%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEU872^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n098.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 300.10s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEU872^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n098.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 11:38:36 CDT 2014
% % CPUTime  : 300.10 
% 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 0x263bef0>, <kernel.Type object at 0x263b7a0>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x263db00>, <kernel.DependentProduct object at 0x263b560>) of role type named cF
% Using role type
% Declaring cF:(a->Prop)
% FOF formula (<kernel.Constant object at 0x263b200>, <kernel.DependentProduct object at 0x263b9e0>) of role type named cE
% Using role type
% Declaring cE:(a->Prop)
% FOF formula (((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X cF))))->(forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), (((or (cE Xt)) (cF Xt))->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X (fun (Xz:a)=> ((or (cE Xz)) (cF Xz))))))) of role conjecture named cDOMLEMMA2_pme
% Conjecture to prove = (((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X cF))))->(forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), (((or (cE Xt)) (cF Xt))->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X (fun (Xz:a)=> ((or (cE Xz)) (cF Xz))))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X cF))))->(forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), (((or (cE Xt)) (cF Xt))->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))))))']
% Parameter a:Type.
% Parameter cF:(a->Prop).
% Parameter cE:(a->Prop).
% Trying to prove (((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X cF))))->(forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), (((or (cE Xt)) (cF Xt))->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) (fun (x:a)=> ((or (cE x)) (cF x))))
% Found (eta_expansion00 (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) b)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) 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)) ((or (cE x1)) (cF x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((or (cE x1)) (cF x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((or (cE x1)) (cF x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((or (cE x1)) (cF x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (cE x)) (cF x))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((or (cE x1)) (cF x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((or (cE x1)) (cF x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((or (cE x1)) (cF x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((or (cE x1)) (cF x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (cE x)) (cF x))))
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz))))
% Found (eq_ref0 (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) b)
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz))))
% Found (eq_ref0 (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx:(a->Prop)), (((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz)))))->(forall (Xt:a), ((cE Xt)->((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))))))))):(((eq Prop) (forall (Xx:(a->Prop)), (((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz)))))->(forall (Xt:a), ((cE Xt)->((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))))))))) (forall (Xx:(a->Prop)), (((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz)))))->(forall (Xt:a), ((cE Xt)->((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))))
% Found (eq_ref0 (forall (Xx:(a->Prop)), (((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz)))))->(forall (Xt:a), ((cE Xt)->((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz)))))->(forall (Xt:a), ((cE Xt)->((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz)))))->(forall (Xt:a), ((cE Xt)->((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz)))))->(forall (Xt:a), ((cE Xt)->((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz)))))->(forall (Xt:a), ((cE Xt)->((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz)))))->(forall (Xt:a), ((cE Xt)->((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz)))))->(forall (Xt:a), ((cE Xt)->((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz)))))->(forall (Xt:a), ((cE Xt)->((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))))))))) 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)) ((or (cE x3)) (cF x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((or (cE x3)) (cF x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((or (cE x3)) (cF x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((or (cE x3)) (cF x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (cE x)) (cF 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)) ((or (cE x3)) (cF x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((or (cE x3)) (cF x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((or (cE x3)) (cF x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((or (cE x3)) (cF x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (cE x)) (cF x))))
% Found eq_ref00:=(eq_ref0 (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))):(((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))))))))
% Found (eq_ref0 (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))):(((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))
% Found (eq_ref0 (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) 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)) ((or (cE x3)) (cF x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((or (cE x3)) (cF x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((or (cE x3)) (cF x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((or (cE x3)) (cF x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (cE x)) (cF 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)) ((or (cE x3)) (cF x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((or (cE x3)) (cF x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((or (cE x3)) (cF x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((or (cE x3)) (cF x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (cE x)) (cF x))))
% Found x1:(forall (X0:((a->Prop)->Prop)), (((and (X0 (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X0 Xx)->(forall (Xt:a), ((cE Xt)->(X0 (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X0 cE)))
% Instantiate: b:=(forall (X0:((a->Prop)->Prop)), (((and (X0 (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X0 Xx)->(forall (Xt:a), ((cE Xt)->(X0 (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X0 cE))):Prop
% Found x1 as proof of b
