%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV261^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n186.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 13:33:58 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  : SEV261^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n186.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 08:39:01 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 0x186bb00>, <kernel.Type object at 0x186bc20>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula ((and ((and ((and (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))))) (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))))) (forall (K:((a->Prop)->Prop)) (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> False))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (False->False))))))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S:(a->Prop)), (((and ((and ((or (((eq (a->Prop)) Y) (fun (Xy:a)=> False))) (((eq (a->Prop)) Y) (fun (Xy:a)=> (False->False))))) ((or (((eq (a->Prop)) Z) (fun (Xy:a)=> False))) (((eq (a->Prop)) Z) (fun (Xy:a)=> (False->False)))))) (((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->((or (((eq (a->Prop)) S) (fun (Xy:a)=> False))) (((eq (a->Prop)) S) (fun (Xy:a)=> (False->False))))))) of role conjecture named cINDISCRETE_TOPOLOGY_pme
% Conjecture to prove = ((and ((and ((and (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))))) (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))))) (forall (K:((a->Prop)->Prop)) (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> False))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (False->False))))))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S:(a->Prop)), (((and ((and ((or (((eq (a->Prop)) Y) (fun (Xy:a)=> False))) (((eq (a->Prop)) Y) (fun (Xy:a)=> (False->False))))) ((or (((eq (a->Prop)) Z) (fun (Xy:a)=> False))) (((eq (a->Prop)) Z) (fun (Xy:a)=> (False->False)))))) (((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->((or (((eq (a->Prop)) S) (fun (Xy:a)=> False))) (((eq (a->Prop)) S) (fun (Xy:a)=> (False->False))))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['((and ((and ((and (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))))) (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))))) (forall (K:((a->Prop)->Prop)) (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> False))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (False->False))))))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S:(a->Prop)), (((and ((and ((or (((eq (a->Prop)) Y) (fun (Xy:a)=> False))) (((eq (a->Prop)) Y) (fun (Xy:a)=> (False->False))))) ((or (((eq (a->Prop)) Z) (fun (Xy:a)=> False))) (((eq (a->Prop)) Z) (fun (Xy:a)=> (False->False)))))) (((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->((or (((eq (a->Prop)) S) (fun (Xy:a)=> False))) (((eq (a->Prop)) S) (fun (Xy:a)=> (False->False)))))))']
% Parameter a:Type.
% Trying to prove ((and ((and ((and (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))))) (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))))) (forall (K:((a->Prop)->Prop)) (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> False))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (False->False))))))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S:(a->Prop)), (((and ((and ((or (((eq (a->Prop)) Y) (fun (Xy:a)=> False))) (((eq (a->Prop)) Y) (fun (Xy:a)=> (False->False))))) ((or (((eq (a->Prop)) Z) (fun (Xy:a)=> False))) (((eq (a->Prop)) Z) (fun (Xy:a)=> (False->False)))))) (((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->((or (((eq (a->Prop)) S) (fun (Xy:a)=> False))) (((eq (a->Prop)) S) (fun (Xy:a)=> (False->False)))))))
% Found eq_ref00:=(eq_ref0 (forall (Y:(a->Prop)) (Z:(a->Prop)) (S:(a->Prop)), (((and ((and ((or (((eq (a->Prop)) Y) (fun (Xy:a)=> False))) (((eq (a->Prop)) Y) (fun (Xy:a)=> (False->False))))) ((or (((eq (a->Prop)) Z) (fun (Xy:a)=> False))) (((eq (a->Prop)) Z) (fun (Xy:a)=> (False->False)))))) (((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->((or (((eq (a->Prop)) S) (fun (Xy:a)=> False))) (((eq (a->Prop)) S) (fun (Xy:a)=> (False->False))))))):(((eq Prop) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S:(a->Prop)), (((and ((and ((or (((eq (a->Prop)) Y) (fun (Xy:a)=> False))) (((eq (a->Prop)) Y) (fun (Xy:a)=> (False->False))))) ((or (((eq (a->Prop)) Z) (fun (Xy:a)=> False))) (((eq (a->Prop)) Z) (fun (Xy:a)=> (False->False)))))) (((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->((or (((eq (a->Prop)) S) (fun (Xy:a)=> False))) (((eq (a->Prop)) S) (fun (Xy:a)=> (False->False))))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S:(a->Prop)), (((and ((and ((or (((eq (a->Prop)) Y) (fun (Xy:a)=> False))) (((eq (a->Prop)) Y) (fun (Xy:a)=> (False->False))))) ((or (((eq (a->Prop)) Z) (fun (Xy:a)=> False))) (((eq (a->Prop)) Z) (fun (Xy:a)=> (False->False)))))) (((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->((or (((eq (a->Prop)) S) (fun (Xy:a)=> False))) (((eq (a->Prop)) S) (fun (Xy:a)=> (False->False)))))))
% Found (eq_ref0 (forall (Y:(a->Prop)) (Z:(a->Prop)) (S:(a->Prop)), (((and ((and ((or (((eq (a->Prop)) Y) (fun (Xy:a)=> False))) (((eq (a->Prop)) Y) (fun (Xy:a)=> (False->False))))) ((or (((eq (a->Prop)) Z) (fun (Xy:a)=> False))) (((eq (a->Prop)) Z) (fun (Xy:a)=> (False->False)))))) (((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->((or (((eq (a->Prop)) S) (fun (Xy:a)=> False))) (((eq (a->Prop)) S) (fun (Xy:a)=> (False->False))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S:(a->Prop)), (((and ((and ((or (((eq (a->Prop)) Y) (fun (Xy:a)=> False))) (((eq (a->Prop)) Y) (fun (Xy:a)=> (False->False))))) ((or (((eq (a->Prop)) Z) (fun (Xy:a)=> False))) (((eq (a->Prop)) Z) (fun (Xy:a)=> (False->False)))))) (((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->((or (((eq (a->Prop)) S) (fun (Xy:a)=> False))) (((eq (a->Prop)) S) (fun (Xy:a)=> (False->False))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S:(a->Prop)), (((and ((and ((or (((eq (a->Prop)) Y) (fun (Xy:a)=> False))) (((eq (a->Prop)) Y) (fun (Xy:a)=> (False->False))))) ((or (((eq (a->Prop)) Z) (fun (Xy:a)=> False))) (((eq (a->Prop)) Z) (fun (Xy:a)=> (False->False)))))) (((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->((or (((eq (a->Prop)) S) (fun (Xy:a)=> False))) (((eq (a->Prop)) S) (fun (Xy:a)=> (False->False))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S:(a->Prop)), (((and ((and ((or (((eq (a->Prop)) Y) (fun (Xy:a)=> False))) (((eq (a->Prop)) Y) (fun (Xy:a)=> (False->False))))) ((or (((eq (a->Prop)) Z) (fun (Xy:a)=> False))) (((eq (a->Prop)) Z) (fun (Xy:a)=> (False->False)))))) (((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->((or (((eq (a->Prop)) S) (fun (Xy:a)=> False))) (((eq (a->Prop)) S) (fun (Xy:a)=> (False->False))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S:(a->Prop)), (((and ((and ((or (((eq (a->Prop)) Y) (fun (Xy:a)=> False))) (((eq (a->Prop)) Y) (fun (Xy:a)=> (False->False))))) ((or (((eq (a->Prop)) Z) (fun (Xy:a)=> False))) (((eq (a->Prop)) Z) (fun (Xy:a)=> (False->False)))))) (((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->((or (((eq (a->Prop)) S) (fun (Xy:a)=> False))) (((eq (a->Prop)) S) (fun (Xy:a)=> (False->False))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S:(a->Prop)), (((and ((and ((or (((eq (a->Prop)) Y) (fun (Xy:a)=> False))) (((eq (a->Prop)) Y) (fun (Xy:a)=> (False->False))))) ((or (((eq (a->Prop)) Z) (fun (Xy:a)=> False))) (((eq (a->Prop)) Z) (fun (Xy:a)=> (False->False)))))) (((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->((or (((eq (a->Prop)) S) (fun (Xy:a)=> False))) (((eq (a->Prop)) S) (fun (Xy:a)=> (False->False))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S:(a->Prop)), (((and ((and ((or (((eq (a->Prop)) Y) (fun (Xy:a)=> False))) (((eq (a->Prop)) Y) (fun (Xy:a)=> (False->False))))) ((or (((eq (a->Prop)) Z) (fun (Xy:a)=> False))) (((eq (a->Prop)) Z) (fun (Xy:a)=> (False->False)))))) (((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->((or (((eq (a->Prop)) S) (fun (Xy:a)=> False))) (((eq (a->Prop)) S) (fun (Xy:a)=> (False->False))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S:(a->Prop)), (((and ((and ((or (((eq (a->Prop)) Y) (fun (Xy:a)=> False))) (((eq (a->Prop)) Y) (fun (Xy:a)=> (False->False))))) ((or (((eq (a->Prop)) Z) (fun (Xy:a)=> False))) (((eq (a->Prop)) Z) (fun (Xy:a)=> (False->False)))))) (((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->((or (((eq (a->Prop)) S) (fun (Xy:a)=> False))) (((eq (a->Prop)) S) (fun (Xy:a)=> (False->False))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Y:(a->Prop)) (Z:(a->Prop)) (S:(a->Prop)), (((and ((and ((or (((eq (a->Prop)) Y) (fun (Xy:a)=> False))) (((eq (a->Prop)) Y) (fun (Xy:a)=> (False->False))))) ((or (((eq (a->Prop)) Z) (fun (Xy:a)=> False))) (((eq (a->Prop)) Z) (fun (Xy:a)=> (False->False)))))) (((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->((or (((eq (a->Prop)) S) (fun (Xy:a)=> False))) (((eq (a->Prop)) S) (fun (Xy:a)=> (not False))))))):(((eq Prop) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S:(a->Prop)), (((and ((and ((or (((eq (a->Prop)) Y) (fun (Xy:a)=> False))) (((eq (a->Prop)) Y) (fun (Xy:a)=> (False->False))))) ((or (((eq (a->Prop)) Z) (fun (Xy:a)=> False))) (((eq (a->Prop)) Z) (fun (Xy:a)=> (False->False)))))) (((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->((or (((eq (a->Prop)) S) (fun (Xy:a)=> False))) (((eq (a->Prop)) S) (fun (Xy:a)=> (not False))))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S:(a->Prop)), (((and ((and ((or (((eq (a->Prop)) Y) (fun (Xy:a)=> False))) (((eq (a->Prop)) Y) (fun (Xy:a)=> (False->False))))) ((or (((eq (a->Prop)) Z) (fun (Xy:a)=> False))) (((eq (a->Prop)) Z) (fun (Xy:a)=> (False->False)))))) (((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->((or (((eq (a->Prop)) S) (fun (Xy:a)=> False))) (((eq (a->Prop)) S) (fun (Xy:a)=> (not False)))))))
% Found (eq_ref0 (forall (Y:(a->Prop)) (Z:(a->Prop)) (S:(a->Prop)), (((and ((and ((or (((eq (a->Prop)) Y) (fun (Xy:a)=> False))) (((eq (a->Prop)) Y) (fun (Xy:a)=> (False->False))))) ((or (((eq (a->Prop)) Z) (fun (Xy:a)=> False))) (((eq (a->Prop)) Z) (fun (Xy:a)=> (False->False)))))) (((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->((or (((eq (a->Prop)) S) (fun (Xy:a)=> False))) (((eq (a->Prop)) S) (fun (Xy:a)=> (not False))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S:(a->Prop)), (((and ((and ((or (((eq (a->Prop)) Y) (fun (Xy:a)=> False))) (((eq (a->Prop)) Y) (fun (Xy:a)=> (False->False))))) ((or (((eq (a->Prop)) Z) (fun (Xy:a)=> False))) (((eq (a->Prop)) Z) (fun (Xy:a)=> (False->False)))))) (((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->((or (((eq (a->Prop)) S) (fun (Xy:a)=> False))) (((eq (a->Prop)) S) (fun (Xy:a)=> (not False))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S:(a->Prop)), (((and ((and ((or (((eq (a->Prop)) Y) (fun (Xy:a)=> False))) (((eq (a->Prop)) Y) (fun (Xy:a)=> (False->False))))) ((or (((eq (a->Prop)) Z) (fun (Xy:a)=> False))) (((eq (a->Prop)) Z) (fun (Xy:a)=> (False->False)))))) (((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->((or (((eq (a->Prop)) S) (fun (Xy:a)=> False))) (((eq (a->Prop)) S) (fun (Xy:a)=> (not False))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S:(a->Prop)), (((and ((and ((or (((eq (a->Prop)) Y) (fun (Xy:a)=> False))) (((eq (a->Prop)) Y) (fun (Xy:a)=> (False->False))))) ((or (((eq (a->Prop)) Z) (fun (Xy:a)=> False))) (((eq (a->Prop)) Z) (fun (Xy:a)=> (False->False)))))) (((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->((or (((eq (a->Prop)) S) (fun (Xy:a)=> False))) (((eq (a->Prop)) S) (fun (Xy:a)=> (not False))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S:(a->Prop)), (((and ((and ((or (((eq (a->Prop)) Y) (fun (Xy:a)=> False))) (((eq (a->Prop)) Y) (fun (Xy:a)=> (False->False))))) ((or (((eq (a->Prop)) Z) (fun (Xy:a)=> False))) (((eq (a->Prop)) Z) (fun (Xy:a)=> (False->False)))))) (((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->((or (((eq (a->Prop)) S) (fun (Xy:a)=> False))) (((eq (a->Prop)) S) (fun (Xy:a)=> (not False))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S:(a->Prop)), (((and ((and ((or (((eq (a->Prop)) Y) (fun (Xy:a)=> False))) (((eq (a->Prop)) Y) (fun (Xy:a)=> (False->False))))) ((or (((eq (a->Prop)) Z) (fun (Xy:a)=> False))) (((eq (a->Prop)) Z) (fun (Xy:a)=> (False->False)))))) (((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->((or (((eq (a->Prop)) S) (fun (Xy:a)=> False))) (((eq (a->Prop)) S) (fun (Xy:a)=> (not False))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S:(a->Prop)), (((and ((and ((or (((eq (a->Prop)) Y) (fun (Xy:a)=> False))) (((eq (a->Prop)) Y) (fun (Xy:a)=> (False->False))))) ((or (((eq (a->Prop)) Z) (fun (Xy:a)=> False))) (((eq (a->Prop)) Z) (fun (Xy:a)=> (False->False)))))) (((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->((or (((eq (a->Prop)) S) (fun (Xy:a)=> False))) (((eq (a->Prop)) S) (fun (Xy:a)=> (not False))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S:(a->Prop)), (((and ((and ((or (((eq (a->Prop)) Y) (fun (Xy:a)=> False))) (((eq (a->Prop)) Y) (fun (Xy:a)=> (False->False))))) ((or (((eq (a->Prop)) Z) (fun (Xy:a)=> False))) (((eq (a->Prop)) Z) (fun (Xy:a)=> (False->False)))))) (((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->((or (((eq (a->Prop)) S) (fun (Xy:a)=> False))) (((eq (a->Prop)) S) (fun (Xy:a)=> (not False))))))) b)
% Found or_intror00:=(or_intror0 (((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))):((((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))))
% Found (or_intror0 (((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))) as proof of ((((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))
% Found ((or_intror (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))) as proof of ((((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))
% Found (fun (R:(a->Prop))=> ((or_intror (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xx:a)=> (False->False))))) as proof of ((((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))
% Found (fun (R:(a->Prop))=> ((or_intror (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xx:a)=> (False->False))))) as proof of (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))))
