TSTP Solution File: SEV119^5 by cocATP---0.2.0

%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV119^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n105.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 13:33:45 EDT 2014

% Result   : Timeout 300.04s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV119^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n105.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 08:08:41 CDT 2014
% % CPUTime  : 300.04 
% 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 0x253d488>, <kernel.Type object at 0x253df38>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (PROP:((a->(a->Prop))->Prop)) (R:(a->(a->Prop))) (S:(a->(a->Prop))), (((eq (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))) of role conjecture named cTHM252_pme
% Conjecture to prove = (forall (PROP:((a->(a->Prop))->Prop)) (R:(a->(a->Prop))) (S:(a->(a->Prop))), (((eq (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (PROP:((a->(a->Prop))->Prop)) (R:(a->(a->Prop))) (S:(a->(a->Prop))), (((eq (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))))']
% Parameter a:Type.
% Trying to prove (forall (PROP:((a->(a->Prop))->Prop)) (R:(a->(a->Prop))) (S:(a->(a->Prop))), (((eq (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))))
% 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 (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found ((eq_ref (a->(a->Prop))) b) as proof of (((eq (a->(a->Prop))) b) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found ((eq_ref (a->(a->Prop))) b) as proof of (((eq (a->(a->Prop))) b) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found ((eq_ref (a->(a->Prop))) b) as proof of (((eq (a->(a->Prop))) b) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))):(((eq (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) (fun (x:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (eta_expansion00 (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) as proof of (((eq (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) b)
% Found ((eta_expansion0 (a->Prop)) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) as proof of (((eq (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) b)
% Found (((eta_expansion a) (a->Prop)) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) as proof of (((eq (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) b)
% Found (((eta_expansion a) (a->Prop)) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) as proof of (((eq (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) b)
% Found (((eta_expansion a) (a->Prop)) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) as proof of (((eq (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) b)
% Found x2:(P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found (fun (x2:(P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))))=> x2) as proof of (P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found (fun (x2:(P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))))=> x2) as proof of (P0 (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found x2:(P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found (fun (x2:(P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))))=> x2) as proof of (P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found (fun (x2:(P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))))=> x2) as proof of (P0 (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found x2:(P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found (fun (x2:(P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))))=> x2) as proof of (P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found (fun (x2:(P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))))=> x2) as proof of (P0 (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found x2:(P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found (fun (x2:(P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))))=> x2) as proof of (P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found (fun (x2:(P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))))=> x2) as proof of (P0 (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% 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 (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found ((eq_ref (a->(a->Prop))) b) as proof of (((eq (a->(a->Prop))) b) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found ((eq_ref (a->(a->Prop))) b) as proof of (((eq (a->(a->Prop))) b) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found ((eq_ref (a->(a->Prop))) b) as proof of (((eq (a->(a->Prop))) b) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))):(((eq (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) (fun (x:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (eta_expansion_dep00 (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) as proof of (((eq (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> (a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) as proof of (((eq (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> (a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) as proof of (((eq (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> (a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) as proof of (((eq (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> (a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) as proof of (((eq (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))):(((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) (fun (x0:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))))
% Found (eta_expansion00 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found ((eta_expansion0 Prop) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))):(((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) (fun (x0:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))))
% Found (eta_expansion_dep00 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found x2:(P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found (fun (x2:(P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))))=> x2) as proof of (P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found (fun (x2:(P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))))=> x2) as proof of (P0 (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found x2:(P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found (fun (x2:(P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))))=> x2) as proof of (P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found (fun (x2:(P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))))=> x2) as proof of (P0 (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found eq_ref00:=(eq_ref0 (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))):(((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y))))
% Found (eq_ref0 (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y))))
% Found x2:(P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found (fun (x2:(P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))))=> x2) as proof of (P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found (fun (x2:(P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))))=> x2) as proof of (P0 (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found x01:(P (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y))))
% Found (fun (x01:(P (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))))=> x01) as proof of (P (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y))))