% Found eq_sym:=(fun (T:Type) (a:T) (b:T) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq T) x) a))) ((eq_ref T) a))):(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a)))
% Instantiate: b:=(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a))):Prop
% Found eq_sym as proof of b
% Found eq_sym:=(fun (T:Type) (a:T) (b:T) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq T) x) a))) ((eq_ref T) a))):(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a)))
% Instantiate: b:=(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a))):Prop
% Found eq_sym as proof of b
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) (fun (x:a)=> ((or (cE x)) (cF x))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) b)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) (fun (x:a)=> ((or (cE x)) (cF x))))
% Found (eta_expansion00 (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) b)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx:(a->Prop)), (((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz)))))->(forall (Xt:a), ((cE Xt)->((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))))))))):(((eq Prop) (forall (Xx:(a->Prop)), (((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz)))))->(forall (Xt:a), ((cE Xt)->((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))))))))) (forall (Xx:(a->Prop)), (((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz)))))->(forall (Xt:a), ((cE Xt)->((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))))
% Found (eq_ref0 (forall (Xx:(a->Prop)), (((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz)))))->(forall (Xt:a), ((cE Xt)->((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz)))))->(forall (Xt:a), ((cE Xt)->((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz)))))->(forall (Xt:a), ((cE Xt)->((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz)))))->(forall (Xt:a), ((cE Xt)->((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz)))))->(forall (Xt:a), ((cE Xt)->((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz)))))->(forall (Xt:a), ((cE Xt)->((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz)))))->(forall (Xt:a), ((cE Xt)->((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz)))))->(forall (Xt:a), ((cE Xt)->((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx:(a->Prop)), (((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz)))))->(forall (Xt:a), ((cE Xt)->((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))))))))):(((eq Prop) (forall (Xx:(a->Prop)), (((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz)))))->(forall (Xt:a), ((cE Xt)->((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))))))))) (forall (Xx:(a->Prop)), (((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz)))))->(forall (Xt:a), ((cE Xt)->((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))))
% Found (eq_ref0 (forall (Xx:(a->Prop)), (((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz)))))->(forall (Xt:a), ((cE Xt)->((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz)))))->(forall (Xt:a), ((cE Xt)->((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz)))))->(forall (Xt:a), ((cE Xt)->((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz)))))->(forall (Xt:a), ((cE Xt)->((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz)))))->(forall (Xt:a), ((cE Xt)->((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz)))))->(forall (Xt:a), ((cE Xt)->((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz)))))->(forall (Xt:a), ((cE Xt)->((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz)))))->(forall (Xt:a), ((cE Xt)->((forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))))))))) 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)) ((or (cE x5)) (cF x5)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or (cE x5)) (cF x5)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or (cE x5)) (cF x5)))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((or (cE x5)) (cF x5)))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (cE x)) (cF 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)) ((or (cE x5)) (cF x5)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or (cE x5)) (cF x5)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or (cE x5)) (cF x5)))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((or (cE x5)) (cF x5)))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (cE x)) (cF x))))
% Found eq_ref00:=(eq_ref0 (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))):(((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))))))))
% Found (eq_ref0 (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))):(((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))
% Found (eq_ref0 (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))):(((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))))))))
% Found (eq_ref0 (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))) 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)) ((or (cE x5)) (cF x5)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or (cE x5)) (cF x5)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or (cE x5)) (cF x5)))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((or (cE x5)) (cF x5)))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (cE x)) (cF x))))
% Found eq_ref00:=(eq_ref0 (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))):(((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))
% Found (eq_ref0 (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) 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)) ((or (cE x5)) (cF x5)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or (cE x5)) (cF x5)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or (cE x5)) (cF x5)))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((or (cE x5)) (cF x5)))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (cE x)) (cF x))))
% Found eq_ref00:=(eq_ref0 (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))):(((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))
% Found (eq_ref0 (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))):(((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))))))))
% Found (eq_ref0 (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))))) b)
% Found eq_sym0:=(eq_sym Prop):(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a)))
% Instantiate: b:=(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))):Prop
% Found eq_sym0 as proof of b
% Found eq_sym0:=(eq_sym Prop):(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a)))
% Instantiate: b:=(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))):Prop
% Found eq_sym0 as proof of b
% Found x40:=(x4 (fun (x6:(a->Prop))=> (X (fun (Xz:a)=> ((or (cE Xz)) (x6 Xz)))))):(((and (X (fun (Xz:a)=> ((or (cE Xz)) False)))) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))->(X (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))))