% Found or_introl00:=(or_introl0 (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))):((((eq (a->Prop)) R) (fun (Xx:a)=> False))->((or (((eq (a->Prop)) R) (fun (Xx:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))
% Found (or_introl0 (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))) as proof of ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))
% Found ((or_introl (((eq (a->Prop)) R) (fun (Xx:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))) as proof of ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))
% Found (fun (R:(a->Prop))=> ((or_introl (((eq (a->Prop)) R) (fun (Xx:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))) as proof of ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))
% Found (fun (R:(a->Prop))=> ((or_introl (((eq (a->Prop)) R) (fun (Xx:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))) as proof of (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))))
% Found ((conj20 (fun (R:(a->Prop))=> ((or_introl (((eq (a->Prop)) R) (fun (Xx:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))) (fun (R:(a->Prop))=> ((or_intror (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))))) as proof of ((and (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))))) (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))))
% Found (((conj2 (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))))) (fun (R:(a->Prop))=> ((or_introl (((eq (a->Prop)) R) (fun (Xx:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))) (fun (R:(a->Prop))=> ((or_intror (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))))) as proof of ((and (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))))) (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))))
% Found ((((conj (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))))) (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))))) (fun (R:(a->Prop))=> ((or_introl (((eq (a->Prop)) R) (fun (Xx:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))) (fun (R:(a->Prop))=> ((or_intror (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))))) as proof of ((and (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))))) (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))))
% Found ((((conj (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))))) (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))))) (fun (R:(a->Prop))=> ((or_introl (((eq (a->Prop)) R) (fun (Xx:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))) (fun (R:(a->Prop))=> ((or_intror (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))))) as proof of ((and (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))))) (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))))
% Found or_intror00:=(or_intror0 (((eq (a->Prop)) R) (fun (Xx:a)=> (not False)))):((((eq (a->Prop)) R) (fun (Xx:a)=> (not False)))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xx:a)=> (not False)))))
% Found (or_intror0 (((eq (a->Prop)) R) (fun (Xx:a)=> (not False)))) as proof of ((((eq (a->Prop)) R) (fun (Xx:a)=> (not False)))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))
% Found ((or_intror (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xx:a)=> (not False)))) as proof of ((((eq (a->Prop)) R) (fun (Xx:a)=> (not False)))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))
% Found (fun (R:(a->Prop))=> ((or_intror (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xx:a)=> (not False))))) as proof of ((((eq (a->Prop)) R) (fun (Xx:a)=> (not False)))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))
% Found (fun (R:(a->Prop))=> ((or_intror (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xx:a)=> (not False))))) as proof of (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> (not False)))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))))
% Found or_introl00:=(or_introl0 (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))):((((eq (a->Prop)) R) (fun (Xx:a)=> False))->((or (((eq (a->Prop)) R) (fun (Xx:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))
% Found (or_introl0 (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))) as proof of ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))
% Found ((or_introl (((eq (a->Prop)) R) (fun (Xx:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))) as proof of ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))
% Found (fun (R:(a->Prop))=> ((or_introl (((eq (a->Prop)) R) (fun (Xx:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))) as proof of ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))
% Found (fun (R:(a->Prop))=> ((or_introl (((eq (a->Prop)) R) (fun (Xx:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))) as proof of (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))))
% Found ((conj20 (fun (R:(a->Prop))=> ((or_introl (((eq (a->Prop)) R) (fun (Xx:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))) (fun (R:(a->Prop))=> ((or_intror (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xx:a)=> (not False)))))) as proof of ((and (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))))) (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> (not False)))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))))
% Found (((conj2 (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> (not False)))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))))) (fun (R:(a->Prop))=> ((or_introl (((eq (a->Prop)) R) (fun (Xx:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))) (fun (R:(a->Prop))=> ((or_intror (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xx:a)=> (not False)))))) as proof of ((and (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))))) (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> (not False)))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))))
% Found ((((conj (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))))) (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> (not False)))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))))) (fun (R:(a->Prop))=> ((or_introl (((eq (a->Prop)) R) (fun (Xx:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))) (fun (R:(a->Prop))=> ((or_intror (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xx:a)=> (not False)))))) as proof of ((and (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))))) (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> (not False)))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))))
