%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV183^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n097.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 13:33:51 EDT 2014

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

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV183^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n097.star.cs.uiowa.edu
% % Model    : x86_64 x86_64
% % CPU      : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory   : 32286.75MB
% % OS       : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 08:23:26 CDT 2014
% % CPUTime  : 300.03 
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x275c440>, <kernel.Type object at 0x275c128>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x275cc68>, <kernel.Type object at 0x29b5200>) of role type named b_type
% Using role type
% Declaring b:Type
% FOF formula (<kernel.Constant object at 0x273c560>, <kernel.DependentProduct object at 0x275bf80>) of role type named f
% Using role type
% Declaring f:((a->Prop)->(b->Prop))
% FOF formula (<kernel.Constant object at 0x275c128>, <kernel.DependentProduct object at 0x275b0e0>) of role type named cA
% Using role type
% Declaring cA:((a->Prop)->Prop)
% FOF formula (<kernel.Constant object at 0x275cc68>, <kernel.DependentProduct object at 0x275bf80>) of role type named cB
% Using role type
% Declaring cB:((a->Prop)->Prop)
% FOF formula (<kernel.Constant object at 0x275c440>, <kernel.DependentProduct object at 0x275b560>) of role type named cC
% Using role type
% Declaring cC:((b->Prop)->Prop)
% FOF formula (((and ((and ((and ((and (forall (Xx:(a->Prop)), ((cA Xx)->(cB Xx)))) (forall (X:((b->Prop)->Prop)), ((forall (Xx:(b->Prop)), ((X Xx)->(cC Xx)))->(cC (fun (Xx:b)=> (forall (S:(b->Prop)), ((X S)->(S Xx))))))))) (forall (D:((b->Prop)->Prop)), (((and ((and (forall (Xx:(b->Prop)), ((D Xx)->(cC Xx)))) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> (D Xy))))) (forall (Xy:(b->Prop)) (Xz:(b->Prop)), ((ex (b->Prop)) (fun (Xw:(b->Prop))=> ((and (forall (Xx:b), ((Xy Xx)->(Xw Xx)))) (forall (Xx:b), ((Xz Xx)->(Xw Xx))))))))->(cC (fun (Xx:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (D S)) (S Xx)))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(cC (f Xx)))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:b), ((Xe Xx0)->((f Xy) Xx0)))))))))))))))->((ex ((a->Prop)->(b->Prop))) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx)))))))) of role conjecture named cDOMTHM7_pme
% Conjecture to prove = (((and ((and ((and ((and (forall (Xx:(a->Prop)), ((cA Xx)->(cB Xx)))) (forall (X:((b->Prop)->Prop)), ((forall (Xx:(b->Prop)), ((X Xx)->(cC Xx)))->(cC (fun (Xx:b)=> (forall (S:(b->Prop)), ((X S)->(S Xx))))))))) (forall (D:((b->Prop)->Prop)), (((and ((and (forall (Xx:(b->Prop)), ((D Xx)->(cC Xx)))) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> (D Xy))))) (forall (Xy:(b->Prop)) (Xz:(b->Prop)), ((ex (b->Prop)) (fun (Xw:(b->Prop))=> ((and (forall (Xx:b), ((Xy Xx)->(Xw Xx)))) (forall (Xx:b), ((Xz Xx)->(Xw Xx))))))))->(cC (fun (Xx:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (D S)) (S Xx)))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(cC (f Xx)))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:b), ((Xe Xx0)->((f Xy) Xx0)))))))))))))))->((ex ((a->Prop)->(b->Prop))) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx)))))))):Prop
% Parameter a_DUMMY:a.
% Parameter b_DUMMY:b.
% We need to prove ['(((and ((and ((and ((and (forall (Xx:(a->Prop)), ((cA Xx)->(cB Xx)))) (forall (X:((b->Prop)->Prop)), ((forall (Xx:(b->Prop)), ((X Xx)->(cC Xx)))->(cC (fun (Xx:b)=> (forall (S:(b->Prop)), ((X S)->(S Xx))))))))) (forall (D:((b->Prop)->Prop)), (((and ((and (forall (Xx:(b->Prop)), ((D Xx)->(cC Xx)))) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> (D Xy))))) (forall (Xy:(b->Prop)) (Xz:(b->Prop)), ((ex (b->Prop)) (fun (Xw:(b->Prop))=> ((and (forall (Xx:b), ((Xy Xx)->(Xw Xx)))) (forall (Xx:b), ((Xz Xx)->(Xw Xx))))))))->(cC (fun (Xx:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (D S)) (S Xx)))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(cC (f Xx)))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:b), ((Xe Xx0)->((f Xy) Xx0)))))))))))))))->((ex ((a->Prop)->(b->Prop))) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))))']
% Parameter a:Type.
