%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV037^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n092.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 13:33:37 EDT 2014

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

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV037^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n092.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 07:39:01 CDT 2014
% % CPUTime  : 300.01 
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x23ee7e8>, <kernel.Type object at 0x23eeea8>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x22b3e60>, <kernel.Type object at 0x23eeb00>) of role type named b_type
% Using role type
% Declaring b:Type
% FOF formula (forall (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop))) (Xr:(a->(b->(b->Prop)))), (((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq))->((forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((and ((and (forall (Xx0:b) (Xy0:b), ((((Xr Xx) Xx0) Xy0)->(((Xr Xx) Xy0) Xx0)))) (forall (Xx0:b) (Xy0:b) (Xz:b), (((and (((Xr Xx) Xx0) Xy0)) (((Xr Xx) Xy0) Xz))->(((Xr Xx) Xx0) Xz))))) (((eq (b->(b->Prop))) (Xr Xx)) (Xr Xy)))))->((and ((and (forall (Xx:(a->b)) (Xy:(a->b)), ((forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xy Xy0))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xy Xx0)) (Xx Xy0))))))) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xz Xy0)))))))) (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))))))) of role conjecture named cTHM516_pme
% Conjecture to prove = (forall (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop))) (Xr:(a->(b->(b->Prop)))), (((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq))->((forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((and ((and (forall (Xx0:b) (Xy0:b), ((((Xr Xx) Xx0) Xy0)->(((Xr Xx) Xy0) Xx0)))) (forall (Xx0:b) (Xy0:b) (Xz:b), (((and (((Xr Xx) Xx0) Xy0)) (((Xr Xx) Xy0) Xz))->(((Xr Xx) Xx0) Xz))))) (((eq (b->(b->Prop))) (Xr Xx)) (Xr Xy)))))->((and ((and (forall (Xx:(a->b)) (Xy:(a->b)), ((forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xy Xy0))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xy Xx0)) (Xx Xy0))))))) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xz Xy0)))))))) (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))))))):Prop
% Parameter a_DUMMY:a.
% Parameter b_DUMMY:b.
% We need to prove ['(forall (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop))) (Xr:(a->(b->(b->Prop)))), (((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq))->((forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((and ((and (forall (Xx0:b) (Xy0:b), ((((Xr Xx) Xx0) Xy0)->(((Xr Xx) Xy0) Xx0)))) (forall (Xx0:b) (Xy0:b) (Xz:b), (((and (((Xr Xx) Xx0) Xy0)) (((Xr Xx) Xy0) Xz))->(((Xr Xx) Xx0) Xz))))) (((eq (b->(b->Prop))) (Xr Xx)) (Xr Xy)))))->((and ((and (forall (Xx:(a->b)) (Xy:(a->b)), ((forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xy Xy0))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xy Xx0)) (Xx Xy0))))))) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xz Xy0)))))))) (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))))))']
% Parameter a:Type.
% Parameter b:Type.