% Found (fun (x01:(P (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))))=> x01) as proof of (P0 (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))):(((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) (fun (x0:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))))
% Found (eta_expansion_dep00 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))):(((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) (fun (x0:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))))
% Found (eta_expansion_dep00 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))):(((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) (fun (x0:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))))
% Found (eta_expansion00 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found ((eta_expansion0 Prop) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))):(((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) (fun (x0:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))))
% Found (eta_expansion_dep00 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found eq_ref00:=(eq_ref0 (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))):(((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found (eq_ref0 (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found eq_ref00:=(eq_ref0 (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))):(((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found (eq_ref0 (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found eq_ref00:=(eq_ref0 (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))):(((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found (eq_ref0 (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found eq_ref00:=(eq_ref0 (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))):(((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found (eq_ref0 (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found eq_ref00:=(eq_ref0 (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))):(((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y))))
% Found (eq_ref0 (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y))))
% Found eq_ref00:=(eq_ref0 (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))):(((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y))))
% Found (eq_ref0 (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y))))
% Found x11:(P (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found (fun (x11:(P (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))))=> x11) as proof of (P (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found (fun (x11:(P (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))))=> x11) as proof of (P0 (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found x11:(P (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found (fun (x11:(P (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))))=> x11) as proof of (P (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found (fun (x11:(P (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))))=> x11) as proof of (P0 (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found x11:(P (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found (fun (x11:(P (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))))=> x11) as proof of (P (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found (fun (x11:(P (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))))=> x11) as proof of (P0 (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found x11:(P (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found (fun (x11:(P (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))))=> x11) as proof of (P (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found (fun (x11:(P (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))))=> x11) as proof of (P0 (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found x01:(P (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y))))
% Found (fun (x01:(P (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))))=> x01) as proof of (P (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y))))
% Found (fun (x01:(P (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))))=> x01) as proof of (P0 (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y))))
% Found x:(P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Instantiate: b:=(fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))):(a->(a->Prop))
% Found x as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))):(((eq (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found (eq_ref0 (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) as proof of (((eq (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) b)
% Found ((eq_ref (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) as proof of (((eq (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) b)
% Found ((eq_ref (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) as proof of (((eq (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) b)
% Found ((eq_ref (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) as proof of (((eq (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) b)
% Found x:(P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Instantiate: f:=(fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))):(a->(a->Prop))
% Found x as proof of (P0 f)
% Found eta_expansion000:=(eta_expansion00 (f x0)):(((eq (a->Prop)) (f x0)) (fun (x:a)=> ((f x0) x)))
% Found (eta_expansion00 (f x0)) as proof of (((eq (a->Prop)) (f x0)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x0) Xy)))))
% Found ((eta_expansion0 Prop) (f x0)) as proof of (((eq (a->Prop)) (f x0)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x0) Xy)))))
% Found (((eta_expansion a) Prop) (f x0)) as proof of (((eq (a->Prop)) (f x0)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x0) Xy)))))
% Found (((eta_expansion a) Prop) (f x0)) as proof of (((eq (a->Prop)) (f x0)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x0) Xy)))))
% Found (fun (x0:a)=> (((eta_expansion a) Prop) (f x0))) as proof of (((eq (a->Prop)) (f x0)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x0) Xy)))))
% Found (fun (x0:a)=> (((eta_expansion a) Prop) (f x0))) as proof of (forall (x:a), (((eq (a->Prop)) (f x)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))
% Found x:(P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Instantiate: f:=(fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))):(a->(a->Prop))
% Found x as proof of (P0 f)
% Found eta_expansion000:=(eta_expansion00 (f x0)):(((eq (a->Prop)) (f x0)) (fun (x:a)=> ((f x0) x)))
% Found (eta_expansion00 (f x0)) as proof of (((eq (a->Prop)) (f x0)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x0) Xy)))))
% Found ((eta_expansion0 Prop) (f x0)) as proof of (((eq (a->Prop)) (f x0)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x0) Xy)))))
% Found (((eta_expansion a) Prop) (f x0)) as proof of (((eq (a->Prop)) (f x0)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x0) Xy)))))
% Found (((eta_expansion a) Prop) (f x0)) as proof of (((eq (a->Prop)) (f x0)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x0) Xy)))))
% Found (fun (x0:a)=> (((eta_expansion a) Prop) (f x0))) as proof of (((eq (a->Prop)) (f x0)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x0) Xy)))))
% Found (fun (x0:a)=> (((eta_expansion a) Prop) (f x0))) as proof of (forall (x:a), (((eq (a->Prop)) (f x)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (PROP Xp0))->((Xp0 Xx0) Xy0))))->((Xp
% EOF
%------------------------------------------------------------------------------