%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV262^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n098.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 300.09s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV262^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n098.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 08:39:11 CDT 2014
% % CPUTime  : 300.09 
% 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 0x2612b00>, <kernel.Type object at 0x2612c20>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (T:((a->Prop)->Prop)), ((forall (K:((a->Prop)->Prop)) (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))->(T R)))->(forall (S:(a->Prop)), ((iff (T S)) (forall (Xx:a), ((S Xx)->((ex (a->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))))))))) of role conjecture named cNBHD_THM2_pme
% Conjecture to prove = (forall (T:((a->Prop)->Prop)), ((forall (K:((a->Prop)->Prop)) (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))->(T R)))->(forall (S:(a->Prop)), ((iff (T S)) (forall (Xx:a), ((S Xx)->((ex (a->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))))))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (T:((a->Prop)->Prop)), ((forall (K:((a->Prop)->Prop)) (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))->(T R)))->(forall (S:(a->Prop)), ((iff (T S)) (forall (Xx:a), ((S Xx)->((ex (a->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0))))))))))))']
% Parameter a:Type.
% Trying to prove (forall (T:((a->Prop)->Prop)), ((forall (K:((a->Prop)->Prop)) (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))->(T R)))->(forall (S:(a->Prop)), ((iff (T S)) (forall (Xx:a), ((S Xx)->((ex (a->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0))))))))))))
% Found eq_ref000:=(eq_ref00 K):((K Xx)->(K Xx))
% Found (eq_ref00 K) as proof of ((K Xx)->(T Xx))
% Found ((eq_ref0 Xx) K) as proof of ((K Xx)->(T Xx))
% Found (((eq_ref (a->Prop)) Xx) K) as proof of ((K Xx)->(T Xx))
% Found (((eq_ref (a->Prop)) Xx) K) as proof of ((K Xx)->(T Xx))
% Found (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K)) as proof of ((K Xx)->(T Xx))
% Found (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K)) as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 K):((K Xx)->(K Xx))
% Found (eq_ref00 K) as proof of ((K Xx)->(T Xx))
% Found ((eq_ref0 Xx) K) as proof of ((K Xx)->(T Xx))
% Found (((eq_ref (a->Prop)) Xx) K) as proof of ((K Xx)->(T Xx))
% Found (((eq_ref (a->Prop)) Xx) K) as proof of ((K Xx)->(T Xx))
% Found (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K)) as proof of ((K Xx)->(T Xx))
% Found (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K)) as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 x2):((x2 Xx0)->(x2 Xx0))
% Found (eq_ref00 x2) as proof of ((x2 Xx0)->(S Xx0))
% Found ((eq_ref0 Xx0) x2) as proof of ((x2 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x2) as proof of ((x2 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x2) as proof of ((x2 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x2)) as proof of ((x2 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x2)) as proof of (forall (Xx0:a), ((x2 Xx0)->(S Xx0)))
% Found eq_ref000:=(eq_ref00 x2):((x2 Xx0)->(x2 Xx0))
% Found (eq_ref00 x2) as proof of ((x2 Xx0)->(S Xx0))
% Found ((eq_ref0 Xx0) x2) as proof of ((x2 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x2) as proof of ((x2 Xx0)->(S Xx0))
% Found (((eq_ref a) Xx0) x2) as proof of ((x2 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x2)) as proof of ((x2 Xx0)->(S Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x2)) as proof of (forall (Xx0:a), ((x2 Xx0)->(S Xx0)))
% 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 (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))):(((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0))))))
% Found (eq_ref0 (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (S x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (S x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (S x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (S x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (S x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (S x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (S x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (S x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (S x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (S x)))