% Trying to prove (forall (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop))) (Xr:(a->(b->(b->Prop)))), (((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq))->((forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((and ((and (forall (Xx0:b) (Xy0:b), ((((Xr Xx) Xx0) Xy0)->(((Xr Xx) Xy0) Xx0)))) (forall (Xx0:b) (Xy0:b) (Xz:b), (((and (((Xr Xx) Xx0) Xy0)) (((Xr Xx) Xy0) Xz))->(((Xr Xx) Xx0) Xz))))) (((eq (b->(b->Prop))) (Xr Xx)) (Xr Xy)))))->((and ((and (forall (Xx:(a->b)) (Xy:(a->b)), ((forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xy Xy0))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xy Xx0)) (Xx Xy0))))))) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xz Xy0)))))))) (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq ((a->b)->((a->b)->Prop))) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))):(((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (x:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x Xx)) (Xg Xy))))))
% Found (eta_expansion00 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found ((eta_expansion0 ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found (((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found (((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found (((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))->(P (fun (x:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x Xx)) (Xg Xy)))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eta_expansion00 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eta_expansion0 ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))->(P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))->(P (fun (x:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x Xx)) (Xg Xy)))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eta_expansion00 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eta_expansion0 ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))->(P (fun (x:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x Xx)) (Xg Xy)))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eta_expansion00 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eta_expansion0 ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq ((a->b)->((a->b)->Prop))) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eq_ref00:=(eq_ref0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))):(((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (eq_ref0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found ((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found ((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found ((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found eq_ref00:=(eq_ref0 (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))):(((eq Prop) (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))) (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))))
% Found (eq_ref0 (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))) as proof of (((eq Prop) (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))) b0)
% Found ((eq_ref Prop) (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))) as proof of (((eq Prop) (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))) b0)
% Found ((eq_ref Prop) (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))) as proof of (((eq Prop) (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))) b0)
% Found ((eq_ref Prop) (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))) as proof of (((eq Prop) (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))) b0)
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))->(P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))->(P (fun (x:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x Xx)) (Xg Xy)))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eta_expansion_dep00 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eta_expansion_dep0 (fun (x4:(a->b))=> ((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((((eta_expansion_dep (a->b)) (fun (x4:(a->b))=> ((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((((eta_expansion_dep (a->b)) (fun (x4:(a->b))=> ((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))->(P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b0
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))->(P (fun (x:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x Xx)) (Xg Xy)))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eta_expansion_dep00 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eta_expansion_dep0 (fun (x4:(a->b))=> ((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((((eta_expansion_dep (a->b)) (fun (x4:(a->b))=> ((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((((eta_expansion_dep (a->b)) (fun (x4:(a->b))=> ((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eq_ref00:=(eq_ref0 (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))):(((eq Prop) (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))) (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))))
% Found (eq_ref0 (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))) as proof of (((eq Prop) (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))) b0)
% Found ((eq_ref Prop) (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))) as proof of (((eq Prop) (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))) b0)
% Found ((eq_ref Prop) (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))) as proof of (((eq Prop) (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))) b0)
% Found ((eq_ref Prop) (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))) as proof of (((eq Prop) (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq ((a->b)->((a->b)->Prop))) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))):(((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (x:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x Xx)) (Xg Xy))))))
% Found (eta_expansion00 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found ((eta_expansion0 ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found (((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found (((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found (((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq ((a->b)->((a->b)->Prop))) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))):(((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (x:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x Xx)) (Xg Xy))))))
