%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV141^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n104.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 13:33:48 EDT 2014

% Result   : Timeout 300.05s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV141^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n104.star.cs.uiowa.edu
% % Model    : x86_64 x86_64
% % CPU      : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory   : 32286.75MB
% % OS       : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 08:13:01 CDT 2014
% % CPUTime  : 300.05 
% 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 0x1f165f0>, <kernel.Type object at 0x19a2b00>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (R:(a->(a->Prop))) (S:(a->(a->Prop))), (((eq (a->(a->Prop))) (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))) (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1)))))) of role conjecture named cTHM250_pme
% Conjecture to prove = (forall (R:(a->(a->Prop))) (S:(a->(a->Prop))), (((eq (a->(a->Prop))) (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))) (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1)))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (R:(a->(a->Prop))) (S:(a->(a->Prop))), (((eq (a->(a->Prop))) (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))) (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))))']
% Parameter a:Type.
% Trying to prove (forall (R:(a->(a->Prop))) (S:(a->(a->Prop))), (((eq (a->(a->Prop))) (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))) (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->(a->Prop))) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->(a->Prop))) b) (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1)))))
% Found ((eq_ref (a->(a->Prop))) b) as proof of (((eq (a->(a->Prop))) b) (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1)))))
% Found ((eq_ref (a->(a->Prop))) b) as proof of (((eq (a->(a->Prop))) b) (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1)))))
% Found ((eq_ref (a->(a->Prop))) b) as proof of (((eq (a->(a->Prop))) b) (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1)))))
% Found eq_ref00:=(eq_ref0 (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))):(((eq (a->(a->Prop))) (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))) (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1)))))
% Found (eq_ref0 (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))) as proof of (((eq (a->(a->Prop))) (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))) b)
% Found ((eq_ref (a->(a->Prop))) (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))) as proof of (((eq (a->(a->Prop))) (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))) b)
% Found ((eq_ref (a->(a->Prop))) (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))) as proof of (((eq (a->(a->Prop))) (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))) b)
% Found ((eq_ref (a->(a->Prop))) (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))) as proof of (((eq (a->(a->Prop))) (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))) b)
% Found x2:(P (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1)))))
% Found (fun (x2:(P (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))))=> x2) as proof of (P (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1)))))
% Found (fun (x2:(P (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))))=> x2) as proof of (P0 (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1)))))
% Found x2:(P (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1)))))
% Found (fun (x2:(P (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))))=> x2) as proof of (P (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1)))))
% Found (fun (x2:(P (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))))=> x2) as proof of (P0 (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1)))))
% Found x2:(P (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1)))))
% Found (fun (x2:(P (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))))=> x2) as proof of (P (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1)))))
% Found (fun (x2:(P (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))))=> x2) as proof of (P0 (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1)))))
% Found x2:(P (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1)))))
% Found (fun (x2:(P (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))))=> x2) as proof of (P (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1)))))
% Found (fun (x2:(P (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))))=> x2) as proof of (P0 (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1)))))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->(a->Prop))) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->(a->Prop))) b) (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1)))))
% Found ((eq_ref (a->(a->Prop))) b) as proof of (((eq (a->(a->Prop))) b) (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1)))))
% Found ((eq_ref (a->(a->Prop))) b) as proof of (((eq (a->(a->Prop))) b) (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1)))))
% Found ((eq_ref (a->(a->Prop))) b) as proof of (((eq (a->(a->Prop))) b) (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1)))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))):(((eq (a->(a->Prop))) (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))) (fun (x:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found (eta_expansion00 (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))) as proof of (((eq (a->(a->Prop))) (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))) b)
% Found ((eta_expansion0 (a->Prop)) (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))) as proof of (((eq (a->(a->Prop))) (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))) b)
% Found (((eta_expansion a) (a->Prop)) (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))) as proof of (((eq (a->(a->Prop))) (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))) b)
% Found (((eta_expansion a) (a->Prop)) (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))) as proof of (((eq (a->(a->Prop))) (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))) b)
% Found (((eta_expansion a) (a->Prop)) (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))) as proof of (((eq (a->(a->Prop))) (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))):(((eq (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) (fun (x0:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) x0)))))
% Found (eta_expansion_dep00 (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) as proof of (((eq (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) as proof of (((eq (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) as proof of (((eq (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) as proof of (((eq (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) as proof of (((eq (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found eq_ref00:=(eq_ref0 (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))):(((eq (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found (eq_ref0 (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) as proof of (((eq (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) as proof of (((eq (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) as proof of (((eq (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) as proof of (((eq (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found x2:(P (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1)))))
% Found (fun (x2:(P (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))))=> x2) as proof of (P (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1)))))