% 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 (x2:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))):(((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) (fun (x:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(x Xx0))))) (N Xx))))) (forall (Xx0:a), ((x Xx0)->(S Xx0))))))
% Found (eta_expansion_dep00 (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found ((eta_expansion_dep0 (fun (x3:(a->Prop))=> Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) b)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (S x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (S x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (S x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (S x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (S x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (S x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (S x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (S x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (S x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (S x)))
% 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 eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% 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_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 (x3:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x3: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 (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% 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):(((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 (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% 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 (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% 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 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 (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))):(((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (eq_ref0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) 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 (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))):(((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (eq_ref0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))->(P (fun (x:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x)))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))->(P (fun (x:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x)))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))->(P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eq_ref0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))->(P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eq_ref0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found eq_ref00:=(eq_ref0 (S x2)):(((eq Prop) (S x2)) (S x2))
% Found (eq_ref0 (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found eq_ref00:=(eq_ref0 (S x2)):(((eq Prop) (S x2)) (S x2))
% Found (eq_ref0 (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found eq_ref00:=(eq_ref0 (S x2)):(((eq Prop) (S x2)) (S x2))
% Found (eq_ref0 (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found eq_ref00:=(eq_ref0 (S x2)):(((eq Prop) (S x2)) (S x2))
% Found (eq_ref0 (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found eq_ref000:=(eq_ref00 P):((P (S x2))->(P (S x2)))
% Found (eq_ref00 P) as proof of (P0 (S x2))
% Found ((eq_ref0 (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% Found eq_ref000:=(eq_ref00 P):((P (S x2))->(P (S x2)))
% Found (eq_ref00 P) as proof of (P0 (S x2))
% Found ((eq_ref0 (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% Found eq_ref000:=(eq_ref00 P):((P (S x2))->(P (S x2)))
% Found (eq_ref00 P) as proof of (P0 (S x2))
% Found ((eq_ref0 (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% Found eq_ref000:=(eq_ref00 P):((P (S x2))->(P (S x2)))
% Found (eq_ref00 P) as proof of (P0 (S x2))
