%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEU867^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n113.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 13:33:18 EDT 2014

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

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEU867^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n113.star.cs.uiowa.edu
% % Model    : x86_64 x86_64
% % CPU      : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory   : 32286.75MB
% % OS       : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 11:37:56 CDT 2014
% % CPUTime  : 300.03 
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0xdd85f0>, <kernel.Type object at 0xdd8488>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (Xp:(a->Prop)) (Xq:(a->Prop)), (((and (forall (Xw:((a->Prop)->Prop)), (((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))->(Xw Xp)))) (forall (Xx:a), ((Xq Xx)->(Xp Xx))))->(forall (Xw:((a->Prop)->Prop)), (((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))->(Xw Xq))))) of role conjecture named cTHM160_pme
% Conjecture to prove = (forall (Xp:(a->Prop)) (Xq:(a->Prop)), (((and (forall (Xw:((a->Prop)->Prop)), (((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))->(Xw Xp)))) (forall (Xx:a), ((Xq Xx)->(Xp Xx))))->(forall (Xw:((a->Prop)->Prop)), (((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))->(Xw Xq))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (Xp:(a->Prop)) (Xq:(a->Prop)), (((and (forall (Xw:((a->Prop)->Prop)), (((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))->(Xw Xp)))) (forall (Xx:a), ((Xq Xx)->(Xp Xx))))->(forall (Xw:((a->Prop)->Prop)), (((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))->(Xw Xq)))))']
% Parameter a:Type.
% Trying to prove (forall (Xp:(a->Prop)) (Xq:(a->Prop)), (((and (forall (Xw:((a->Prop)->Prop)), (((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))->(Xw Xp)))) (forall (Xx:a), ((Xq Xx)->(Xp Xx))))->(forall (Xw:((a->Prop)->Prop)), (((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))->(Xw Xq)))))
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (Xq x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (Xq x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (Xq x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (Xq x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (Xq x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (Xq x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (Xq x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (Xq x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (Xq x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (Xq x)))
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (Xq x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (Xq x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (Xq x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (Xq x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (Xq x)))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (Xq x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (Xq x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (Xq x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (Xq x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (Xq x)))
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))):(((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) ((a->Prop)->(a->((Xw Xq)->(Xw Xq)))))
% Found (eq_ref0 ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) as proof of (((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) as proof of (((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) as proof of (((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) as proof of (((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) b)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (Xq x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (Xq x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (Xq x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (Xq x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (Xq x)))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (Xq x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (Xq x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (Xq x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (Xq x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (Xq x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq))))))):(((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq))))))) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq)))))))
% Found (eq_ref0 ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq))))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq))))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq))))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq))))))) b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq)))))):(((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq)))))) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq))))))
% Found (eq_ref0 ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq)))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq)))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq)))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq)))))) b)
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) (Xq x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (Xq x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (Xq x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) (Xq x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) (Xq x)))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) (Xq x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (Xq x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (Xq x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) (Xq x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) (Xq x)))
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))):(((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) ((a->Prop)->(a->((Xw Xq)->(Xw Xq)))))
% Found (eq_ref0 ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) as proof of (((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) as proof of (((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) as proof of (((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) as proof of (((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) b)
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) (Xq x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (Xq x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (Xq x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) (Xq x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) (Xq x)))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) (Xq x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (Xq x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (Xq x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) (Xq x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) (Xq x)))
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))):(((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) ((a->Prop)->(a->((Xw Xq)->(Xw Xq)))))
% Found (eq_ref0 ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) as proof of (((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) as proof of (((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) as proof of (((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) as proof of (((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))):(((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) ((a->Prop)->(a->((Xw Xq)->(Xw Xq)))))