% Found (fun (x2:(P (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))))=> x2) as proof of (P0 (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1)))))
% Found x2:(P (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1)))))
% Found (fun (x2:(P (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))))=> x2) as proof of (P (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1)))))
% Found (fun (x2:(P (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))))=> x2) as proof of (P0 (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1)))))
% Found x01:(P (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found (fun (x01:(P (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))))=> x01) as proof of (P (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found (fun (x01:(P (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))))=> x01) as proof of (P0 (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found x01:(P (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found (fun (x01:(P (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))))=> x01) as proof of (P (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found (fun (x01:(P (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))))=> x01) as proof of (P0 (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found x01:(P (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found (fun (x01:(P (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))))=> x01) as proof of (P (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found (fun (x01:(P (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))))=> x01) as proof of (P0 (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found x01:(P (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found (fun (x01:(P (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))))=> x01) as proof of (P (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found (fun (x01:(P (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))))=> x01) as proof of (P0 (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found eq_ref00:=(eq_ref0 (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) y)))):(((eq Prop) (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) y)))) (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) y))))
% Found (eq_ref0 (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) y)))) as proof of (((eq Prop) (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) y)))) b)
% Found ((eq_ref Prop) (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) y)))) as proof of (((eq Prop) (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) y)))) b)
% Found ((eq_ref Prop) (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) y)))) as proof of (((eq Prop) (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) y)))) b)
% Found ((eq_ref Prop) (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) y)))) as proof of (((eq Prop) (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) y)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) y))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) y))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) y))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) y))))
% Found x2:(P (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1)))))
% Found (fun (x2:(P (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))))=> x2) as proof of (P (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1)))))
% Found (fun (x2:(P (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1))))))=> x2) as proof of (P0 (fun (Xx1:a) (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 Xx1) Xy1)))))
% Found x01:(P (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found (fun (x01:(P (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))))=> x01) as proof of (P (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found (fun (x01:(P (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))))=> x01) as proof of (P0 (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found x01:(P (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found (fun (x01:(P (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))))=> x01) as proof of (P (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found (fun (x01:(P (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))))=> x01) as proof of (P0 (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found x01:(P (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) y))))
% Found (fun (x01:(P (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) y)))))=> x01) as proof of (P (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) y))))
% Found (fun (x01:(P (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) y)))))=> x01) as proof of (P0 (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) y))))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))):(((eq (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) (fun (x0:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) x0)))))
% Found (eta_expansion00 (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) as proof of (((eq (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) b)
% Found ((eta_expansion0 Prop) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) as proof of (((eq (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) as proof of (((eq (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) as proof of (((eq (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) as proof of (((eq (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))):(((eq (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) (fun (x0:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) x0)))))
% Found (eta_expansion00 (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) as proof of (((eq (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) b)
% Found ((eta_expansion0 Prop) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) as proof of (((eq (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) as proof of (((eq (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) as proof of (((eq (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) as proof of (((eq (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found eq_ref00:=(eq_ref0 (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))):(((eq (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found (eq_ref0 (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) as proof of (((eq (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) as proof of (((eq (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) as proof of (((eq (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) as proof of (((eq (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))):(((eq (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) (fun (x0:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) x0)))))
% Found (eta_expansion00 (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) as proof of (((eq (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) b)
% Found ((eta_expansion0 Prop) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) as proof of (((eq (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) as proof of (((eq (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) as proof of (((eq (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) as proof of (((eq (a->Prop)) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((R Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((S Xx0) Xy0)->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))->((Xp10 Xx) Xy))))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy1:a)=> (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((or ((R Xx) Xy)) ((S Xx) Xy))->((Xp1 Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp1 Xx) Xy)) ((Xp1 Xy) Xz))->((Xp1 Xx) Xz))))->((Xp1 x) Xy1)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy1:a)=> 
% EOF
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