% Found ((eq_ref0 (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% 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 (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))):(((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (eq_ref0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) 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 (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))):(((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (eq_ref0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))->(P (fun (x:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x)))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))->(P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eq_ref0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))->(P (fun (x:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x)))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))->(P (fun (x:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x)))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found eq_ref00:=(eq_ref0 (S x2)):(((eq Prop) (S x2)) (S x2))
% Found (eq_ref0 (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found eq_ref00:=(eq_ref0 (S x2)):(((eq Prop) (S x2)) (S x2))
% Found (eq_ref0 (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found eq_ref00:=(eq_ref0 (S x2)):(((eq Prop) (S x2)) (S x2))
% Found (eq_ref0 (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found eq_ref00:=(eq_ref0 (S x2)):(((eq Prop) (S x2)) (S x2))
% Found (eq_ref0 (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found eq_ref000:=(eq_ref00 P):((P (S x2))->(P (S x2)))
% Found (eq_ref00 P) as proof of (P0 (S x2))
% Found ((eq_ref0 (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% Found eq_ref000:=(eq_ref00 P):((P (S x2))->(P (S x2)))
% Found (eq_ref00 P) as proof of (P0 (S x2))
% Found ((eq_ref0 (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% 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_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref000:=(eq_ref00 P):((P (S x2))->(P (S x2)))
% Found (eq_ref00 P) as proof of (P0 (S x2))
% Found ((eq_ref0 (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% Found eq_ref000:=(eq_ref00 P):((P (S x2))->(P (S x2)))
% Found (eq_ref00 P) as proof of (P0 (S x2))
% Found ((eq_ref0 (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% Found x0:(T S)
% Instantiate: x3:=S:(a->Prop)
% Found x0 as proof of (T x3)
% Found eq_ref00:=(eq_ref0 (((eq (a->Prop)) S) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))):(((eq Prop) (((eq (a->Prop)) S) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))) (((eq (a->Prop)) S) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))
% Found (eq_ref0 (((eq (a->Prop)) S) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))) as proof of (((eq Prop) (((eq (a->Prop)) S) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) S) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))) as proof of (((eq Prop) (((eq (a->Prop)) S) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) S) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))) as proof of (((eq Prop) (((eq (a->Prop)) S) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) S) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))) as proof of (((eq Prop) (((eq (a->Prop)) S) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))) b)
% Found x0:(T S)
% Instantiate: x3:=S:(a->Prop)
% Found x0 as proof of (T x3)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq ((a->Prop)->Prop)) a0) (fun (x:(a->Prop))=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x3:(a->Prop))=> Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq ((a->Prop)->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found ((eq_ref ((a->Prop)->Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found ((eq_ref ((a->Prop)->Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found ((eq_ref ((a->Prop)->Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found eq_ref000:=(eq_ref00 x3):((x3 Xx0)->(x3 Xx0))
% Found (eq_ref00 x3) as proof of ((x3 Xx0)->(x2 Xx0))
% Found ((eq_ref0 Xx0) x3) as proof of ((x3 Xx0)->(x2 Xx0))
% Found (((eq_ref a) Xx0) x3) as proof of ((x3 Xx0)->(x2 Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x3)) as proof of ((x3 Xx0)->(x2 Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x3)) as proof of (forall (Xx0:a), ((x3 Xx0)->(x2 Xx0)))