% Found ((((conj (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))))) (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> (not False)))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))))) (fun (R:(a->Prop))=> ((or_introl (((eq (a->Prop)) R) (fun (Xx:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))) (fun (R:(a->Prop))=> ((or_intror (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xx:a)=> (not False)))))) as proof of ((and (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))))) (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> (not False)))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion_dep00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) 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)=> (False->False)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (False->False)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (False->False)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (False->False)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion_dep00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) 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)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eta_expansion000:=(eta_expansion00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion0 Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) 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)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) 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)=> (not False)))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (not False)))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (not False)))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (not False)))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (not False)))
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref00:=(eq_ref0 (forall (K:((a->Prop)->Prop)) (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> False))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (False->False))))))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))))):(((eq Prop) (forall (K:((a->Prop)->Prop)) (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> False))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (False->False))))))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))))) (forall (K:((a->Prop)->Prop)) (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> False))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (False->False))))))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))))))
% Found (eq_ref0 (forall (K:((a->Prop)->Prop)) (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> False))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (False->False))))))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))))) as proof of (((eq Prop) (forall (K:((a->Prop)->Prop)) (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> False))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (False->False))))))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))))) b)
% Found ((eq_ref Prop) (forall (K:((a->Prop)->Prop)) (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> False))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (False->False))))))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))))) as proof of (((eq Prop) (forall (K:((a->Prop)->Prop)) (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> False))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (False->False))))))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))))) b)
% Found ((eq_ref Prop) (forall (K:((a->Prop)->Prop)) (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> False))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (False->False))))))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))))) as proof of (((eq Prop) (forall (K:((a->Prop)->Prop)) (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> False))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (False->False))))))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))))) b)
% Found ((eq_ref Prop) (forall (K:((a->Prop)->Prop)) (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> False))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (False->False))))))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))))) as proof of (((eq Prop) (forall (K:((a->Prop)->Prop)) (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> False))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (False->False))))))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (K:((a->Prop)->Prop)) (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> False))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (False->False))))))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (not False))))))):(((eq Prop) (forall (K:((a->Prop)->Prop)) (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> False))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (False->False))))))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (not False))))))) (forall (K:((a->Prop)->Prop)) (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> False))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (False->False))))))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (not False)))))))
% Found (eq_ref0 (forall (K:((a->Prop)->Prop)) (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> False))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (False->False))))))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (not False))))))) as proof of (((eq Prop) (forall (K:((a->Prop)->Prop)) (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> False))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (False->False))))))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (not False))))))) b)
% Found ((eq_ref Prop) (forall (K:((a->Prop)->Prop)) (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> False))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (False->False))))))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (not False))))))) as proof of (((eq Prop) (forall (K:((a->Prop)->Prop)) (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> False))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (False->False))))))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (not False))))))) b)
% Found ((eq_ref Prop) (forall (K:((a->Prop)->Prop)) (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> False))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (False->False))))))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (not False))))))) as proof of (((eq Prop) (forall (K:((a->Prop)->Prop)) (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> False))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (False->False))))))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (not False))))))) b)
% Found ((eq_ref Prop) (forall (K:((a->Prop)->Prop)) (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> False))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (False->False))))))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (not False))))))) as proof of (((eq Prop) (forall (K:((a->Prop)->Prop)) (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> False))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (False->False))))))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))->((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (((eq (a->Prop)) R) (fun (Xy:a)=> (not False))))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 R):(((eq (a->Prop)) R) (fun (x:a)=> (R x)))
% Found (eta_expansion_dep00 R) as proof of (((eq (a->Prop)) R) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) R) as proof of (((eq (a->Prop)) R) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) R) as proof of (((eq (a->Prop)) R) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) R) as proof of (((eq (a->Prop)) R) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) R) as proof of (((eq (a->Prop)) R) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (False->False)))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (False->False)))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (False->False)))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (False->False)))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (False->False)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 R):(((eq (a->Prop)) R) (fun (x:a)=> (R x)))
