (f0 x7)) ((w x3)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))
% Found (fun (x7:b)=> ((eq_ref Prop) (f0 x7))) as proof of (((eq Prop) (f0 x7)) ((w x3)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx))))
% Found (fun (x7:b)=> ((eq_ref Prop) (f0 x7))) as proof of (forall (x:b), (((eq Prop) (f0 x)) ((w x3)->((((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x3 Xt0)) (((eq a) Xz) (f Xt0)))))))->(S Xx)))))
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found x4:(((eq a) Xx) (f x1))
% Instantiate: a0:=Xx:a;b0:=(f x1):a
% Found x4 as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq (b->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (b->Prop)) a0) b0)
% Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) b0)
% Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) b0)
% Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (b->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (b->Prop)) b0) (fun (x50:b)=> (S Xx)))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (x50:b)=> (S Xx)))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (x50:b)=> (S Xx)))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (x50:b)=> (S Xx)))
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) Xx)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx)
% Found (x40 ((eq_ref a) Xx)) as proof of (((eq a) b0) Xx)
% Found ((x4 (fun (x6:a)=> (((eq a) x6) Xx))) ((eq_ref a) Xx)) as proof of (((eq a) b0) Xx)
% Found ((x4 (fun (x6:a)=> (((eq a) x6) Xx))) ((eq_ref a) Xx)) as proof of (((eq a) b0) Xx)
% Found x6:(((eq a) Xx) (f x3))
% Instantiate: a0:=Xx:a;b0:=(f x3):a
% Found x6 as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found x6:(((eq a) Xx) (f x1))
% Instantiate: a0:=Xx:a;b0:=(f x1):a
% Found x6 as proof of (((eq a) a0) b0)
% Found x4:(((eq a) Xx) (f x1))
% Instantiate: a0:=Xx:a;b0:=(f x1):a
% Found x4 as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (b->Prop)) a0) (fun (x:b)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (b->Prop)) a0) b0)
% Found ((eta_expansion_dep0 (fun (x6:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) b0)
% Found (((eta_expansion_dep b) (fun (x6:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) b0)
% Found (((eta_expansion_dep b) (fun (x6:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) b0)
% Found (((eta_expansion_dep b) (fun (x6:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (b->Prop)) b0) (fun (x:b)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (b->Prop)) b0) (fun (x50:b)=> (forall (x500:(b->Prop)), (((and (w x500)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x500 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx)))))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (x50:b)=> (forall (x500:(b->Prop)), (((and (w x500)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x500 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx)))))
% Found (((eta_expansion b) Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (x50:b)=> (forall (x500:(b->Prop)), (((and (w x500)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x500 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx)))))
% Found (((eta_expansion b) Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (x50:b)=> (forall (x500:(b->Prop)), (((and (w x500)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x500 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx)))))
% Found (((eta_expansion b) Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (x50:b)=> (forall (x500:(b->Prop)), (((and (w x500)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x500 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx)))))
% Found eq_ref00:=(eq_ref0 b0):(((eq (b->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (b->Prop)) b0) (fun (x50:b)=> (forall (x500:(b->Prop)), (((and (w x500)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x500 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx)))))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (x50:b)=> (forall (x500:(b->Prop)), (((and (w x500)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x500 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx)))))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (x50:b)=> (forall (x500:(b->Prop)), (((and (w x500)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x500 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx)))))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (x50:b)=> (forall (x500:(b->Prop)), (((and (w x500)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x500 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx)))))
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (b->Prop)) a0) (fun (x:b)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (b->Prop)) a0) b0)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (b->Prop)) a0) b0)
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) b0)
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) b0)
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (b->Prop)) b0) (fun (x:b)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (b->Prop)) b0) (fun (x70:b)=> (S Xx)))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (x70:b)=> (S Xx)))
% Found (((eta_expansion b) Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (x70:b)=> (S Xx)))
% Found (((eta_expansion b) Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (x70:b)=> (S Xx)))
% Found (((eta_expansion b) Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (x70:b)=> (S Xx)))
% Found eq_ref00:=(eq_ref0 b0):(((eq (b->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (b->Prop)) b0) (fun (x70:b)=> (S Xx)))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (x70:b)=> (S Xx)))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (x70:b)=> (S Xx)))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (x70:b)=> (S Xx)))
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (b->Prop)) a0) (fun (x:b)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (b->Prop)) a0) b0)
% Found ((eta_expansion_dep0 (fun (x7:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) b0)
% Found (((eta_expansion_dep b) (fun (x7:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) b0)
% Found (((eta_expansion_dep b) (fun (x7:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) b0)
% Found (((eta_expansion_dep b) (fun (x7:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (b->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (b->Prop)) b0) (fun (x60:b)=> (((and (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx))))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (x60:b)=> (((and (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx))))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (x60:b)=> (((and (w x5)) (((eq (a->Prop)) S) (fun (Xz:a)=> ((ex b) (fun (Xt0:b)=> ((and (x5 Xt0)) (((eq a) Xz) (f Xt0))))))))->(S Xx))))
% Found ((eq_ref (b->Prop)) b0) as pro
% EOF
%------------------------------------------------------------------------------