% Instantiate: b:=(((and (X (fun (Xz:a)=> ((or (cE Xz)) False)))) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))->(X (fun (Xz:a)=> ((or (cE Xz)) (cF Xz))))):Prop
% Found x40 as proof of b
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: b:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of b
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: b:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of b
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: b:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of b
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: b:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of b
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: b:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of b
% Found eq_sym0:=(eq_sym Prop):(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a)))
% Instantiate: b:=(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))):Prop
% Found eq_sym0 as proof of b
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: b:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of b
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: b:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of b
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or False) (cF Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) (fun (Xz:a)=> ((or False) (cF Xz))))
% Found (eq_ref0 (fun (Xz:a)=> ((or False) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) b)
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or (cE Xz)) False))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) (fun (Xz:a)=> ((or (cE Xz)) False)))
% Found (eq_ref0 (fun (Xz:a)=> ((or (cE Xz)) False))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) b)
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or False) (cF Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) (fun (Xz:a)=> ((or False) (cF Xz))))
% Found (eq_ref0 (fun (Xz:a)=> ((or False) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) b)
% Found x3:(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))
% Instantiate: b:=(fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))):(a->Prop)
% Found x3 as proof of (P b)
% Found x3:(X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))
% Instantiate: b:=(fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))):(a->Prop)
% Found x3 as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) (fun (x:a)=> ((or (cE x)) ((or (Xx x)) (((eq a) Xt) x)))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) b)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) (fun (x:a)=> ((or ((or (Xx x)) (((eq a) Xt) x))) (cF x))))
% Found (eta_expansion00 (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) b)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) 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)) ((or False) (cF x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((or False) (cF x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((or False) (cF x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((or False) (cF x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or False) (cF 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)) ((or False) (cF x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((or False) (cF x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((or False) (cF x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((or False) (cF x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or False) (cF 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)) ((or (cE x3)) False))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((or (cE x3)) False))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((or (cE x3)) False))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((or (cE x3)) False))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (cE x)) False)))
% 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)) ((or (cE x3)) False))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((or (cE x3)) False))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((or (cE x3)) False))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((or (cE x3)) False))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (cE x)) False)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((or False) (cF x2)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((or False) (cF x2)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((or False) (cF x2)))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((or False) (cF x2)))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or False) (cF x))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((or False) (cF x2)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((or False) (cF x2)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((or False) (cF x2)))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((or False) (cF x2)))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or False) (cF x))))
% Found eq_ref00:=(eq_ref0 (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))):(((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))
% Found (eq_ref0 (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))):(((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False))))))))
% Found (eq_ref0 (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))) b)
% Found x3:(X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))
% Instantiate: f:=(fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))):(a->Prop)
% Found x3 as proof of (P f)
% Found x3:(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))
% Instantiate: f:=(fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))):(a->Prop)
% Found x3 as proof of (P f)
% Found x3:(X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))
% Instantiate: f:=(fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))):(a->Prop)
% Found x3 as proof of (P f)
% Found x3:(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))
% Instantiate: f:=(fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))):(a->Prop)
% Found x3 as proof of (P f)
% Found eq_ref00:=(eq_ref0 (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))):(((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))
% Found (eq_ref0 (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found x200:(forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))
% Found x200 as proof of (forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or (cE Xz)) False))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) (fun (x:a)=> ((or (cE x)) False)))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or (cE Xz)) False))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) b)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) b)