% Found ((conj30 x0) (fun (Xx0:a)=> (((eq_ref a) Xx0) x3))) as proof of ((and (T x3)) (forall (Xx0:a), ((x3 Xx0)->(x2 Xx0))))
% Found (((conj3 (forall (Xx0:a), ((x3 Xx0)->(x2 Xx0)))) x0) (fun (Xx0:a)=> (((eq_ref a) Xx0) x3))) as proof of ((and (T x3)) (forall (Xx0:a), ((x3 Xx0)->(x2 Xx0))))
% Found ((((conj (T x3)) (forall (Xx0:a), ((x3 Xx0)->(x2 Xx0)))) x0) (fun (Xx0:a)=> (((eq_ref a) Xx0) x3))) as proof of ((and (T x3)) (forall (Xx0:a), ((x3 Xx0)->(x2 Xx0))))
% Found ((((conj (T x3)) (forall (Xx0:a), ((x3 Xx0)->(x2 Xx0)))) x0) (fun (Xx0:a)=> (((eq_ref a) Xx0) x3))) as proof of ((and (T x3)) (forall (Xx0:a), ((x3 Xx0)->(x2 Xx0))))
% Found x1:(S Xx)
% Found x1 as proof of (x3 Xx)
% Found ((conj20 ((((conj (T x3)) (forall (Xx0:a), ((x3 Xx0)->(x2 Xx0)))) x0) (fun (Xx0:a)=> (((eq_ref a) Xx0) x3)))) x1) as proof of ((and ((and (T x3)) (forall (Xx0:a), ((x3 Xx0)->(x2 Xx0))))) (x3 Xx))
% Found (((conj2 (x3 Xx)) ((((conj (T x3)) (forall (Xx0:a), ((x3 Xx0)->(x2 Xx0)))) x0) (fun (Xx0:a)=> (((eq_ref a) Xx0) x3)))) x1) as proof of ((and ((and (T x3)) (forall (Xx0:a), ((x3 Xx0)->(x2 Xx0))))) (x3 Xx))
% Found ((((conj ((and (T x3)) (forall (Xx0:a), ((x3 Xx0)->(x2 Xx0))))) (x3 Xx)) ((((conj (T x3)) (forall (Xx0:a), ((x3 Xx0)->(x2 Xx0)))) x0) (fun (Xx0:a)=> (((eq_ref a) Xx0) x3)))) x1) as proof of ((and ((and (T x3)) (forall (Xx0:a), ((x3 Xx0)->(x2 Xx0))))) (x3 Xx))
% Found ((((conj ((and (T x3)) (forall (Xx0:a), ((x3 Xx0)->(x2 Xx0))))) (x3 Xx)) ((((conj (T x3)) (forall (Xx0:a), ((x3 Xx0)->(x2 Xx0)))) x0) (fun (Xx0:a)=> (((eq_ref a) Xx0) x3)))) x1) as proof of ((and ((and (T x3)) (forall (Xx0:a), ((x3 Xx0)->(x2 Xx0))))) (x3 Xx))
% Found (ex_intro010 ((((conj ((and (T x3)) (forall (Xx0:a), ((x3 Xx0)->(x2 Xx0))))) (x3 Xx)) ((((conj (T x3)) (forall (Xx0:a), ((x3 Xx0)->(x2 Xx0)))) x0) (fun (Xx0:a)=> (((eq_ref a) Xx0) x3)))) x1)) as proof of ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(x2 Xx0))))) (N Xx))))
% Found ((ex_intro01 S) ((((conj ((and (T S)) (forall (Xx0:a), ((S Xx0)->(x2 Xx0))))) (S Xx)) ((((conj (T S)) (forall (Xx0:a), ((S Xx0)->(x2 Xx0)))) x0) (fun (Xx0:a)=> (((eq_ref a) Xx0) S)))) x1)) as proof of ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(x2 Xx0))))) (N Xx))))
% Found (((ex_intro0 (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(x2 Xx0))))) (N Xx)))) S) ((((conj ((and (T S)) (forall (Xx0:a), ((S Xx0)->(x2 Xx0))))) (S Xx)) ((((conj (T S)) (forall (Xx0:a), ((S Xx0)->(x2 Xx0)))) x0) (fun (Xx0:a)=> (((eq_ref a) Xx0) S)))) x1)) as proof of ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(x2 Xx0))))) (N Xx))))
% Found (((ex_intro0 (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(x2 Xx0))))) (N Xx)))) S) ((((conj ((and (T S)) (forall (Xx0:a), ((S Xx0)->(x2 Xx0))))) (S Xx)) ((((conj (T S)) (forall (Xx0:a), ((S Xx0)->(x2 Xx0)))) x0) (fun (Xx0:a)=> (((eq_ref a) Xx0) S)))) x1)) as proof of ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(x2 Xx0))))) (N Xx))))
% Found ((conj10 (((ex_intro0 (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(x2 Xx0))))) (N Xx)))) S) ((((conj ((and (T S)) (forall (Xx0:a), ((S Xx0)->(x2 Xx0))))) (S Xx)) ((((conj (T S)) (forall (Xx0:a), ((S Xx0)->(x2 Xx0)))) x0) (fun (Xx0:a)=> (((eq_ref a) Xx0) S)))) x1))) (fun (Xx0:a)=> (((eq_ref a) Xx0) x2))) as proof of ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(x2 Xx0))))) (N Xx))))) (forall (Xx0:a), ((x2 Xx0)->(S Xx0))))
% Found (((conj1 (forall (Xx0:a), ((x2 Xx0)->(S Xx0)))) (((ex_intro0 (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(x2 Xx0))))) (N Xx)))) S) ((((conj ((and (T S)) (forall (Xx0:a), ((S Xx0)->(x2 Xx0))))) (S Xx)) ((((conj (T S)) (forall (Xx0:a), ((S Xx0)->(x2 Xx0)))) x0) (fun (Xx0:a)=> (((eq_ref a) Xx0) S)))) x1))) (fun (Xx0:a)=> (((eq_ref a) Xx0) x2))) as proof of ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(x2 Xx0))))) (N Xx))))) (forall (Xx0:a), ((x2 Xx0)->(S Xx0))))
% Found ((((conj ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(x2 Xx0))))) (N Xx))))) (forall (Xx0:a), ((x2 Xx0)->(S Xx0)))) (((ex_intro0 (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(x2 Xx0))))) (N Xx)))) S) ((((conj ((and (T S)) (forall (Xx0:a), ((S Xx0)->(x2 Xx0))))) (S Xx)) ((((conj (T S)) (forall (Xx0:a), ((S Xx0)->(x2 Xx0)))) x0) (fun (Xx0:a)=> (((eq_ref a) Xx0) S)))) x1))) (fun (Xx0:a)=> (((eq_ref a) Xx0) x2))) as proof of ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(x2 Xx0))))) (N Xx))))) (forall (Xx0:a), ((x2 Xx0)->(S Xx0))))