% Found (eq_ref0 ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) as proof of (((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) as proof of (((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) as proof of (((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) as proof of (((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) Xq)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xq)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xq)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xq)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) Xq)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xq)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xq)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xq)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) Xq)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xq)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xq)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xq)
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((a->Prop)->(a->((Xw Xq)->(Xw Xq)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((a->Prop)->(a->((Xw Xq)->(Xw Xq)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((a->Prop)->(a->((Xw Xq)->(Xw Xq)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((a->Prop)->(a->((Xw Xq)->(Xw Xq)))))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) Xq)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xq)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xq)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xq)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) Xq)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xq)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xq)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xq)
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq)))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((a->Prop)->(a->((Xw Xq)->(Xw Xq)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((a->Prop)->(a->((Xw Xq)->(Xw Xq)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((a->Prop)->(a->((Xw Xq)->(Xw Xq)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((a->Prop)->(a->((Xw Xq)->(Xw Xq)))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((a->Prop)->(a->((Xw Xq)->(Xw Xq)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((a->Prop)->(a->((Xw Xq)->(Xw Xq)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((a->Prop)->(a->((Xw Xq)->(Xw Xq)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((a->Prop)->(a->((Xw Xq)->(Xw Xq)))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((a->Prop)->(a->((Xw Xq)->(Xw Xq)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((a->Prop)->(a->((Xw Xq)->(Xw Xq)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((a->Prop)->(a->((Xw Xq)->(Xw Xq)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((a->Prop)->(a->((Xw Xq)->(Xw Xq)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 f):(((eq (a->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 f):(((eq (a->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 f):(((eq (a->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found x0:((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))
% Instantiate: b:=(forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))):Prop;f:=(fun (Xx:a)=> False):(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P f)->(P f))))):(((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) ((a->Prop)->(a->((P f)->(P f)))))
% Found (eq_ref0 ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found x0:((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))
% Instantiate: b:=(forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))):Prop;f:=(fun (Xx:a)=> False):(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P f)->(P f))))):(((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) ((a->Prop)->(a->((P f)->(P f)))))
% Found (eq_ref0 ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (b x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (b x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (b x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (b x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (b x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (b x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 f):(((eq (a->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P b)->(P b))))):(((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) ((a->Prop)->(a->((P b)->(P b)))))
% Found (eq_ref0 ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (b x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (b x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 f):(((eq (a->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P b)->(P b))))):(((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) ((a->Prop)->(a->((P b)->(P b)))))
% Found (eq_ref0 ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 f):(((eq (a->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (b x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (b x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found x0:((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))
% Instantiate: b:=(forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))):Prop;f:=(fun (Xx:a)=> False):(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P f)->(P f))))):(((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) ((a->Prop)->(a->((P f)->(P f)))))
% Found (eq_ref0 ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found x0:((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))
% Instantiate: b:=(forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))):Prop;f:=(fun (Xx:a)=> False):(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P f)->(P f))))):(((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) ((a->Prop)->(a->((P f)->(P f)))))
% Found (eq_ref0 ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P f)->(P f))))):(((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) ((a->Prop)->(a->((P f)->(P f)))))
% Found (eq_ref0 ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P f)->(P f))))):(((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) ((a->Prop)->(a->((P f)->(P f)))))
% Found (eq_ref0 ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (b x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (b x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (b x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (b x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P b)->(P b))))):(((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) ((a->Prop)->(a->((P b)->(P b)))))
% Found (eq_ref0 ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (b x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (b x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (b x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P b)->(P b))))):(((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) ((a->Prop)->(a->((P b)->(P b)))))
% Found (eq_ref0 ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P f)->(P f))))):(((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) ((a->Prop)->(a->((P f)->(P f)))))
% Found (eq_ref0 ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P f)->(P f))))):(((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) ((a->Prop)->(a->((P f)->(P f)))))
% Found (eq_ref0 ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))):(((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) ((a->Prop)->(a->((Xw Xq)->(Xw Xq)))))
% Found (eq_ref0 ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) as proof of (((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) as proof of (((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) as proof of (((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) as proof of (((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) b0)
% Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) (f x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq)))))):(((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq)))))) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq))))))