% Parameter b:Type.
% Parameter f:((a->Prop)->(b->Prop)).
% Parameter cA:((a->Prop)->Prop).
% Parameter cB:((a->Prop)->Prop).
% Parameter cC:((b->Prop)->Prop).
% Trying to prove (((and ((and ((and ((and (forall (Xx:(a->Prop)), ((cA Xx)->(cB Xx)))) (forall (X:((b->Prop)->Prop)), ((forall (Xx:(b->Prop)), ((X Xx)->(cC Xx)))->(cC (fun (Xx:b)=> (forall (S:(b->Prop)), ((X S)->(S Xx))))))))) (forall (D:((b->Prop)->Prop)), (((and ((and (forall (Xx:(b->Prop)), ((D Xx)->(cC Xx)))) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> (D Xy))))) (forall (Xy:(b->Prop)) (Xz:(b->Prop)), ((ex (b->Prop)) (fun (Xw:(b->Prop))=> ((and (forall (Xx:b), ((Xy Xx)->(Xw Xx)))) (forall (Xx:b), ((Xz Xx)->(Xw Xx))))))))->(cC (fun (Xx:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (D S)) (S Xx)))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(cC (f Xx)))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:b), ((Xe Xx0)->((f Xy) Xx0)))))))))))))))->((ex ((a->Prop)->(b->Prop))) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))))
% Found eq_ref00:=(eq_ref0 (f Xx)):(((eq (b->Prop)) (f Xx)) (f Xx))
% Found (eq_ref0 (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x0 Xx))
% Found ((eq_ref (b->Prop)) (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x0 Xx))
% Found ((eq_ref (b->Prop)) (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x0 Xx))
% Found (fun (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of (((eq (b->Prop)) (f Xx)) (x0 Xx))
% Found (fun (Xx:(a->Prop)) (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of ((cA Xx)->(((eq (b->Prop)) (f Xx)) (x0 Xx)))
% Found (fun (Xx:(a->Prop)) (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (x0 Xx))))
% Found eq_ref00:=(eq_ref0 (f Xx)):(((eq (b->Prop)) (f Xx)) (f Xx))
% Found (eq_ref0 (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x2 Xx))
% Found ((eq_ref (b->Prop)) (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x2 Xx))
% Found ((eq_ref (b->Prop)) (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x2 Xx))
% Found (fun (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of (((eq (b->Prop)) (f Xx)) (x2 Xx))
% Found (fun (Xx:(a->Prop)) (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of ((cA Xx)->(((eq (b->Prop)) (f Xx)) (x2 Xx)))
% Found (fun (Xx:(a->Prop)) (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (x2 Xx))))
% Found eq_ref00:=(eq_ref0 (f Xx)):(((eq (b->Prop)) (f Xx)) (f Xx))
% Found (eq_ref0 (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x0 Xx))
% Found ((eq_ref (b->Prop)) (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x0 Xx))
% Found ((eq_ref (b->Prop)) (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x0 Xx))
% Found (fun (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of (((eq (b->Prop)) (f Xx)) (x0 Xx))
% Found (fun (Xx:(a->Prop)) (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of ((cA Xx)->(((eq (b->Prop)) (f Xx)) (x0 Xx)))
% Found (fun (Xx:(a->Prop)) (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (x0 Xx))))
% Found eq_ref00:=(eq_ref0 (f Xx)):(((eq (b->Prop)) (f Xx)) (f Xx))
% Found (eq_ref0 (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x4 Xx))
% Found ((eq_ref (b->Prop)) (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x4 Xx))
% Found ((eq_ref (b->Prop)) (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x4 Xx))
% Found (fun (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of (((eq (b->Prop)) (f Xx)) (x4 Xx))
% Found (fun (Xx:(a->Prop)) (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of ((cA Xx)->(((eq (b->Prop)) (f Xx)) (x4 Xx)))
% Found (fun (Xx:(a->Prop)) (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (x4 Xx))))
% Found eq_ref00:=(eq_ref0 (f Xx)):(((eq (b->Prop)) (f Xx)) (f Xx))
% Found (eq_ref0 (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x0 Xx))
% Found ((eq_ref (b->Prop)) (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x0 Xx))
% Found ((eq_ref (b->Prop)) (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x0 Xx))
% Found (fun (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of (((eq (b->Prop)) (f Xx)) (x0 Xx))
% Found (fun (Xx:(a->Prop)) (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of ((cA Xx)->(((eq (b->Prop)) (f Xx)) (x0 Xx)))
% Found (fun (Xx:(a->Prop)) (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (x0 Xx))))
% Found eq_ref00:=(eq_ref0 (f Xx)):(((eq (b->Prop)) (f Xx)) (f Xx))
% Found (eq_ref0 (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x2 Xx))
% Found ((eq_ref (b->Prop)) (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x2 Xx))
% Found ((eq_ref (b->Prop)) (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x2 Xx))
% Found (fun (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of (((eq (b->Prop)) (f Xx)) (x2 Xx))
% Found (fun (Xx:(a->Prop)) (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of ((cA Xx)->(((eq (b->Prop)) (f Xx)) (x2 Xx)))