% Found ((((conj ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(x2 Xx0))))) (N Xx))))) (forall (Xx0:a), ((x2 Xx0)->(S Xx0)))) (((ex_intro0 (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(x2 Xx0))))) (N Xx)))) S) ((((conj ((and (T S)) (forall (Xx0:a), ((S Xx0)->(x2 Xx0))))) (S Xx)) ((((conj (T S)) (forall (Xx0:a), ((S Xx0)->(x2 Xx0)))) x0) (fun (Xx0:a)=> (((eq_ref a) Xx0) S)))) x1))) (fun (Xx0:a)=> (((eq_ref a) Xx0) x2))) as proof of ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(x2 Xx0))))) (N Xx))))) (forall (Xx0:a), ((x2 Xx0)->(S Xx0))))
% Found (ex_intro000 ((((conj ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(x2 Xx0))))) (N Xx))))) (forall (Xx0:a), ((x2 Xx0)->(S Xx0)))) (((ex_intro0 (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(x2 Xx0))))) (N Xx)))) S) ((((conj ((and (T S)) (forall (Xx0:a), ((S Xx0)->(x2 Xx0))))) (S Xx)) ((((conj (T S)) (forall (Xx0:a), ((S Xx0)->(x2 Xx0)))) x0) (fun (Xx0:a)=> (((eq_ref a) Xx0) S)))) x1))) (fun (Xx0:a)=> (((eq_ref a) Xx0) x2)))) as proof of ((ex (a->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0))))))
% Found ((ex_intro00 S) ((((conj ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(S Xx0))))) (N Xx))))) (forall (Xx0:a), ((S Xx0)->(S Xx0)))) (((ex_intro0 (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(S Xx0))))) (N Xx)))) S) ((((conj ((and (T S)) (forall (Xx0:a), ((S Xx0)->(S Xx0))))) (S Xx)) ((((conj (T S)) (forall (Xx0:a), ((S Xx0)->(S Xx0)))) x0) (fun (Xx0:a)=> (((eq_ref a) Xx0) S)))) x1))) (fun (Xx0:a)=> (((eq_ref a) Xx0) S)))) as proof of ((ex (a->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0))))))
% Found (((ex_intro0 (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) S) ((((conj ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(S Xx0))))) (N Xx))))) (forall (Xx0:a), ((S Xx0)->(S Xx0)))) (((ex_intro0 (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(S Xx0))))) (N Xx)))) S) ((((conj ((and (T S)) (forall (Xx0:a), ((S Xx0)->(S Xx0))))) (S Xx)) ((((conj (T S)) (forall (Xx0:a), ((S Xx0)->(S Xx0)))) x0) (fun (Xx0:a)=> (((eq_ref a) Xx0) S)))) x1))) (fun (Xx0:a)=> (((eq_ref a) Xx0) S)))) as proof of ((ex (a->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0))))))
% Found ((((ex_intro (a->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) S) ((((conj ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(S Xx0))))) (N Xx))))) (forall (Xx0:a), ((S Xx0)->(S Xx0)))) ((((ex_intro (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(S Xx0))))) (N Xx)))) S) ((((conj ((and (T S)) (forall (Xx0:a), ((S Xx0)->(S Xx0))))) (S Xx)) ((((conj (T S)) (forall (Xx0:a), ((S Xx0)->(S Xx0)))) x0) (fun (Xx0:a)=> (((eq_ref a) Xx0) S)))) x1))) (fun (Xx0:a)=> (((eq_ref a) Xx0) S)))) as proof of ((ex (a->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0))))))
% Found (fun (x1:(S Xx))=> ((((ex_intro (a->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) S) ((((conj ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(S Xx0))))) (N Xx))))) (forall (Xx0:a), ((S Xx0)->(S Xx0)))) ((((ex_intro (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(S Xx0))))) (N Xx)))) S) ((((conj ((and (T S)) (forall (Xx0:a), ((S Xx0)->(S Xx0))))) (S Xx)) ((((conj (T S)) (forall (Xx0:a), ((S Xx0)->(S Xx0)))) x0) (fun (Xx0:a)=> (((eq_ref a) Xx0) S)))) x1))) (fun (Xx0:a)=> (((eq_ref a) Xx0) S))))) as proof of ((ex (a->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0))))))
% Found (fun (Xx:a) (x1:(S Xx))=> ((((ex_intro (a->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) S) ((((conj ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(S Xx0))))) (N Xx))))) (forall (Xx0:a), ((S Xx0)->(S Xx0)))) ((((ex_intro (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(S Xx0))))) (N Xx)))) S) ((((conj ((and (T S)) (forall (Xx0:a), ((S Xx0)->(S Xx0))))) (S Xx)) ((((conj (T S)) (forall (Xx0:a), ((S Xx0)->(S Xx0)))) x0) (fun (Xx0:a)=> (((eq_ref a) Xx0) S)))) x1))) (fun (Xx0:a)=> (((eq_ref a) Xx0) S))))) as proof of ((S Xx)->((ex (a->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))))