% Found (eta_expansion00 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found ((eta_expansion0 ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found (((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found (((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found (((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))->(P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq ((a->b)->((a->b)->Prop))) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eq_ref00:=(eq_ref0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))):(((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (eq_ref0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found ((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found ((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found ((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))->(P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b0
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))->(P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))->(P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))->(P (fun (x:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x Xx)) (Xg Xy)))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eta_expansion_dep00 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eta_expansion_dep0 (fun (x6:(a->b))=> ((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((((eta_expansion_dep (a->b)) (fun (x6:(a->b))=> ((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((((eta_expansion_dep (a->b)) (fun (x6:(a->b))=> ((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))->(P (fun (x:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x Xx)) (Xg Xy)))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eta_expansion_dep00 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eta_expansion_dep0 (fun (x6:(a->b))=> ((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((((eta_expansion_dep (a->b)) (fun (x6:(a->b))=> ((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((((eta_expansion_dep (a->b)) (fun (x6:(a->b))=> ((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))->(P (fun (x:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x Xx)) (Xg Xy)))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eta_expansion_dep00 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eta_expansion_dep0 (fun (x4:(a->b))=> ((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((((eta_expansion_dep (a->b)) (fun (x4:(a->b))=> ((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((((eta_expansion_dep (a->b)) (fun (x4:(a->b))=> ((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))->(P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))):(((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) (fun (x:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (x Xy))))))
% Found (eta_expansion_dep00 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) b0)
% Found ((eta_expansion_dep0 (fun (x3:(a->b))=> Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) b0)
% Found (((eta_expansion_dep (a->b)) (fun (x3:(a->b))=> Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) b0)
% Found (((eta_expansion_dep (a->b)) (fun (x3:(a->b))=> Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) b0)
% Found (((eta_expansion_dep (a->b)) (fun (x3:(a->b))=> Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq ((a->b)->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq ((a->b)->Prop)) b0) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->Prop)) b0) as proof of (((eq ((a->b)->Prop)) b0) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->Prop)) b0) as proof of (((eq ((a->b)->Prop)) b0) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->Prop)) b0) as proof of (((eq ((a->b)->Prop)) b0) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found eq_ref00:=(eq_ref0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))):(((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found (eq_ref0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) b0)
% Found ((eq_ref ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) b0)
% Found ((eq_ref ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) b0)
% Found ((eq_ref ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq ((a->b)->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq ((a->b)->Prop)) b0) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->Prop)) b0) as proof of (((eq ((a->b)->Prop)) b0) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->Prop)) b0) as proof of (((eq ((a->b)->Prop)) b0) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->Prop)) b0) as proof of (((eq ((a->b)->Prop)) b0) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found eq_ref00:=(eq_ref0 (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))):(((eq Prop) (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))) (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))))
% Found (eq_ref0 (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))) as proof of (((eq Prop) (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))) b0)
% Found ((eq_ref Prop) (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))) as proof of (((eq Prop) (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))) b0)
% Found ((eq_ref Prop) (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))) as proof of (((eq Prop) (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))) b0)
% Found ((eq_ref Prop) (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))) as proof of (((eq Prop) (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))) b0)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))->(P (fun (x:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (x Xy)))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found ((eta_expansion_dep00 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found (((eta_expansion_dep0 (fun (x3:(a->b))=> Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found ((((eta_expansion_dep (a->b)) (fun (x3:(a->b))=> Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found ((((eta_expansion_dep (a->b)) (fun (x3:(a->b))=> Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))->(P (fun (x:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (x Xy)))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found ((eta_expansion_dep00 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found (((eta_expansion_dep0 (fun (x3:(a->b))=> Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found ((((eta_expansion_dep (a->b)) (fun (x3:(a->b))=> Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found ((((eta_expansion_dep (a->b)) (fun (x3:(a->b))=> Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))->(P (fun (x:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (x Xy)))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found ((eta_expansion_dep00 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found (((eta_expansion_dep0 (fun (x3:(a->b))=> Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found ((((eta_expansion_dep (a->b)) (fun (x3:(a->b))=> Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found ((((eta_expansion_dep (a->b)) (fun (x3:(a->b))=> Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))->(P (fun (x:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (x Xy)))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found ((eta_expansion_dep00 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found (((eta_expansion_dep0 (fun (x3:(a->b))=> Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found ((((eta_expansion_dep (a->b)) (fun (x3:(a->b))=> Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found ((((eta_expansion_dep (a->b)) (fun (x3:(a->b))=> Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))->(P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b0