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or False) (cF Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) (fun (Xz:a)=> ((or False) (cF Xz))))
% Found (eq_ref0 (fun (Xz:a)=> ((or False) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((or (cE Xz)) False))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) (fun (x:a)=> ((or (cE x)) False)))
% Found (eta_expansion00 (fun (Xz:a)=> ((or (cE Xz)) False))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) b)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or (cE Xz)) False))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (cE Xz)) False))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (cE Xz)) False))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (cE Xz)) False))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) b)
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or False) (cF Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) (fun (Xz:a)=> ((or False) (cF Xz))))
% Found (eq_ref0 (fun (Xz:a)=> ((or False) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) (fun (x:a)=> ((or ((or (Xx x)) (((eq a) Xt) x))) (cF x))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) b)
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) b)
% Found x21:=(x2 x20):(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))
% Instantiate: b:=(fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))):(a->Prop)
% Found (x2 x20) as proof of (P b)
% Found (x2 x20) as proof of (P 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)) ((or ((or (Xx x5)) (((eq a) Xt) x5))) (cF x5)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or ((or (Xx x5)) (((eq a) Xt) x5))) (cF x5)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or ((or (Xx x5)) (((eq a) Xt) x5))) (cF x5)))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((or ((or (Xx x5)) (((eq a) Xt) x5))) (cF x5)))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or ((or (Xx x)) (((eq a) Xt) x))) (cF 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)) ((or ((or (Xx x5)) (((eq a) Xt) x5))) (cF x5)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or ((or (Xx x5)) (((eq a) Xt) x5))) (cF x5)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or ((or (Xx x5)) (((eq a) Xt) x5))) (cF x5)))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((or ((or (Xx x5)) (((eq a) Xt) x5))) (cF x5)))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or ((or (Xx x)) (((eq a) Xt) x))) (cF 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)) ((or (cE x5)) ((or (Xx x5)) (((eq a) Xt) x5))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or (cE x5)) ((or (Xx x5)) (((eq a) Xt) x5))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or (cE x5)) ((or (Xx x5)) (((eq a) Xt) x5))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((or (cE x5)) ((or (Xx x5)) (((eq a) Xt) x5))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (cE x)) ((or (Xx x)) (((eq a) Xt) 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)) ((or (cE x5)) ((or (Xx x5)) (((eq a) Xt) x5))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or (cE x5)) ((or (Xx x5)) (((eq a) Xt) x5))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or (cE x5)) ((or (Xx x5)) (((eq a) Xt) x5))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((or (cE x5)) ((or (Xx x5)) (((eq a) Xt) x5))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (cE x)) ((or (Xx x)) (((eq a) Xt) x)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or False) (cF Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) (fun (x:a)=> ((or False) (cF x))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or False) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) b)
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) b)
% Found x200:(forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))
% Found x200 as proof of (forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or (cE Xz)) False))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) (fun (Xz:a)=> ((or (cE Xz)) False)))
% Found (eq_ref0 (fun (Xz:a)=> ((or (cE Xz)) False))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) b)
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or False) (cF Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) (fun (Xz:a)=> ((or False) (cF Xz))))
% Found (eq_ref0 (fun (Xz:a)=> ((or False) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))):(((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz))))))))))))
% Found (eq_ref0 (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))):(((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz))))))))))))
% Found (eq_ref0 (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or False) (cF Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) (fun (x:a)=> ((or False) (cF x))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or False) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) b)
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) b)
% Found x200:(forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))
% Found x200 as proof of (forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))
% Found x21:=(x2 x20):(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))
% Instantiate: f:=(fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))):(a->Prop)
% Found (x2 x20) as proof of (P f)
% Found (x2 x20) as proof of (P f)
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((or ((or (Xx x4)) (((eq a) Xt) x4))) (cF x4)))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((or ((or (Xx x4)) (((eq a) Xt) x4))) (cF x4)))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((or ((or (Xx x4)) (((eq a) Xt) x4))) (cF x4)))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((or ((or (Xx x4)) (((eq a) Xt) x4))) (cF x4)))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or ((or (Xx x)) (((eq a) Xt) x))) (cF x))))
% Found x21:=(x2 x20):(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))
% Instantiate: f:=(fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))):(a->Prop)
% Found (x2 x20) as proof of (P f)
% Found (x2 x20) as proof of (P f)
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((or ((or (Xx x4)) (((eq a) Xt) x4))) (cF x4)))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((or ((or (Xx x4)) (((eq a) Xt) x4))) (cF x4)))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((or ((or (Xx x4)) (((eq a) Xt) x4))) (cF x4)))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((or ((or (Xx x4)) (((eq a) Xt) x4))) (cF x4)))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or ((or (Xx x)) (((eq a) Xt) x))) (cF x))))
% Found eq_ref00:=(eq_ref0 (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))):(((eq Prop) (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))) (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))))