% Found (eta_expansion_dep00 R) as proof of (((eq (a->Prop)) R) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) R) as proof of (((eq (a->Prop)) R) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) R) as proof of (((eq (a->Prop)) R) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) R) as proof of (((eq (a->Prop)) R) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) R) as proof of (((eq (a->Prop)) R) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found x1:(((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Instantiate: b:=(fun (Xx:a)=> ((and (Y Xx)) (Z Xx))):(a->Prop)
% Found x1 as proof of (((eq (a->Prop)) S) b)
% Found x1:(((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Instantiate: b:=(fun (Xx:a)=> ((and (Y Xx)) (Z Xx))):(a->Prop)
% Found x1 as proof of (((eq (a->Prop)) S) b)
% Found eq_ref000:=(eq_ref00 P):((P R)->(P R))
% Found (eq_ref00 P) as proof of (P0 R)
% Found ((eq_ref0 R) P) as proof of (P0 R)
% Found (((eq_ref (a->Prop)) R) P) as proof of (P0 R)
% Found (((eq_ref (a->Prop)) R) P) as proof of (P0 R)
% Found eq_ref000:=(eq_ref00 P):((P R)->(P R))
% Found (eq_ref00 P) as proof of (P0 R)
% Found ((eq_ref0 R) P) as proof of (P0 R)
% Found (((eq_ref (a->Prop)) R) P) as proof of (P0 R)
% Found (((eq_ref (a->Prop)) R) P) as proof of (P0 R)
% 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)=> (False->False)))
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (False->False)))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (False->False)))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (False->False)))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (False->False)))
% 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)=> False))
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found x1:(((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Instantiate: b:=(fun (Xx:a)=> ((and (Y Xx)) (Z Xx))):(a->Prop)
% Found x1 as proof of (((eq (a->Prop)) S) b)
% Found x1:(((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Instantiate: b:=(fun (Xx:a)=> ((and (Y Xx)) (Z Xx))):(a->Prop)
% Found x1 as proof of (((eq (a->Prop)) S) b)
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (False->False)))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (False->False)))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (False->False)))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (False->False)))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (False->False)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion_dep00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% 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)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (False->False)))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (False->False)))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (False->False)))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (False->False)))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (False->False)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 R):(((eq (a->Prop)) R) (fun (x:a)=> (R x)))
% Found (eta_expansion_dep00 R) as proof of (((eq (a->Prop)) R) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) R) as proof of (((eq (a->Prop)) R) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) R) as proof of (((eq (a->Prop)) R) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) R) as proof of (((eq (a->Prop)) R) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) R) as proof of (((eq (a->Prop)) R) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found eta_expansion_dep000:=(eta_expansion_dep00 R):(((eq (a->Prop)) R) (fun (x:a)=> (R x)))
% Found (eta_expansion_dep00 R) as proof of (((eq (a->Prop)) R) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) R) as proof of (((eq (a->Prop)) R) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) R) as proof of (((eq (a->Prop)) R) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) R) as proof of (((eq (a->Prop)) R) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) R) as proof of (((eq (a->Prop)) R) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (not False)))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (not False)))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (not False)))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (not False)))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (not False)))
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found x1:(((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Instantiate: b:=(fun (Xx:a)=> ((and (Y Xx)) (Z Xx))):(a->Prop)
% Found x1 as proof of (((eq (a->Prop)) S) b)
% Found x1:(((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Instantiate: b:=(fun (Xx:a)=> ((and (Y Xx)) (Z Xx))):(a->Prop)
% Found x1 as proof of (((eq (a->Prop)) S) b)
% Found eq_ref000:=(eq_ref00 P):((P R)->(P R))
% Found (eq_ref00 P) as proof of (P0 R)
% Found ((eq_ref0 R) P) as proof of (P0 R)
% Found (((eq_ref (a->Prop)) R) P) as proof of (P0 R)
% Found (((eq_ref (a->Prop)) R) P) as proof of (P0 R)
% Found eq_ref000:=(eq_ref00 P):((P R)->(P R))
% Found (eq_ref00 P) as proof of (P0 R)
% Found ((eq_ref0 R) P) as proof of (P0 R)
% Found (((eq_ref (a->Prop)) R) P) as proof of (P0 R)
% Found (((eq_ref (a->Prop)) R) P) as proof of (P0 R)
% Found eq_ref000:=(eq_ref00 P):((P R)->(P R))
% Found (eq_ref00 P) as proof of (P0 R)
% Found ((eq_ref0 R) P) as proof of (P0 R)
% Found (((eq_ref (a->Prop)) R) P) as proof of (P0 R)
% Found (((eq_ref (a->Prop)) R) P) as proof of (P0 R)
% Found eq_ref000:=(eq_ref00 P):((P R)->(P R))
% Found (eq_ref00 P) as proof of (P0 R)
% Found ((eq_ref0 R) P) as proof of (P0 R)
% Found (((eq_ref (a->Prop)) R) P) as proof of (P0 R)
% Found (((eq_ref (a->Prop)) R) P) as proof of (P0 R)
% Found eq_ref000:=(eq_ref00 P):((P R)->(P R))
% Found (eq_ref00 P) as proof of (P0 R)
% Found ((eq_ref0 R) P) as proof of (P0 R)
% Found (((eq_ref (a->Prop)) R) P) as proof of (P0 R)
% Found (((eq_ref (a->Prop)) R) P) as proof of (P0 R)
% Found eq_ref000:=(eq_ref00 P):((P R)->(P R))
% Found (eq_ref00 P) as proof of (P0 R)
% Found ((eq_ref0 R) P) as proof of (P0 R)
% Found (((eq_ref (a->Prop)) R) P) as proof of (P0 R)
% Found (((eq_ref (a->Prop)) R) P) as proof of (P0 R)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% 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)=> (not False)))
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (not False)))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (not False)))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (not False)))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (not False)))
% Found eq_ref00:=(eq_ref0 (((eq (a->Prop)) S) (fun (Xy:a)=> (False->False)))):(((eq Prop) (((eq (a->Prop)) S) (fun (Xy:a)=> (False->False)))) (((eq (a->Prop)) S) (fun (Xy:a)=> (False->False))))