% Found (fun (Xx:(a->Prop)) (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (x2 Xx))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))):(((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) (fun (x:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (x Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((x Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((x Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((x Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (x Xx)))))))
% Found (eta_expansion00 (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) b0)
% Found (((eta_expansion ((a->Prop)->(b->Prop))) Prop) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) b0)
% Found (((eta_expansion ((a->Prop)->(b->Prop))) Prop) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) b0)
% Found (((eta_expansion ((a->Prop)->(b->Prop))) Prop) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))):(((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) (fun (x:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (x Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((x Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((x Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((x Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (x Xx)))))))
% Found (eta_expansion00 (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) b0)
% Found (((eta_expansion ((a->Prop)->(b->Prop))) Prop) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) b0)
% Found (((eta_expansion ((a->Prop)->(b->Prop))) Prop) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) b0)
% Found (((eta_expansion ((a->Prop)->(b->Prop))) Prop) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) b0)
% Found eq_ref00:=(eq_ref0 (f Xx)):(((eq (b->Prop)) (f Xx)) (f Xx))
% Found (eq_ref0 (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x6 Xx))
% Found ((eq_ref (b->Prop)) (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x6 Xx))
% Found ((eq_ref (b->Prop)) (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x6 Xx))
% Found (fun (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of (((eq (b->Prop)) (f Xx)) (x6 Xx))
% Found (fun (Xx:(a->Prop)) (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of ((cA Xx)->(((eq (b->Prop)) (f Xx)) (x6 Xx)))
% Found (fun (Xx:(a->Prop)) (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (x6 Xx))))
% Found eq_ref00:=(eq_ref0 (f Xx)):(((eq (b->Prop)) (f Xx)) (f Xx))
% Found (eq_ref0 (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x0 Xx))
% Found ((eq_ref (b->Prop)) (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x0 Xx))
% Found ((eq_ref (b->Prop)) (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x0 Xx))
% Found (fun (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of (((eq (b->Prop)) (f Xx)) (x0 Xx))
% Found (fun (Xx:(a->Prop)) (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of ((cA Xx)->(((eq (b->Prop)) (f Xx)) (x0 Xx)))
% Found (fun (Xx:(a->Prop)) (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (x0 Xx))))
% Found eq_ref00:=(eq_ref0 (f Xx)):(((eq (b->Prop)) (f Xx)) (f Xx))
% Found (eq_ref0 (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x2 Xx))
% Found ((eq_ref (b->Prop)) (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x2 Xx))
% Found ((eq_ref (b->Prop)) (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x2 Xx))
% Found (fun (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of (((eq (b->Prop)) (f Xx)) (x2 Xx))
% Found (fun (Xx:(a->Prop)) (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of ((cA Xx)->(((eq (b->Prop)) (f Xx)) (x2 Xx)))
% Found (fun (Xx:(a->Prop)) (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (x2 Xx))))
% Found eq_ref00:=(eq_ref0 (f Xx)):(((eq (b->Prop)) (f Xx)) (f Xx))
% Found (eq_ref0 (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x4 Xx))
% Found ((eq_ref (b->Prop)) (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x4 Xx))
% Found ((eq_ref (b->Prop)) (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x4 Xx))
% Found (fun (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of (((eq (b->Prop)) (f Xx)) (x4 Xx))
% Found (fun (Xx:(a->Prop)) (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of ((cA Xx)->(((eq (b->Prop)) (f Xx)) (x4 Xx)))
% Found (fun (Xx:(a->Prop)) (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (x4 Xx))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))):(((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) (fun (x:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (x Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((x Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((x Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((x Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (x Xx)))))))
% Found (eta_expansion00 (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) b0)
% Found (((eta_expansion ((a->Prop)->(b->Prop))) Prop) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) b0)
% Found (((eta_expansion ((a->Prop)->(b->Prop))) Prop) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) b0)
% Found (((eta_expansion ((a->Prop)->(b->Prop))) Prop) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) b0)
% Found eq_ref00:=(eq_ref0 (f Xx)):(((eq (b->Prop)) (f Xx)) (f Xx))
% Found (eq_ref0 (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x8 Xx))