% Found (eq_ref0 (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))):(((eq Prop) (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))) (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))))
% Found (eq_ref0 (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))) b)
% Found x5:(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))
% Instantiate: b:=(fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))):(a->Prop)
% Found x5 as proof of (P b)
% Found x5:(X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))
% Instantiate: b:=(fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))):(a->Prop)
% Found x5 as proof of (P b)
% Found x5:(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))
% Instantiate: b:=(fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))):(a->Prop)
% Found x5 as proof of (P b)
% Found x5:(X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))
% Instantiate: b:=(fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))):(a->Prop)
% Found x5 as proof of (P 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)) ((or (cE x5)) False))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or (cE x5)) False))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or (cE x5)) False))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((or (cE x5)) False))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (cE x)) False)))
% 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)) ((or False) (cF x5)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or False) (cF x5)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or False) (cF x5)))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((or False) (cF x5)))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or False) (cF 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)) ((or (cE x5)) False))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or (cE x5)) False))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or (cE x5)) False))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((or (cE x5)) False))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (cE x)) False)))
% 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)) ((or False) (cF x5)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or False) (cF x5)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or False) (cF x5)))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((or False) (cF x5)))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or False) (cF x))))
% Found eq_sym0:=(eq_sym Prop):(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a)))
% Instantiate: b:=(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))):Prop
% Found eq_sym0 as proof of b
% Found eq_sym0:=(eq_sym Prop):(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a)))
% Instantiate: b:=(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))):Prop
% Found eq_sym0 as proof of 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)) ((or False) (cF x5)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or False) (cF x5)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or False) (cF x5)))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((or False) (cF x5)))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or False) (cF 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)) ((or (cE x5)) False))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or (cE x5)) False))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or (cE x5)) False))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((or (cE x5)) False))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (cE x)) False)))
% 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)) ((or False) (cF x5)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or False) (cF x5)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or False) (cF x5)))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((or False) (cF x5)))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or False) (cF 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)) ((or (cE x5)) False))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or (cE x5)) False))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or (cE x5)) False))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((or (cE x5)) False))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (cE x)) False)))
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))
% Found (eq_ref0 (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) (fun (x:a)=> ((or ((or (Xx x)) (((eq a) Xt) x))) (cF x))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) b)
% Found ((eta_expansion_dep0 (fun (x8:a)=> Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) (fun (x:a)=> ((or ((or (Xx x)) (((eq a) Xt) x))) (cF x))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) b)
% Found ((eta_expansion_dep0 (fun (x8:a)=> Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) (fun (x:a)=> ((or (cE x)) ((or (Xx x)) (((eq a) Xt) x)))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) b)
% Found ((eta_expansion_dep0 (fun (x8:a)=> Prop)) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) b)
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) b)
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) b)
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx0:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))):(((eq Prop) (forall (Xx0:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) (forall (Xx0:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))
% Found (eq_ref0 (forall (Xx0:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx0:(a->Prop)), ((X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))):(((eq Prop) (forall (Xx0:(a->Prop)), ((X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))) (forall (Xx0:(a->Prop)), ((X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))
% Found (eq_ref0 (forall (Xx0:(a->Prop)), ((X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))) b)
% Found x5:(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))
% Instantiate: b:=(fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))):(a->Prop)
% Found x5 as proof of (P b)
% Found x5:(X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))
% Instantiate: b:=(fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))):(a->Prop)
% Found x5 as proof of (P b)
% Found eq_sym0:=(eq_sym Prop):(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a)))
% Instantiate: b:=(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))):Prop
% Found eq_sym0 as proof of b
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((or False) (cF x4)))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((or False) (cF x4)))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((or False) (cF x4)))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((or False) (cF x4)))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or False) (cF x))))