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))->(P (fun (x:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x Xx)) (Xg Xy)))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eta_expansion00 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eta_expansion0 ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))->(P (fun (x:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x Xx)) (Xg Xy)))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eta_expansion_dep00 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eta_expansion_dep0 (fun (x4:(a->b))=> ((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((((eta_expansion_dep (a->b)) (fun (x4:(a->b))=> ((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((((eta_expansion_dep (a->b)) (fun (x4:(a->b))=> ((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq ((a->b)->((a->b)->Prop))) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eq_ref00:=(eq_ref0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))):(((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (eq_ref0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found ((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found ((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found ((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))->(P (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found ((eq_ref0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found (((eq_ref ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found (((eq_ref ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))->(P (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found ((eq_ref0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found (((eq_ref ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found (((eq_ref ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq ((a->b)->((a->b)->Prop))) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))):(((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (x:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x Xx)) (Xg Xy))))))
% Found (eta_expansion00 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found ((eta_expansion0 ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found (((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found (((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found (((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (y Xy))))):(((eq Prop) (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (y Xy))))) (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (y Xy)))))
% Found (eq_ref0 (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (y Xy))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (y Xy))))) b0)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (y Xy))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (y Xy))))) b0)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (y Xy))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (y Xy))))) b0)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (y Xy))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (y Xy))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (y Xy)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (y Xy)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (y Xy)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (y Xy)))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))->(P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eq_ref000:=(eq_ref00 P):((P (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (y Xy)))))->(P (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (y Xy))))))
% Found (eq_ref00 P) as proof of (P0 (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (y Xy)))))
% Found ((eq_ref0 (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (y Xy))))) P) as proof of (P0 (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (y Xy)))))
% Found (((eq_ref Prop) (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (y Xy))))) P) as proof of (P0 (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (y Xy)))))
% Found (((eq_ref Prop) (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (y Xy))))) P) as proof of (P0 (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (y Xy)))))
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b0:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b0
% Found eq_ref00:=(eq_ref0 (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xz Xy0))))))):(((eq Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xz Xy0))))))) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xz Xy0)))))))
% Found (eq_ref0 (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xz Xy0))))))) as proof of (((eq Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xz Xy0))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xz Xy0))))))) as proof of (((eq Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xz Xy0))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xz Xy0))))))) as proof of (((eq Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xz Xy0))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xz Xy0))))))) as proof of (((eq Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xz Xy0))))))) b0)
% Found x20:=(x2 (fun (x3:(a->(a->Prop)))=> (P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))))):((P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))->(P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))))
% Found (x2 (fun (x3:(a->(a->Prop)))=> (P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))))) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (x2 (fun (x3:(a->(a->Prop)))=> (P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))))) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))->(P (fun (x:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x Xx)) (Xg Xy)))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eta_expansion_dep00 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eta_expansion_dep0 (fun (x4:(a->b))=> ((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((((eta_expansion_dep (a->b)) (fun (x4:(a->b))=> ((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((((eta_expansion_dep (a->b)) (fun (x4:(a->b))=> ((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found x1:(P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Instantiate: b0:=(fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))):((a->b)->((a->b)->Prop))