% Found (eq_ref0 (((eq (a->Prop)) S) (fun (Xy:a)=> (False->False)))) as proof of (((eq Prop) (((eq (a->Prop)) S) (fun (Xy:a)=> (False->False)))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) S) (fun (Xy:a)=> (False->False)))) as proof of (((eq Prop) (((eq (a->Prop)) S) (fun (Xy:a)=> (False->False)))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) S) (fun (Xy:a)=> (False->False)))) as proof of (((eq Prop) (((eq (a->Prop)) S) (fun (Xy:a)=> (False->False)))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) S) (fun (Xy:a)=> (False->False)))) as proof of (((eq Prop) (((eq (a->Prop)) S) (fun (Xy:a)=> (False->False)))) b)
% Found classic0:=(classic (((eq (a->Prop)) S) (fun (Xy:a)=> False))):((or (((eq (a->Prop)) S) (fun (Xy:a)=> False))) (not (((eq (a->Prop)) S) (fun (Xy:a)=> False))))
% Found (classic (((eq (a->Prop)) S) (fun (Xy:a)=> False))) as proof of (P b)
% Found (classic (((eq (a->Prop)) S) (fun (Xy:a)=> False))) as proof of (P b)
% Found (classic (((eq (a->Prop)) S) (fun (Xy:a)=> False))) as proof of (P b)
% Found x1:(((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Instantiate: b:=(fun (Xx:a)=> ((and (Y Xx)) (Z Xx))):(a->Prop)
% Found x1 as proof of (((eq (a->Prop)) S) b)
% Found x1:(((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Instantiate: b:=(fun (Xx:a)=> ((and (Y Xx)) (Z Xx))):(a->Prop)
% Found x1 as proof of (((eq (a->Prop)) S) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion_dep00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (not False)))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (not False)))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (not False)))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (not False)))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (not False)))
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found eta_expansion000:=(eta_expansion00 (fun (Xy:a)=> False)):(((eq (a->Prop)) (fun (Xy:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xy:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xy:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xy:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xy:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xy:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xy:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xy:a)=> False)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found eq_ref00:=(eq_ref0 (fun (Xy:a)=> (False->False))):(((eq (a->Prop)) (fun (Xy:a)=> (False->False))) (fun (Xy:a)=> (False->False)))
% Found (eq_ref0 (fun (Xy:a)=> (False->False))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (False->False))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> (False->False))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (False->False))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> (False->False))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (False->False))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> (False->False))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (False->False))) 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)=> False))
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (not False)))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (not False)))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (not False)))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (not False)))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (not False)))
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P R)->(P R))
% Found (eq_ref00 P) as proof of (P0 R)
% Found ((eq_ref0 R) P) as proof of (P0 R)
% Found (((eq_ref (a->Prop)) R) P) as proof of (P0 R)
% Found (((eq_ref (a->Prop)) R) P) as proof of (P0 R)
% Found eq_ref000:=(eq_ref00 P):((P R)->(P R))
% Found (eq_ref00 P) as proof of (P0 R)
% Found ((eq_ref0 R) P) as proof of (P0 R)
% Found (((eq_ref (a->Prop)) R) P) as proof of (P0 R)
% Found (((eq_ref (a->Prop)) R) P) as proof of (P0 R)
% Found eq_ref000:=(eq_ref00 P):((P R)->(P R))
% Found (eq_ref00 P) as proof of (P0 R)
% Found ((eq_ref0 R) P) as proof of (P0 R)
% Found (((eq_ref (a->Prop)) R) P) as proof of (P0 R)
% Found (((eq_ref (a->Prop)) R) P) as proof of (P0 R)
% Found eq_ref000:=(eq_ref00 P):((P R)->(P R))
% Found (eq_ref00 P) as proof of (P0 R)
% Found ((eq_ref0 R) P) as proof of (P0 R)
% Found (((eq_ref (a->Prop)) R) P) as proof of (P0 R)
% Found (((eq_ref (a->Prop)) R) P) as proof of (P0 R)
% Found eq_ref00:=(eq_ref0 (((eq (a->Prop)) S) (fun (Xy:a)=> (not False)))):(((eq Prop) (((eq (a->Prop)) S) (fun (Xy:a)=> (not False)))) (((eq (a->Prop)) S) (fun (Xy:a)=> (not False))))
% Found (eq_ref0 (((eq (a->Prop)) S) (fun (Xy:a)=> (not False)))) as proof of (((eq Prop) (((eq (a->Prop)) S) (fun (Xy:a)=> (not False)))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) S) (fun (Xy:a)=> (not False)))) as proof of (((eq Prop) (((eq (a->Prop)) S) (fun (Xy:a)=> (not False)))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) S) (fun (Xy:a)=> (not False)))) as proof of (((eq Prop) (((eq (a->Prop)) S) (fun (Xy:a)=> (not False)))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) S) (fun (Xy:a)=> (not False)))) as proof of (((eq Prop) (((eq (a->Prop)) S) (fun (Xy:a)=> (not False)))) b)
% Found classic0:=(classic (((eq (a->Prop)) S) (fun (Xy:a)=> False))):((or (((eq (a->Prop)) S) (fun (Xy:a)=> False))) (not (((eq (a->Prop)) S) (fun (Xy:a)=> False))))
% Found (classic (((eq (a->Prop)) S) (fun (Xy:a)=> False))) as proof of (P b)
% Found (classic (((eq (a->Prop)) S) (fun (Xy:a)=> False))) as proof of (P b)
% Found (classic (((eq (a->Prop)) S) (fun (Xy:a)=> False))) as proof of (P b)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found eta_expansion000:=(eta_expansion00 (fun (Xy:a)=> False)):(((eq (a->Prop)) (fun (Xy:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xy:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xy:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xy:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xy:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xy:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xy:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xy:a)=> False)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found eq_ref00:=(eq_ref0 (fun (Xy:a)=> (not False))):(((eq (a->Prop)) (fun (Xy:a)=> (not False))) (fun (Xy:a)=> (not False)))
% Found (eq_ref0 (fun (Xy:a)=> (not False))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (not False))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> (not False))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (not False))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> (not False))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (not False))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> (not False))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (not False))) b)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xy:a)=> (False->False)))->(P (fun (x:a)=> (False->False))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xy:a)=> (False->False)))