% Found ((eq_ref (b->Prop)) (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x8 Xx))
% Found ((eq_ref (b->Prop)) (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x8 Xx))
% Found (fun (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of (((eq (b->Prop)) (f Xx)) (x8 Xx))
% Found (fun (Xx:(a->Prop)) (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of ((cA Xx)->(((eq (b->Prop)) (f Xx)) (x8 Xx)))
% Found (fun (Xx:(a->Prop)) (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (x8 Xx))))
% Found eq_ref00:=(eq_ref0 (f Xx)):(((eq (b->Prop)) (f Xx)) (f Xx))
% Found (eq_ref0 (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x0 Xx))
% Found ((eq_ref (b->Prop)) (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x0 Xx))
% Found ((eq_ref (b->Prop)) (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x0 Xx))
% Found (fun (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of (((eq (b->Prop)) (f Xx)) (x0 Xx))
% Found (fun (Xx:(a->Prop)) (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of ((cA Xx)->(((eq (b->Prop)) (f Xx)) (x0 Xx)))
% Found (fun (Xx:(a->Prop)) (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (x0 Xx))))
% Found eq_ref00:=(eq_ref0 (f Xx)):(((eq (b->Prop)) (f Xx)) (f Xx))
% Found (eq_ref0 (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x2 Xx))
% Found ((eq_ref (b->Prop)) (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x2 Xx))
% Found ((eq_ref (b->Prop)) (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x2 Xx))
% Found (fun (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of (((eq (b->Prop)) (f Xx)) (x2 Xx))
% Found (fun (Xx:(a->Prop)) (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of ((cA Xx)->(((eq (b->Prop)) (f Xx)) (x2 Xx)))
% Found (fun (Xx:(a->Prop)) (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (x2 Xx))))
% Found eq_ref00:=(eq_ref0 (f Xx)):(((eq (b->Prop)) (f Xx)) (f Xx))
% Found (eq_ref0 (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x4 Xx))
% Found ((eq_ref (b->Prop)) (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x4 Xx))
% Found ((eq_ref (b->Prop)) (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x4 Xx))
% Found (fun (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of (((eq (b->Prop)) (f Xx)) (x4 Xx))
% Found (fun (Xx:(a->Prop)) (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of ((cA Xx)->(((eq (b->Prop)) (f Xx)) (x4 Xx)))
% Found (fun (Xx:(a->Prop)) (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (x4 Xx))))
% Found eq_ref00:=(eq_ref0 (f Xx)):(((eq (b->Prop)) (f Xx)) (f Xx))
% Found (eq_ref0 (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x6 Xx))
% Found ((eq_ref (b->Prop)) (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x6 Xx))
% Found ((eq_ref (b->Prop)) (f Xx)) as proof of (((eq (b->Prop)) (f Xx)) (x6 Xx))
% Found (fun (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of (((eq (b->Prop)) (f Xx)) (x6 Xx))
% Found (fun (Xx:(a->Prop)) (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of ((cA Xx)->(((eq (b->Prop)) (f Xx)) (x6 Xx)))
% Found (fun (Xx:(a->Prop)) (x00:(cA Xx))=> ((eq_ref (b->Prop)) (f Xx))) as proof of (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (x6 Xx))))
% Found eq_ref00:=(eq_ref0 (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))):(((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx)))))))
% Found (eq_ref0 (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) b0)
% Found ((eq_ref (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) b0)
% Found ((eq_ref (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) b0)
% Found ((eq_ref (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))):(((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) (fun (x:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (x Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((x Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((x Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((x Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (x Xx)))))))
% Found (eta_expansion_dep00 (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) b0)
% Found ((eta_expansion_dep0 (fun (x9:((a->Prop)->(b->Prop)))=> Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) b0)
% Found (((eta_expansion_dep ((a->Prop)->(b->Prop))) (fun (x9:((a->Prop)->(b->Prop)))=> Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) b0)
% Found (((eta_expansion_dep ((a->Prop)->(b->Prop))) (fun (x9:((a->Prop)->(b->Prop)))=> Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(a->Prop)), ((cA Xx)->(((eq (b->Prop)) (f Xx)) (Xg Xx))))))) b0)
% Found (((eta_expansion_dep ((a->Prop)->(b->Prop))) (fun (x9:((a->Prop)->(b->Prop)))=> Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((and ((and (forall (Xx:(a->Prop)), ((cB Xx)->(cC (Xg Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->(cB Xx)))) (forall (Xx:(a->Prop)), (((and (cB Xx)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:b), ((Xe Xx0)->((Xg Xy) Xx0)))))))))))))))) (forall (Xx:(
% EOF
%------------------------------------------------------------------------------