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))):(((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (eq_ref0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found ((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found ((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found ((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found eq_ref00:=(eq_ref0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))):(((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found (eq_ref0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) b0)
% Found ((eq_ref ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) b0)
% Found ((eq_ref ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) b0)
% Found ((eq_ref ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq ((a->b)->Prop)) b0) (fun (x:(a->b))=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq ((a->b)->Prop)) b0) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found ((eta_expansion_dep0 (fun (x5:(a->b))=> Prop)) b0) as proof of (((eq ((a->b)->Prop)) b0) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found (((eta_expansion_dep (a->b)) (fun (x5:(a->b))=> Prop)) b0) as proof of (((eq ((a->b)->Prop)) b0) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found (((eta_expansion_dep (a->b)) (fun (x5:(a->b))=> Prop)) b0) as proof of (((eq ((a->b)->Prop)) b0) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found (((eta_expansion_dep (a->b)) (fun (x5:(a->b))=> Prop)) b0) as proof of (((eq ((a->b)->Prop)) b0) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq ((a->b)->Prop)) b0) (fun (x:(a->b))=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq ((a->b)->Prop)) b0) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found ((eta_expansion_dep0 (fun (x5:(a->b))=> Prop)) b0) as proof of (((eq ((a->b)->Prop)) b0) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found (((eta_expansion_dep (a->b)) (fun (x5:(a->b))=> Prop)) b0) as proof of (((eq ((a->b)->Prop)) b0) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found (((eta_expansion_dep (a->b)) (fun (x5:(a->b))=> Prop)) b0) as proof of (((eq ((a->b)->Prop)) b0) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found (((eta_expansion_dep (a->b)) (fun (x5:(a->b))=> Prop)) b0) as proof of (((eq ((a->b)->Prop)) b0) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found eq_ref00:=(eq_ref0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))):(((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found (eq_ref0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) b0)
% Found ((eq_ref ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) b0)
% Found ((eq_ref ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) b0)
% Found ((eq_ref ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) b0)
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))->(P (fun (x:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x Xx)) (Xg Xy)))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eta_expansion00 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eta_expansion0 ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))->(P (fun (x:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x Xx)) (Xg Xy)))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eta_expansion_dep00 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eta_expansion_dep0 (fun (x6:(a->b))=> ((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((((eta_expansion_dep (a->b)) (fun (x6:(a->b))=> ((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((((eta_expansion_dep (a->b)) (fun (x6:(a->b))=> ((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))->(P (fun (x:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (x Xy)))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found ((eta_expansion00 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found (((eta_expansion0 Prop) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found ((((eta_expansion (a->b)) Prop) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found ((((eta_expansion (a->b)) Prop) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))->(P (fun (x:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (x Xy)))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found ((eta_expansion_dep00 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found (((eta_expansion_dep0 (fun (x5:(a->b))=> Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found ((((eta_expansion_dep (a->b)) (fun (x5:(a->b))=> Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found ((((eta_expansion_dep (a->b)) (fun (x5:(a->b))=> Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b0:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b0
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))->(P (fun (x:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (x Xy)))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found ((eta_expansion_dep00 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found (((eta_expansion_dep0 (fun (x5:(a->b))=> Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found ((((eta_expansion_dep (a->b)) (fun (x5:(a->b))=> Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found ((((eta_expansion_dep (a->b)) (fun (x5:(a->b))=> Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))->(P (fun (x:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (x Xy)))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found ((eta_expansion00 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found (((eta_expansion0 Prop) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found ((((eta_expansion (a->b)) Prop) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found ((((eta_expansion (a->b)) Prop) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b0:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b0
% Found x1:(P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Instantiate: f:=(fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))):((a->b)->((a->b)->Prop))
% Found x1 as proof of (P0 f)
% Found x1:(P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Instantiate: f:=(fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))):((a->b)->((a->b)->Prop))
% Found x1 as proof of (P0 f)
% Found x20:=(x2 (fun (x3:(a->(a->Prop)))=> (P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))))):((P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))->(P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))))