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((or False) (cF x4)))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((or False) (cF x4)))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((or False) (cF x4)))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((or False) (cF x4)))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or False) (cF 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)) ((or False) (cF x5)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or False) (cF x5)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or False) (cF x5)))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((or False) (cF x5)))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or False) (cF 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)) ((or False) (cF x5)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or False) (cF x5)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or False) (cF x5)))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((or False) (cF x5)))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or False) (cF 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)) ((or (cE x5)) False))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or (cE x5)) False))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or (cE x5)) False))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((or (cE x5)) False))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (cE x)) False)))
% 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)) ((or (cE x5)) False))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or (cE x5)) False))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or (cE x5)) False))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((or (cE x5)) False))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (cE x)) False)))
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) (fun (x:a)=> ((or (cE x)) ((or (Xx x)) (((eq a) Xt) x)))))
% Found (eta_expansion00 (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) b)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) (fun (x:a)=> ((or ((or (Xx x)) (((eq a) Xt) x))) (cF x))))
% Found (eta_expansion00 (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) b)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx0:(a->Prop)), ((X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))):(((eq Prop) (forall (Xx0:(a->Prop)), ((X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))) (forall (Xx0:(a->Prop)), ((X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))
% Found (eq_ref0 (forall (Xx0:(a->Prop)), ((X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))):(((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False))))))))
% Found (eq_ref0 (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))):(((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))
% Found (eq_ref0 (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))):(((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))
% Found (eq_ref0 (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))):(((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False))))))))
% Found (eq_ref0 (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))) b)
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((or False) (cF x4)))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((or False) (cF x4)))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((or False) (cF x4)))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((or False) (cF x4)))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or False) (cF x))))
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((or False) (cF x4)))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((or False) (cF x4)))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((or False) (cF x4)))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((or False) (cF x4)))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or False) (cF x))))
% Found eq_ref00:=(eq_ref0 (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))):(((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))
% Found (eq_ref0 (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))):(((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False))))))))
% Found (eq_ref0 (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (Xx Xz)) False)))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))):(((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))
% Found (eq_ref0 (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% 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 eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz))))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz))))
% Found eq_ref00:=(eq_ref0 (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))):(((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))
% Found (eq_ref0 (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or False) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or False) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))) b)
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or False) (cF Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) (fun (Xz:a)=> ((or False) (cF Xz))))
% Found (eq_ref0 (fun (Xz:a)=> ((or False) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) b)
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or (cE Xz)) False))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) (fun (Xz:a)=> ((or (cE Xz)) False)))
% Found (eq_ref0 (fun (Xz:a)=> ((or (cE Xz)) False))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (cE Xz)) False))) b)
% Found x400:(forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))
% Found x400 as proof of (forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))
% Found x20:(forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))
% Found x20 as proof of (forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))
% Found x200:(forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))
% Found x200 as proof of (forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or False) (cF Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) (fun (Xz:a)=> ((or False) (cF Xz))))
% Found (eq_ref0 (fun (Xz:a)=> ((or False) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or False) (cF Xz)))) b)
% Found x20:=(x2 (fun (x4:(a->Prop))=> (X (fun (Xz:a)=> ((or (cE Xz)) (x4 Xz)))))):(((and (X (fun (Xz:a)=> ((or (cE Xz)) False)))) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))->(X (fun (Xz:a)=> ((or (cE Xz)) (cF Xz)))))