% Found (eq_ref0 ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq)))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq)))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq)))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq)))))) 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)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% 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)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eta_expansion000:=(eta_expansion00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 f):(((eq (a->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq))))))):(((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq))))))) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq)))))))
% Found (eq_ref0 ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq))))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq))))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq))))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(Xw Xq)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(Xw Xq))))))) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) 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)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% 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)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found x0:((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))
% Instantiate: b:=(forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))):Prop;f:=(fun (Xx:a)=> False):(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P f)->(P f))))):(((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) ((a->Prop)->(a->((P f)->(P f)))))
% Found (eq_ref0 ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found x0:((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))
% Instantiate: b:=(forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))):Prop;f:=(fun (Xx:a)=> False):(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P f)->(P f))))):(((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) ((a->Prop)->(a->((P f)->(P f)))))
% Found (eq_ref0 ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found x0:((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))
% Instantiate: b:=(forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))):Prop;f:=(fun (Xx:a)=> False):(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P f)->(P f))))):(((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) ((a->Prop)->(a->((P f)->(P f)))))
% Found (eq_ref0 ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found x0:((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))
% Instantiate: b:=(forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))):Prop;f:=(fun (Xx:a)=> False):(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P f)->(P f))))):(((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) ((a->Prop)->(a->((P f)->(P f)))))
% Found (eq_ref0 ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))):(((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))))
% Found (eq_ref0 ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))) b0)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P b)->(P b))))):(((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) ((a->Prop)->(a->((P b)->(P b)))))
% Found (eq_ref0 ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P b)->(P b))))):(((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) ((a->Prop)->(a->((P b)->(P b)))))
% Found (eq_ref0 ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 f):(((eq (a->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eta_expansion000:=(eta_expansion00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))):(((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))))
% Found (eq_ref0 ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))) b0)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P b)->(P b))))):(((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) ((a->Prop)->(a->((P b)->(P b)))))
% Found (eq_ref0 ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) 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)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% 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)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P b)->(P b))))):(((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) ((a->Prop)->(a->((P b)->(P b)))))
% Found (eq_ref0 ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) 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)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% 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)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 f):(((eq (a->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))):(((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))
% Found (eq_ref0 ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 f):(((eq (a->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% 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)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% 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)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found conj00:=(conj0 (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))):((Xw (fun (Xx:a)=> False))->((forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))->((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))))
% Found (conj0 (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) as proof of ((Xw (fun (Xx:a)=> False))->((forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))->(P0 b0)))
% Found ((conj (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) as proof of ((Xw (fun (Xx:a)=> False))->((forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))->(P0 b0)))
% Found ((conj (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) as proof of ((Xw (fun (Xx:a)=> False))->((forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))->(P0 b0)))
% Found ((conj (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) as proof of ((Xw (fun (Xx:a)=> False))->((forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))->(P0 b0)))
% Found ((conj (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) as proof of ((Xw (fun (Xx:a)=> False))->((forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))->(P0 b0)))
% Found (and_rect10 ((conj (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) as proof of (P0 b0)
% Found ((and_rect1 (P0 b0)) ((conj (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) as proof of (P0 b0)
% Found (((fun (P1:Type) (x3:((Xw (fun (Xx:a)=> False))->((forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))->P1)))=> (((((and_rect (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) P1) x3) x0)) (P0 b0)) ((conj (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) as proof of (P0 b0)
% Found (((fun (P1:Type) (x3:((Xw (fun (Xx:a)=> False))->((forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))->P1)))=> (((((and_rect (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) P1) x3) x0)) (P0 b0)) ((conj (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) as proof of (P0 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)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% 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)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found x0:((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))
% Instantiate: b:=(forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))):Prop;f:=(fun (Xx:a)=> False):(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P f)->(P f))))):(((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) ((a->Prop)->(a->((P f)->(P f)))))
% Found (eq_ref0 ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found x0:((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))
% Instantiate: b:=(forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))):Prop;f:=(fun (Xx:a)=> False):(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P f)->(P f))))):(((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) ((a->Prop)->(a->((P f)->(P f)))))