% Found ((eta_expansion_dep00 (fun (Xy:a)=> (False->False))) P) as proof of (P0 (fun (Xy:a)=> (False->False)))
% Found (((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xy:a)=> (False->False))) P) as proof of (P0 (fun (Xy:a)=> (False->False)))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (False->False))) P) as proof of (P0 (fun (Xy:a)=> (False->False)))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (False->False))) P) as proof of (P0 (fun (Xy:a)=> (False->False)))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xy:a)=> (False->False)))->(P (fun (x:a)=> (False->False))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xy:a)=> (False->False)))
% Found ((eta_expansion_dep00 (fun (Xy:a)=> (False->False))) P) as proof of (P0 (fun (Xy:a)=> (False->False)))
% Found (((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xy:a)=> (False->False))) P) as proof of (P0 (fun (Xy:a)=> (False->False)))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (False->False))) P) as proof of (P0 (fun (Xy:a)=> (False->False)))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (False->False))) P) as proof of (P0 (fun (Xy:a)=> (False->False)))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xy:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xy:a)=> False))
% Found ((eta_expansion_dep00 (fun (Xy:a)=> False)) P) as proof of (P0 (fun (Xy:a)=> False))
% Found (((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xy:a)=> False)) P) as proof of (P0 (fun (Xy:a)=> False))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> False)) P) as proof of (P0 (fun (Xy:a)=> False))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> False)) P) as proof of (P0 (fun (Xy:a)=> False))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xy:a)=> False))->(P (fun (Xy:a)=> False)))
% Found (eq_ref00 P) as proof of (P0 (fun (Xy:a)=> False))
% Found ((eq_ref0 (fun (Xy:a)=> False)) P) as proof of (P0 (fun (Xy:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> False)) P) as proof of (P0 (fun (Xy:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> False)) P) as proof of (P0 (fun (Xy:a)=> False))
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref00:=(eq_ref0 (S x0)):(((eq Prop) (S x0)) (S x0))
% Found (eq_ref0 (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (False->False))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (False->False))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (False->False))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (False->False))
% Found eq_ref00:=(eq_ref0 (S x0)):(((eq Prop) (S x0)) (S x0))
% Found (eq_ref0 (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (False->False))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (False->False))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (False->False))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (False->False))
% Found eq_ref00:=(eq_ref0 (S x0)):(((eq Prop) (S x0)) (S x0))
% Found (eq_ref0 (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 (S x0)):(((eq Prop) (S x0)) (S x0))
% Found (eq_ref0 (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found x1:(((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))))
% Instantiate: b:=(fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))):(a->Prop)
% Found x1 as proof of (((eq (a->Prop)) R) b)
% Found x1:(((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))))
% Instantiate: b:=(fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))):(a->Prop)
% Found x1 as proof of (((eq (a->Prop)) R) b)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P (S x0))->(P (S x0)))
% Found (eq_ref00 P) as proof of (P0 (S x0))
% Found ((eq_ref0 (S x0)) P) as proof of (P0 (S x0))
% Found (((eq_ref Prop) (S x0)) P) as proof of (P0 (S x0))
% Found (((eq_ref Prop) (S x0)) P) as proof of (P0 (S x0))
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P (S x0))->(P (S x0)))
% Found (eq_ref00 P) as proof of (P0 (S x0))
% Found ((eq_ref0 (S x0)) P) as proof of (P0 (S x0))
% Found (((eq_ref Prop) (S x0)) P) as proof of (P0 (S x0))
% Found (((eq_ref Prop) (S x0)) P) as proof of (P0 (S x0))
% Found eq_ref000:=(eq_ref00 P):((P (S x0))->(P (S x0)))
% Found (eq_ref00 P) as proof of (P0 (S x0))
% Found ((eq_ref0 (S x0)) P) as proof of (P0 (S x0))
% Found (((eq_ref Prop) (S x0)) P) as proof of (P0 (S x0))
% Found (((eq_ref Prop) (S x0)) P) as proof of (P0 (S x0))
% Found eq_ref000:=(eq_ref00 P):((P (S x0))->(P (S x0)))
% Found (eq_ref00 P) as proof of (P0 (S x0))
% Found ((eq_ref0 (S x0)) P) as proof of (P0 (S x0))
% Found (((eq_ref Prop) (S x0)) P) as proof of (P0 (S x0))
% Found (((eq_ref Prop) (S x0)) P) as proof of (P0 (S x0))
% 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)=> (False->False)))
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (False->False)))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (False->False)))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (False->False)))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (False->False)))
% 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)=> False))
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xy:a)=> False))->(P (fun (Xy:a)=> False)))
% Found (eq_ref00 P) as proof of (P0 (fun (Xy:a)=> False))
% Found ((eq_ref0 (fun (Xy:a)=> False)) P) as proof of (P0 (fun (Xy:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> False)) P) as proof of (P0 (fun (Xy:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> False)) P) as proof of (P0 (fun (Xy:a)=> False))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xy:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xy:a)=> False))
% Found ((eta_expansion_dep00 (fun (Xy:a)=> False)) P) as proof of (P0 (fun (Xy:a)=> False))
% Found (((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xy:a)=> False)) P) as proof of (P0 (fun (Xy:a)=> False))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> False)) P) as proof of (P0 (fun (Xy:a)=> False))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> False)) P) as proof of (P0 (fun (Xy:a)=> False))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xy:a)=> (not False)))->(P (fun (x:a)=> (not False))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xy:a)=> (not False)))
% Found ((eta_expansion_dep00 (fun (Xy:a)=> (not False))) P) as proof of (P0 (fun (Xy:a)=> (not False)))
% Found (((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xy:a)=> (not False))) P) as proof of (P0 (fun (Xy:a)=> (not False)))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (not False))) P) as proof of (P0 (fun (Xy:a)=> (not False)))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (not False))) P) as proof of (P0 (fun (Xy:a)=> (not False)))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xy:a)=> (not False)))->(P (fun (x:a)=> (not False))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xy:a)=> (not False)))
% Found ((eta_expansion00 (fun (Xy:a)=> (not False))) P) as proof of (P0 (fun (Xy:a)=> (not False)))
% Found (((eta_expansion0 Prop) (fun (Xy:a)=> (not False))) P) as proof of (P0 (fun (Xy:a)=> (not False)))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (not False))) P) as proof of (P0 (fun (Xy:a)=> (not False)))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (not False))) P) as proof of (P0 (fun (Xy:a)=> (not False)))