% Found (x2 (fun (x3:(a->(a->Prop)))=> (P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))))) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (x2 (fun (x3:(a->(a->Prop)))=> (P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))))) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq ((a->b)->((a->b)->Prop))) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))):(((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (x:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x Xx)) (Xg Xy))))))
% Found (eta_expansion00 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found ((eta_expansion0 ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found (((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found (((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found (((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))->(P (fun (x:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x Xx)) (Xg Xy)))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eta_expansion00 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eta_expansion0 ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eta_expansion000:=(eta_expansion00 (f x2)):(((eq ((a->b)->Prop)) (f x2)) (fun (x:(a->b))=> ((f x2) x)))
% Found (eta_expansion00 (f x2)) as proof of (((eq ((a->b)->Prop)) (f x2)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x2 Xx)) (Xg Xy))))))
% Found ((eta_expansion0 Prop) (f x2)) as proof of (((eq ((a->b)->Prop)) (f x2)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x2 Xx)) (Xg Xy))))))
% Found (((eta_expansion (a->b)) Prop) (f x2)) as proof of (((eq ((a->b)->Prop)) (f x2)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x2 Xx)) (Xg Xy))))))
% Found (((eta_expansion (a->b)) Prop) (f x2)) as proof of (((eq ((a->b)->Prop)) (f x2)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x2 Xx)) (Xg Xy))))))
% Found (fun (x2:(a->b))=> (((eta_expansion (a->b)) Prop) (f x2))) as proof of (((eq ((a->b)->Prop)) (f x2)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x2 Xx)) (Xg Xy))))))
% Found (fun (x2:(a->b))=> (((eta_expansion (a->b)) Prop) (f x2))) as proof of (forall (x:(a->b)), (((eq ((a->b)->Prop)) (f x)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x Xx)) (Xg Xy)))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (f x2)):(((eq ((a->b)->Prop)) (f x2)) (fun (x:(a->b))=> ((f x2) x)))
% Found (eta_expansion_dep00 (f x2)) as proof of (((eq ((a->b)->Prop)) (f x2)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x2 Xx)) (Xg Xy))))))
% Found ((eta_expansion_dep0 (fun (x4:(a->b))=> Prop)) (f x2)) as proof of (((eq ((a->b)->Prop)) (f x2)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x2 Xx)) (Xg Xy))))))
% Found (((eta_expansion_dep (a->b)) (fun (x4:(a->b))=> Prop)) (f x2)) as proof of (((eq ((a->b)->Prop)) (f x2)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x2 Xx)) (Xg Xy))))))
% Found (((eta_expansion_dep (a->b)) (fun (x4:(a->b))=> Prop)) (f x2)) as proof of (((eq ((a->b)->Prop)) (f x2)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x2 Xx)) (Xg Xy))))))
% Found (fun (x2:(a->b))=> (((eta_expansion_dep (a->b)) (fun (x4:(a->b))=> Prop)) (f x2))) as proof of (((eq ((a->b)->Prop)) (f x2)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x2 Xx)) (Xg Xy))))))
% Found (fun (x2:(a->b))=> (((eta_expansion_dep (a->b)) (fun (x4:(a->b))=> Prop)) (f x2))) as proof of (forall (x:(a->b)), (((eq ((a->b)->Prop)) (f x)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x Xx)) (Xg Xy)))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))->(P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq ((a->b)->((a->b)->Prop))) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))):(((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (x:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x Xx)) (Xg Xy))))))
% Found (eta_expansion00 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found ((eta_expansion0 ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found (((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found (((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found (((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq ((a->b)->((a->b)->Prop))) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))):(((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (x:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x Xx)) (Xg Xy))))))
% Found (eta_expansion00 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found ((eta_expansion0 ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found (((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found (((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found (((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq ((a->b)->((a->b)->Prop))) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))):(((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (x:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x Xx)) (Xg Xy))))))
% Found (eta_expansion00 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found ((eta_expansion0 ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found (((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found (((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found (((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found x1:(P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Instantiate: f:=(fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))):((a->b)->((a->b)->Prop))
% Found x1 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 b0):(((eq ((a->b)->((a->b)->Prop))) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eq_ref00:=(eq_ref0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))):(((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (eq_ref0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found ((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found ((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found ((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))->(P (fun (x:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (x Xy)))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found ((eta_expansion00 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found (((eta_expansion0 Prop) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found ((((eta_expansion (a->b)) Prop) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found ((((eta_expansion (a->b)) Prop) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))->(P (fun (x:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (x Xy)))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found ((eta_expansion00 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found (((eta_expansion0 Prop) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found ((((eta_expansion (a->b)) Prop) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found ((((eta_expansion (a->b)) Prop) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (Xg Xy))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))->(P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eq_ref00:=(eq_ref0 (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xz Xy0))))))):(((eq Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xz Xy0))))))) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xz Xy0)))))))