% Found (eq_ref0 ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))):(((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))))
% Found (eq_ref0 ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))):(((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))))
% Found (eq_ref0 ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))):(((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))
% Found (eq_ref0 ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))) b0)
% Found x0:((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))
% Instantiate: b:=(forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))):Prop;f:=(fun (Xx:a)=> False):(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P f)->(P f))))):(((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) ((a->Prop)->(a->((P f)->(P f)))))
% Found (eq_ref0 ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found x0:((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))
% Instantiate: b:=(forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))):Prop;f:=(fun (Xx:a)=> False):(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P f)->(P f))))):(((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) ((a->Prop)->(a->((P f)->(P f)))))
% Found (eq_ref0 ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P f)->(P f))))):(((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) ((a->Prop)->(a->((P f)->(P f)))))
% Found (eq_ref0 ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P f)->(P f))))):(((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) ((a->Prop)->(a->((P f)->(P f)))))
% Found (eq_ref0 ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))):(((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))))
% Found (eq_ref0 ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))) b0)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P f)->(P f))))):(((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) ((a->Prop)->(a->((P f)->(P f)))))
% Found (eq_ref0 ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P f)->(P f))))):(((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) ((a->Prop)->(a->((P f)->(P f)))))
% Found (eq_ref0 ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P b)->(P b))))):(((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) ((a->Prop)->(a->((P b)->(P b)))))
% Found (eq_ref0 ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))):(((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))))
% Found (eq_ref0 ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))):(((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))))
% Found (eq_ref0 ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P b)->(P b))))):(((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) ((a->Prop)->(a->((P b)->(P b)))))
% Found (eq_ref0 ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found x0:((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))
% Instantiate: b:=(forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))):Prop;f:=(fun (Xx:a)=> False):(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P f)->(P f))))):(((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) ((a->Prop)->(a->((P f)->(P f)))))
% Found (eq_ref0 ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found x0:((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))
% Instantiate: b:=(forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))):Prop;f:=(fun (Xx:a)=> False):(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P f)->(P f))))):(((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) ((a->Prop)->(a->((P f)->(P f)))))
% Found (eq_ref0 ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))):(((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) ((a->Prop)->(a->((Xw Xq)->(Xw Xq)))))
% Found (eq_ref0 ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) as proof of (((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) as proof of (((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) as proof of (((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) as proof of (((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) b0)
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))):(((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))))
% Found (eq_ref0 ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))):(((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))
% Found (eq_ref0 ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))):(((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))
% Found (eq_ref0 ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P b)->(P b))))):(((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) ((a->Prop)->(a->((P b)->(P b)))))
% Found (eq_ref0 ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P b)->(P b))))):(((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) ((a->Prop)->(a->((P b)->(P b)))))
% Found (eq_ref0 ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P b)->(P b))))):(((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) ((a->Prop)->(a->((P b)->(P b)))))
% Found (eq_ref0 ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) 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)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% 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)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))):(((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))
% Found (eq_ref0 ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))) 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)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% 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)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))):(((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))
% Found (eq_ref0 ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))):(((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))
% Found (eq_ref0 ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) b)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found conj00:=(conj0 (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))):((Xw (fun (Xx:a)=> False))->((forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))->((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))))
% Found (conj0 (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) as proof of ((Xw (fun (Xx:a)=> False))->((forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))->(P0 b0)))
% Found ((conj (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) as proof of ((Xw (fun (Xx:a)=> False))->((forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))->(P0 b0)))
% Found ((conj (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) as proof of ((Xw (fun (Xx:a)=> False))->((forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))->(P0 b0)))
% Found ((conj (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) as proof of ((Xw (fun (Xx:a)=> False))->((forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))->(P0 b0)))
% Found ((conj (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) as proof of ((Xw (fun (Xx:a)=> False))->((forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))->(P0 b0)))
% Found (and_rect10 ((conj (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) as proof of (P0 b0)
% Found ((and_rect1 (P0 b0)) ((conj (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) as proof of (P0 b0)
% Found (((fun (P1:Type) (x3:((Xw (fun (Xx:a)=> False))->((forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))->P1)))=> (((((and_rect (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) P1) x3) x0)) (P0 b0)) ((conj (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) as proof of (P0 b0)