% Found x1:(((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))))
% Instantiate: b:=(fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))):(a->Prop)
% Found x1 as proof of (((eq (a->Prop)) R) b)
% Found x1:(((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))))
% Instantiate: b:=(fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))):(a->Prop)
% Found x1 as proof of (((eq (a->Prop)) R) b)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P S)->(P S))
% Found (eq_ref00 P) as proof of (P0 S)
% Found ((eq_ref0 S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found (((eq_ref (a->Prop)) S) P) as proof of (P0 S)
% Found eq_ref00:=(eq_ref0 R):(((eq (a->Prop)) R) R)
% Found (eq_ref0 R) as proof of (((eq (a->Prop)) R) b)
% Found ((eq_ref (a->Prop)) R) as proof of (((eq (a->Prop)) R) b)
% Found ((eq_ref (a->Prop)) R) as proof of (((eq (a->Prop)) R) b)
% Found ((eq_ref (a->Prop)) R) as proof of (((eq (a->Prop)) R) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found eq_ref00:=(eq_ref0 R):(((eq (a->Prop)) R) R)
% Found (eq_ref0 R) as proof of (((eq (a->Prop)) R) b)
% Found ((eq_ref (a->Prop)) R) as proof of (((eq (a->Prop)) R) b)
% Found ((eq_ref (a->Prop)) R) as proof of (((eq (a->Prop)) R) b)
% Found ((eq_ref (a->Prop)) R) as proof of (((eq (a->Prop)) R) 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)=> (False->False)))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (False->False)))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (False->False)))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (False->False)))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (False->False)))
% Found eq_ref00:=(eq_ref0 (S x0)):(((eq Prop) (S x0)) (S x0))
% Found (eq_ref0 (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (not False))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (not False))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (not False))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (not False))
% Found eq_ref00:=(eq_ref0 (S x0)):(((eq Prop) (S x0)) (S x0))
% Found (eq_ref0 (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (not False))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (not False))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (not False))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (not False))
% Found eq_ref00:=(eq_ref0 (S x0)):(((eq Prop) (S x0)) (S x0))
% Found (eq_ref0 (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 (S x0)):(((eq Prop) (S x0)) (S x0))
% Found (eq_ref0 (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% 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)=> (False->False)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (False->False)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (False->False)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (False->False)))
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found x1:(((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Instantiate: b:=(fun (Xx:a)=> ((and (Y Xx)) (Z Xx))):(a->Prop)
% Found x1 as proof of (((eq (a->Prop)) S) b)
% Found x1:(((eq (a->Prop)) S) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Instantiate: b:=(fun (Xx:a)=> ((and (Y Xx)) (Z Xx))):(a->Prop)
% Found x1 as proof of (((eq (a->Prop)) S) b)
% Found eq_ref000:=(eq_ref00 P):((P R)->(P R))
% Found (eq_ref00 P) as proof of (P0 R)
% Found ((eq_ref0 R) P) as proof of (P0 R)
% Found (((eq_ref (a->Prop)) R) P) as proof of (P0 R)
% Found (((eq_ref (a->Prop)) R) P) as proof of (P0 R)
% Found eq_ref000:=(eq_ref00 P):((P R)->(P R))
% Found (eq_ref00 P) as proof of (P0 R)
% Found ((eq_ref0 R) P) as proof of (P0 R)
% Found (((eq_ref (a->Prop)) R) P) as proof of (P0 R)
% Found (((eq_ref (a->Prop)) R) P) as proof of (P0 R)
% Found eq_ref00:=(eq_ref0 (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))):(((eq Prop) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False))))
% Found (eq_ref0 (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))) as proof of (((eq Prop) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))) as proof of (((eq Prop) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))) as proof of (((eq Prop) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))) as proof of (((eq Prop) (((eq (a->Prop)) R) (fun (Xy:a)=> (False->False)))) b)
% Found classic0:=(classic (((eq (a->Prop)) R) (fun (Xy:a)=> False))):((or (((eq (a->Prop)) R) (fun (Xy:a)=> False))) (not (((eq (a->Prop)) R) (fun (Xy:a)=> False))))
% Found (classic (((eq (a->Prop)) R) (fun (Xy:a)=> False))) as proof of (P b)
% Found (classic (((eq (a->Prop)) R) (fun (Xy:a)=> False))) as proof of (P b)
% Found (classic (((eq (a->Prop)) R) (fun (Xy:a)=> False))) as proof of (P b)
% Found x1:(((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))))
% Instantiate: b:=(fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))):(a->Prop)
% Found x1 as proof of (((eq (a->Prop)) R) b)
% Found x1:(((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))))
% Instantiate: b:=(fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))):(a->Prop)
% Found x1 as proof of (((eq (a->Prop)) R) b)
% Found eq_ref000:=(eq_ref00 P):((P (S x0))->(P (S x0)))
% Found (eq_ref00 P) as proof of (P0 (S x0))
% Found ((eq_ref0 (S x0)) P) as proof of (P0 (S x0))
% Found (((eq_ref Prop) (S x0)) P) as proof of (P0 (S x0))
% Found (((eq_ref Prop) (S x0)) P) as proof of (P0 (S x0))
% Found eq_ref000:=(eq_ref00 P):((P (S x0))->(P (S x0)))
% Found (eq_ref00 P) as proof of (P0 (S x0))
% Found ((eq_ref0 (S x0)) P) as proof of (P0 (S x0))
% Found (((eq_ref Prop) (S x0)) P) as proof of (P0 (S x0))
% Found (((eq_ref Prop) (S x0)) P) as proof of (P0 (S x0))
% Found eq_ref000:=(eq_ref00 P):((P (S x0))->(P (S x0)))
% Found (eq_ref00 P) as proof of (P0 (S x0))
% Found ((eq_ref0 (S x0)) P) as proof of (P0 (S x0))
% Found (((eq_ref Prop) (S x0)) P) as proof of (P0 (S x0))
% Found (((eq_ref Prop) (S x0)) P) as proof of (P0 (S x0))
% Found eq_ref000:=(eq_ref00 P):((P (S x0))->(P (S x0)))
% Found (eq_ref00 P) as proof of (P0 (S x0))
% Found ((eq_ref0 (S x0)) P) as proof of (P0 (S x0))
% Found (((eq_ref Prop) (S x0)) P) as proof of (P0 (S x0))
% Found (((eq_ref Prop) (S x0)) P) as proof of (P0 (S x0))
% 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)=> (False->False)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (False->False)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (False->False)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (False->False)))
% 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)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% 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)=> False))
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> False))
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (not False)))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (not False)))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (not False)))
% Found (((eta_expansion a) Prop) b) as proof of 
% EOF
%------------------------------------------------------------------------------