% Found (eq_ref0 (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xz Xy0))))))) as proof of (((eq Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xz Xy0))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xz Xy0))))))) as proof of (((eq Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xz Xy0))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xz Xy0))))))) as proof of (((eq Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xz Xy0))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xz Xy0))))))) as proof of (((eq Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xr Xx0) (Xx Xx0)) (Xz Xy0))))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x3 Xx)) (y Xy)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x3 Xx)) (y Xy)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x3 Xx)) (y Xy)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x3 Xx)) (y Xy)))))
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (y Xy))))):(((eq Prop) (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (y Xy))))) (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (y Xy)))))
% Found (eq_ref0 (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (y Xy))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (y Xy))))) b0)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (y Xy))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (y Xy))))) b0)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (y Xy))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (y Xy))))) b0)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (y Xy))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (y Xy))))) b0)
% Found x3:(P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Instantiate: b0:=(fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))):((a->b)->((a->b)->Prop))
% Found x3 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq ((a->b)->((a->b)->Prop))) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (((eq ((a->b)->((a->b)->Prop))) b0) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eq_ref00:=(eq_ref0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))):(((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (eq_ref0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found ((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found ((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found ((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found eta_expansion:=(fun (A:Type) (B:Type)=> ((eta_expansion_dep A) (fun (x1:A)=> B))):(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x))))
% Instantiate: b0:=(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x)))):Prop
% Found eta_expansion as proof of b0
% Found eta_expansion000:=(eta_expansion00 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))):(((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (x:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x Xx)) (Xg Xy))))))
% Found (eta_expansion00 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found ((eta_expansion0 ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found (((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found (((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found (((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found eq_ref000:=(eq_ref00 P):((P (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (y Xy)))))->(P (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (y Xy))))))
% Found (eq_ref00 P) as proof of (P0 (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (y Xy)))))
% Found ((eq_ref0 (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (y Xy))))) P) as proof of (P0 (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (y Xy)))))
% Found (((eq_ref Prop) (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (y Xy))))) P) as proof of (P0 (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (y Xy)))))
% Found (((eq_ref Prop) (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (y Xy))))) P) as proof of (P0 (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x3 Xx)) (y Xy)))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))->(P (fun (x:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x Xx)) (Xg Xy)))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eta_expansion_dep00 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eta_expansion_dep0 (fun (x6:(a->b))=> ((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((((eta_expansion_dep (a->b)) (fun (x6:(a->b))=> ((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((((eta_expansion_dep (a->b)) (fun (x6:(a->b))=> ((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))->(P (fun (x:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x Xx)) (Xg Xy)))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eta_expansion00 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eta_expansion0 ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((((eta_expansion (a->b)) ((a->b)->Prop)) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq ((a->b)->((a->b)->Prop))) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (P b0)
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (P b0)
% Found ((eq_ref ((a->b)->((a->b)->Prop))) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))):(((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (eq_ref0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found ((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found ((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found ((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq ((a->b)->Prop)) b0) (fun (x:(a->b))=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq ((a->b)->Prop)) b0) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found ((eta_expansion_dep0 (fun (x3:(a->b))=> Prop)) b0) as proof of (((eq ((a->b)->Prop)) b0) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found (((eta_expansion_dep (a->b)) (fun (x3:(a->b))=> Prop)) b0) as proof of (((eq ((a->b)->Prop)) b0) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found (((eta_expansion_dep (a->b)) (fun (x3:(a->b))=> Prop)) b0) as proof of (((eq ((a->b)->Prop)) b0) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found (((eta_expansion_dep (a->b)) (fun (x3:(a->b))=> Prop)) b0) as proof of (((eq ((a->b)->Prop)) b0) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))):(((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) (fun (x:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (x Xy))))))