% Instantiate: b:=(((and (X (fun (Xz:a)=> ((or (cE Xz)) False)))) (forall (Xx:(a->Prop)), ((X (fun (Xz:a)=> ((or (cE Xz)) (Xx Xz))))->(forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (cE Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))->(X (fun (Xz:a)=> ((or (cE Xz)) (cF Xz))))):Prop
% Found x20 as proof of b
% Found x20:=(x2 (fun (x5:(a->Prop))=> (forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (x5 Xz)))))))):(((and (forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz))))))))))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))))))
% Instantiate: b:=(((and (forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) False)))))) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz))))))))))))->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))))):Prop
% Found x20 as proof of b
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz))))
% Found (eq_ref0 (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) b)
% Found x41:=(x4 x40):(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))
% Found (x4 x40) as proof of (P b)
% Found (x4 x40) as proof of (P b)
% Found (x4 x40) as proof of (P b)
% Found x400:(forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))
% Found x400 as proof of (forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) (fun (x:a)=> ((or ((or (Xx x)) (((eq a) Xt) x))) (cF x))))
% Found (eta_expansion00 (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) b)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (cF Xz)))) b)
% Found x40:=(x4 x20):(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))
% Instantiate: b:=(fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))):(a->Prop)
% Found (x4 x20) as proof of (P b)
% Found (x4 x20) as proof of (P b)
% Found x20:(forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))
% Found x20 as proof of (forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))
% Found x200:(forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))
% Found x200 as proof of (forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))
% Found x400:(forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))
% Found x400 as proof of (forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))
% Found eq_ref00:=(eq_ref0 (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))):(((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz))))))))))))
% Found (eq_ref0 (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))):(((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz))))))))))))
% Found (eq_ref0 (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))):(((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz))))))))))))
% Found (eq_ref0 (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))):(((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz))))))))))))
% Found (eq_ref0 (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))) b)
% Found x20:(forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))
% Found x20 as proof of (forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))
% Found x200:(forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt0:a), ((cF Xt0)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))->(X00 cF)))
% Found x200 as proof of (forall (X00:((a->Prop)->Prop)), (((and (X00 (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X00 Xx0)->(forall (Xt:a), ((cF Xt)->(X00 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X00 cF)))
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b
% Found eq_ref00:=(eq_ref0 (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))):(((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz))))))))))))
% Found (eq_ref0 (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz))))))->(forall (Xt:a), ((cF Xt)->(forall (Xt0:a), ((cE Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt0) Xz))) ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))):(((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz))))))))))))
% Found (eq_ref0 (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xt:a), ((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))->(forall (Xt:a), ((cE Xt)->(forall (Xt0:a), ((cF Xt0)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt) Xz))) ((or (Xx Xz)) (((eq a) Xt0) Xz)))))))))))) 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) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz))))
% Found x41:=(x4 x40):(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))
% Instantiate: f:=(fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))):(a->Prop)
% Found (x4 x40) as proof of (P f)
% Found (x4 x40) as proof of (P f)
% Found eq_ref00:=(eq_ref0 (f x6)):(((eq Prop) (f x6)) (f x6))
% Found (eq_ref0 (f x6)) as proof of (((eq Prop) (f x6)) ((or ((or (Xx x6)) (((eq a) Xt) x6))) (cF x6)))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) ((or ((or (Xx x6)) (((eq a) Xt) x6))) (cF x6)))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) ((or ((or (Xx x6)) (((eq a) Xt) x6))) (cF x6)))
% Found (fun (x6:a)=> ((eq_ref Prop) (f x6))) as proof of (((eq Prop) (f x6)) ((or ((or (Xx x6)) (((eq a) Xt) x6))) (cF x6)))
% Found (fun (x6:a)=> ((eq_ref Prop) (f x6))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or ((or (Xx x)) (((eq a) Xt) x))) (cF x))))
% Found x41:=(x4 x40):(X (fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))))
% Instantiate: f:=(fun (Xz:a)=> ((or (Xx Xz)) (cF Xz))):(a->Prop)
% Found (x4 x40) as proof of (P f)
% Found (x4 x40) as proof of (P f)
% Found eq_ref00:=(eq_ref0 (f x6)):(((eq Prop) (f x6)) (f x6))
% Found (eq_ref0 (f x6)) as proof of (((eq Prop) (f x6)) ((or ((or (Xx x6)) (((eq a) Xt) x6))) (cF x6)))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) ((or ((or (Xx x6)) (((eq a) Xt) x6))) (cF x6)))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) ((or ((or (Xx x6)) (((eq a) Xt) x6))) (cF x6)))
% Found (fun (x6:a)=> ((eq_ref Prop) (f x6))) as proof of (((eq Prop) (f x6)) ((or ((or (Xx x6)) (((eq a) Xt) x6))) (cF x6)))
% Found (fun (x6:a)=> ((eq_ref Prop) (f x6))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or ((or (Xx x)) (((eq a) Xt) x))) (cF x))))
% 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 eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz))))
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz))))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz))))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz))))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (cE Xz)) (cF Xz))))
% Found eq_ref00:=(eq_ref0 (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))):(((eq Prop) (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))) (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))))
% Found (eq_ref0 (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))):(((eq Prop) (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))) (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))))