% Found (((fun (P1:Type) (x3:((Xw (fun (Xx:a)=> False))->((forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))->P1)))=> (((((and_rect (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) P1) x3) x0)) (P0 b0)) ((conj (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) as proof of (P0 b0)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 f):(((eq (a->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P b)->(P b))))):(((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) ((a->Prop)->(a->((P b)->(P b)))))
% Found (eq_ref0 ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) 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)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% 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)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))):(((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b))))))
% Found (eq_ref0 ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P b))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P b)))))) 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)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% 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)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (((eq Prop) (f0 x5)) (f x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found x0:((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))
% Instantiate: b:=(forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))):Prop;f:=(fun (Xx:a)=> False):(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P f)->(P f))))):(((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) ((a->Prop)->(a->((P f)->(P f)))))
% Found (eq_ref0 ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found x0:((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))
% Instantiate: b:=(forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))):Prop;f:=(fun (Xx:a)=> False):(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P f)->(P f))))):(((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) ((a->Prop)->(a->((P f)->(P f)))))
% Found (eq_ref0 ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))):(((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))))
% Found (eq_ref0 ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))):(((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))))
% Found (eq_ref0 ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) b)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b0)
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found x0:((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))
% Instantiate: b:=(forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))):Prop;f:=(fun (Xx:a)=> False):(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P f)->(P f))))):(((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) ((a->Prop)->(a->((P f)->(P f)))))
% Found (eq_ref0 ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found x0:((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))
% Instantiate: b:=(forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))):Prop;f:=(fun (Xx:a)=> False):(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P f)->(P f))))):(((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) ((a->Prop)->(a->((P f)->(P f)))))
% Found (eq_ref0 ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P b)->(P b))))):(((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) ((a->Prop)->(a->((P b)->(P b)))))
% Found (eq_ref0 ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P f)->(P f))))):(((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) ((a->Prop)->(a->((P f)->(P f)))))
% Found (eq_ref0 ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P b)->(P b))))):(((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) ((a->Prop)->(a->((P b)->(P b)))))
% Found (eq_ref0 ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P f)->(P f))))):(((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) ((a->Prop)->(a->((P f)->(P f)))))
% Found (eq_ref0 ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P f)->(P f))))):(((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) ((a->Prop)->(a->((P f)->(P f)))))
% Found (eq_ref0 ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P f)->(P f))))):(((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) ((a->Prop)->(a->((P f)->(P f)))))
% Found (eq_ref0 ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))):(((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))))
% Found (eq_ref0 ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))):(((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))))
% Found (eq_ref0 ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) as proof of (((eq Prop) ((a->Prop)->(a->(((Xw (fun (Xx:a)=> False))->((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f)))->((Xw (fun (Xx0:a)=> False))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))) b)
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) (b x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found x0:((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))
% Instantiate: b:=(forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))):Prop;f:=(fun (Xx:a)=> False):(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P f)->(P f))))):(((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) ((a->Prop)->(a->((P f)->(P f)))))
% Found (eq_ref0 ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found x0:((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))
% Instantiate: b:=(forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))):Prop;f:=(fun (Xx:a)=> False):(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P f)->(P f))))):(((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) ((a->Prop)->(a->((P f)->(P f)))))
% Found (eq_ref0 ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P f)->(P f))))) as proof of (((eq Prop) ((a->Prop)->(a->((P f)->(P f))))) b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))):(((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) ((a->Prop)->(a->((Xw Xq)->(Xw Xq)))))
% Found (eq_ref0 ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) as proof of (((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) as proof of (((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) as proof of (((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) as proof of (((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) b0)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P b)->(P b))))):(((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) ((a->Prop)->(a->((P b)->(P b)))))
% Found (eq_ref0 ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))):(((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) ((a->Prop)->(a->((Xw Xq)->(Xw Xq)))))
% Found (eq_ref0 ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) as proof of (((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) as proof of (((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) as proof of (((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) as proof of (((eq Prop) ((a->Prop)->(a->((Xw Xq)->(Xw Xq))))) b0)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->((P b)->(P b))))):(((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) ((a->Prop)->(a->((P b)->(P b)))))
% Found (eq_ref0 ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found ((eq_ref Prop) ((a->Prop)->(a->((P b)->(P b))))) as proof of (((eq Prop) ((a->Prop)->(a->((P b)->(P b))))) b0)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))):(((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))
% Found (eq_ref0 ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))):(((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f))))))
% Found (eq_ref0 ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx))))))->(P f))->((forall (Xr0:(a->Prop)) (Xx0:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or (Xr0 Xt)) (((eq a) Xt) Xx0))))))->(P f)))))) as proof of (((eq Prop) ((a->Prop)->(a->(((forall (Xr0:(a->Prop)) (Xx:a), ((Xw Xr0)->(Xw (fun (Xt:a)=> ((or 
% EOF
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