% Found (eta_expansion00 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) b0)
% Found (((eta_expansion (a->b)) Prop) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) b0)
% Found (((eta_expansion (a->b)) Prop) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) b0)
% Found (((eta_expansion (a->b)) Prop) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))):(((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) (fun (x:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (x Xy))))))
% Found (eta_expansion_dep00 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) b0)
% Found ((eta_expansion_dep0 (fun (x3:(a->b))=> Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) b0)
% Found (((eta_expansion_dep (a->b)) (fun (x3:(a->b))=> Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) b0)
% Found (((eta_expansion_dep (a->b)) (fun (x3:(a->b))=> Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) b0)
% Found (((eta_expansion_dep (a->b)) (fun (x3:(a->b))=> Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq ((a->b)->Prop)) b0) (fun (x:(a->b))=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq ((a->b)->Prop)) b0) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq ((a->b)->Prop)) b0) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found (((eta_expansion (a->b)) Prop) b0) as proof of (((eq ((a->b)->Prop)) b0) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found (((eta_expansion (a->b)) Prop) b0) as proof of (((eq ((a->b)->Prop)) b0) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found (((eta_expansion (a->b)) Prop) b0) as proof of (((eq ((a->b)->Prop)) b0) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found eq_ref00:=(eq_ref0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))):(((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found (eq_ref0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) b0)
% Found ((eq_ref ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) b0)
% Found ((eq_ref ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) b0)
% Found ((eq_ref ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq ((a->b)->Prop)) b0) (fun (x:(a->b))=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq ((a->b)->Prop)) b0) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found ((eta_expansion_dep0 (fun (x3:(a->b))=> Prop)) b0) as proof of (((eq ((a->b)->Prop)) b0) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found (((eta_expansion_dep (a->b)) (fun (x3:(a->b))=> Prop)) b0) as proof of (((eq ((a->b)->Prop)) b0) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found (((eta_expansion_dep (a->b)) (fun (x3:(a->b))=> Prop)) b0) as proof of (((eq ((a->b)->Prop)) b0) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found (((eta_expansion_dep (a->b)) (fun (x3:(a->b))=> Prop)) b0) as proof of (((eq ((a->b)->Prop)) b0) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found eq_ref00:=(eq_ref0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))):(((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found (eq_ref0 (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) b0)
% Found ((eq_ref ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) b0)
% Found ((eq_ref ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) b0)
% Found ((eq_ref ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) as proof of (((eq ((a->b)->Prop)) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy)))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq ((a->b)->Prop)) b0) (fun (x:(a->b))=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq ((a->b)->Prop)) b0) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found ((eta_expansion_dep0 (fun (x3:(a->b))=> Prop)) b0) as proof of (((eq ((a->b)->Prop)) b0) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found (((eta_expansion_dep (a->b)) (fun (x3:(a->b))=> Prop)) b0) as proof of (((eq ((a->b)->Prop)) b0) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found (((eta_expansion_dep (a->b)) (fun (x3:(a->b))=> Prop)) b0) as proof of (((eq ((a->b)->Prop)) b0) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found (((eta_expansion_dep (a->b)) (fun (x3:(a->b))=> Prop)) b0) as proof of (((eq ((a->b)->Prop)) b0) (fun (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((Xr Xx) (x1 Xx)) (Xg Xy))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))->(P (fun (x:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x Xx)) (Xg Xy)))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eta_expansion_dep00 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eta_expansion_dep0 (fun (x4:(a->b))=> ((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((((eta_expansion_dep (a->b)) (fun (x4:(a->b))=> ((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((((eta_expansion_dep (a->b)) (fun (x4:(a->b))=> ((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))->(P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found eq_ref00:=(eq_ref0 ((f x2) y)):(((eq Prop) ((f x2) y)) ((f x2) y))
% Found (eq_ref0 ((f x2) y)) as proof of (((eq Prop) ((f x2) y)) (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x2 Xx)) (y Xy)))))
% Found ((eq_ref Prop) ((f x2) y)) as proof of (((eq Prop) ((f x2) y)) (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x2 Xx)) (y Xy)))))
% Found ((eq_ref Prop) ((f x2) y)) as proof of (((eq Prop) ((f x2) y)) (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x2 Xx)) (y Xy)))))
% Found (fun (y:(a->b))=> ((eq_ref Prop) ((f x2) y))) as proof of (((eq Prop) ((f x2) y)) (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x2 Xx)) (y Xy)))))
% Found (fun (x2:(a->b)) (y:(a->b))=> ((eq_ref Prop) ((f x2) y))) as proof of (forall (y:(a->b)), (((eq Prop) ((f x2) y)) (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x2 Xx)) (y Xy))))))
% Found (fun (x2:(a->b)) (y:(a->b))=> ((eq_ref Prop) ((f x2) y))) as proof of (forall (x:(a->b)) (y:(a->b)), (((eq Prop) ((f x) y)) (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (x Xx)) (y Xy))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))->(P (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found ((eq_ref0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy))))))
% Found (((eq_ref ((a->b)->((a->b)->Prop))) (fun (Xf:(a->b)) (Xg:(a->b))=> (forall (Xx:a) (Xy:a), (((Xq Xx) Xy)->(((Xr Xx) (Xf Xx)) (Xg Xy)))))) P) as proof of (P0 (fun (Xf:(
% EOF
%------------------------------------------------------------------------------