% Found (eq_ref0 (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))):(((eq Prop) (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))) (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))))
% Found (eq_ref0 (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))):(((eq Prop) (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))) (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))))
% Found (eq_ref0 (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))) 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)) ((or False) (cF x5)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or False) (cF x5)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or False) (cF x5)))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((or False) (cF x5)))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or False) (cF 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)) ((or (cE x5)) False))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or (cE x5)) False))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or (cE x5)) False))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((or (cE x5)) False))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (cE x)) False)))
% 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)) ((or (cE x5)) False))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or (cE x5)) False))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or (cE x5)) False))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((or (cE x5)) False))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (cE x)) False)))
% 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)) ((or False) (cF x5)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or False) (cF x5)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or False) (cF x5)))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((or False) (cF x5)))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or False) (cF x))))
% Found or_comm_i:=(fun (A:Prop) (B:Prop) (H:((or A) B))=> ((((((or_ind A) B) ((or B) A)) ((or_intror B) A)) ((or_introl B) A)) H)):(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A))):Prop
% Found or_comm_i as proof of b
% Found or_comm_i:=(fun (A:Prop) (B:Prop) (H:((or A) B))=> ((((((or_ind A) B) ((or B) A)) ((or_intror B) A)) ((or_introl B) A)) H)):(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A))):Prop
% Found or_comm_i as proof of b
% Found eq_ref00:=(eq_ref0 (f x7)):(((eq Prop) (f x7)) (f x7))
% Found (eq_ref0 (f x7)) as proof of (((eq Prop) (f x7)) ((or (cE x7)) ((or (Xx x7)) (((eq a) Xt) x7))))
% Found ((eq_ref Prop) (f x7)) as proof of (((eq Prop) (f x7)) ((or (cE x7)) ((or (Xx x7)) (((eq a) Xt) x7))))
% Found ((eq_ref Prop) (f x7)) as proof of (((eq Prop) (f x7)) ((or (cE x7)) ((or (Xx x7)) (((eq a) Xt) x7))))
% Found (fun (x7:a)=> ((eq_ref Prop) (f x7))) as proof of (((eq Prop) (f x7)) ((or (cE x7)) ((or (Xx x7)) (((eq a) Xt) x7))))
% Found (fun (x7:a)=> ((eq_ref Prop) (f x7))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (cE x)) ((or (Xx x)) (((eq a) Xt) x)))))
% Found eq_ref00:=(eq_ref0 (f x7)):(((eq Prop) (f x7)) (f x7))
% Found (eq_ref0 (f x7)) as proof of (((eq Prop) (f x7)) ((or ((or (Xx x7)) (((eq a) Xt) x7))) (cF x7)))
% Found ((eq_ref Prop) (f x7)) as proof of (((eq Prop) (f x7)) ((or ((or (Xx x7)) (((eq a) Xt) x7))) (cF x7)))
% Found ((eq_ref Prop) (f x7)) as proof of (((eq Prop) (f x7)) ((or ((or (Xx x7)) (((eq a) Xt) x7))) (cF x7)))
% Found (fun (x7:a)=> ((eq_ref Prop) (f x7))) as proof of (((eq Prop) (f x7)) ((or ((or (Xx x7)) (((eq a) Xt) x7))) (cF x7)))
% Found (fun (x7:a)=> ((eq_ref Prop) (f x7))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or ((or (Xx x)) (((eq a) Xt) x))) (cF x))))
% Found eq_ref00:=(eq_ref0 (f x7)):(((eq Prop) (f x7)) (f x7))
% Found (eq_ref0 (f x7)) as proof of (((eq Prop) (f x7)) ((or ((or (Xx x7)) (((eq a) Xt) x7))) (cF x7)))
% Found ((eq_ref Prop) (f x7)) as proof of (((eq Prop) (f x7)) ((or ((or (Xx x7)) (((eq a) Xt) x7))) (cF x7)))
% Found ((eq_ref Prop) (f x7)) as proof of (((eq Prop) (f x7)) ((or ((or (Xx x7)) (((eq a) Xt) x7))) (cF x7)))
% Found (fun (x7:a)=> ((eq_ref Prop) (f x7))) as proof of (((eq Prop) (f x7)) ((or ((or (Xx x7)) (((eq a) Xt) x7))) (cF x7)))
% Found (fun (x7:a)=> ((eq_ref Prop) (f x7))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or ((or (Xx x)) (((eq a) Xt) x))) (cF x))))
% Found eq_ref00:=(eq_ref0 (f x7)):(((eq Prop) (f x7)) (f x7))
% Found (eq_ref0 (f x7)) as proof of (((eq Prop) (f x7)) ((or (cE x7)) ((or (Xx x7)) (((eq a) Xt) x7))))
% Found ((eq_ref Prop) (f x7)) as proof of (((eq Prop) (f x7)) ((or (cE x7)) ((or (Xx x7)) (((eq a) Xt) x7))))
% Found ((eq_ref Prop) (f x7)) as proof of (((eq Prop) (f x7)) ((or (cE x7)) ((or (Xx x7)) (((eq a) Xt) x7))))
% Found (fun (x7:a)=> ((eq_ref Prop) (f x7))) as proof of (((eq Prop) (f x7)) ((or (cE x7)) ((or (Xx x7)) (((eq a) Xt) x7))))
% Found (fun (x7:a)=> ((eq_ref Prop) (f x7))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (cE x)) ((or (Xx x)) (((eq a) Xt) x)))))
% Found eq_ref00:=(eq_ref0 (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))):(((eq Prop) (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))) (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz)))))))))))
% Found (eq_ref0 (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), (((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) (Xx0 Xz)))))->(forall (Xt0:a), ((cF Xt0)->((cE Xt)->(X (fun (Xz:a)=> ((or ((or (Xx Xz)) (((eq a) Xt) Xz))) ((or (Xx0 Xz)) (((eq a) Xt0) Xz))))))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))):(((eq Prop) (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))) (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))))
% Found (eq_ref0 (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), (((cF Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) ((or (Xx Xz)) (((eq a) Xt) Xz))))))->(forall (Xt0:a), ((cE Xt0)->((cF Xt)->(X (fun (Xz:a)=> ((or ((or (Xx0 Xz)) (((eq a) Xt0) Xz))) ((or (Xx Xz)) (((eq a) Xt) Xz))))))))))) b)
% Found eq_ref00:=(eq_ref0 (f x7)):(((eq Prop) (f x7)) (f x7))
% Found (eq_ref0 (f x7)) as proof of (((eq Prop) (f x7)) ((or (cE x7)) ((or (Xx x7)) (((eq a) Xt) x7))))
% Found ((eq_ref Prop) (f x7)) as proof of (((eq Prop) (f x7)) ((or (cE x7)) ((or (Xx x7)) (((eq a) Xt) x7))))
% Found ((eq_ref Prop) (f x7)) as proof of (((eq Prop) (f x7)) ((or (cE x7)) ((or (Xx x7)) (((eq a) Xt) x7))))
% Found (fun (x7:a)=> ((eq_ref Prop) (f x7))) as proof of (((eq Prop) (f x7)) ((or (cE x7)) ((or (Xx x7)) (((eq a) Xt) x7))))
% Found (fun (x7:a)=> ((eq_ref Prop) (f x7))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (cE x)) ((or (Xx x)) 
% EOF
%------------------------------------------------------------------------------