% Found (fun (x0:(T S)) (Xx:a) (x1:(S Xx))=> ((((ex_intro (a->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) S) ((((conj ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(S Xx0))))) (N Xx))))) (forall (Xx0:a), ((S Xx0)->(S Xx0)))) ((((ex_intro (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(S Xx0))))) (N Xx)))) S) ((((conj ((and (T S)) (forall (Xx0:a), ((S Xx0)->(S Xx0))))) (S Xx)) ((((conj (T S)) (forall (Xx0:a), ((S Xx0)->(S Xx0)))) x0) (fun (Xx0:a)=> (((eq_ref a) Xx0) S)))) x1))) (fun (Xx0:a)=> (((eq_ref a) Xx0) S))))) as proof of (forall (Xx:a), ((S Xx)->((ex (a->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0))))))))
% Found (fun (x0:(T S)) (Xx:a) (x1:(S Xx))=> ((((ex_intro (a->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))) S) ((((conj ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(S Xx0))))) (N Xx))))) (forall (Xx0:a), ((S Xx0)->(S Xx0)))) ((((ex_intro (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(S Xx0))))) (N Xx)))) S) ((((conj ((and (T S)) (forall (Xx0:a), ((S Xx0)->(S Xx0))))) (S Xx)) ((((conj (T S)) (forall (Xx0:a), ((S Xx0)->(S Xx0)))) x0) (fun (Xx0:a)=> (((eq_ref a) Xx0) S)))) x1))) (fun (Xx0:a)=> (((eq_ref a) Xx0) S))))) as proof of ((T S)->(forall (Xx:a), ((S Xx)->((ex (a->Prop)) (fun (R:(a->Prop))=> ((and ((ex (a->Prop)) (fun (N:(a->Prop))=> ((and ((and (T N)) (forall (Xx0:a), ((N Xx0)->(R Xx0))))) (N Xx))))) (forall (Xx0:a), ((R Xx0)->(S Xx0)))))))))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))):(((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x2))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))):(((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x2))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))->(P ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))->(P ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))):(((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x2))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))):(((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x2))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))):(((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x2))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x2))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))):(((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found eq_ref000:=(eq_ref00 P):((P ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))->(P ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))->(P ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))):(((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x2))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))):(((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x2))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found eq_ref00:=(eq_ref0 (S x2)):(((eq Prop) (S x2)) (S x2))
% Found (eq_ref0 (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found eq_ref00:=(eq_ref0 (S x2)):(((eq Prop) (S x2)) (S x2))
% Found (eq_ref0 (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found eq_ref00:=(eq_ref0 (S x2)):(((eq Prop) (S x2)) (S x2))
% Found (eq_ref0 (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found eq_ref00:=(eq_ref0 (S x2)):(((eq Prop) (S x2)) (S x2))
% Found (eq_ref0 (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) 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 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 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 (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% 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:=(x K):(forall (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))->(T R)))
% Instantiate: b:=(forall (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))->(T R))):Prop
% Found x1 as proof of b
% Found eq_ref000:=(eq_ref00 K):((K Xx)->(K Xx))
% Found (eq_ref00 K) as proof of ((K Xx)->(T Xx))
% Found ((eq_ref0 Xx) K) as proof of ((K Xx)->(T Xx))
% Found (((eq_ref (a->Prop)) Xx) K) as proof of ((K Xx)->(T Xx))
% Found (((eq_ref (a->Prop)) Xx) K) as proof of ((K Xx)->(T Xx))
% Found (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K)) as proof of ((K Xx)->(T Xx))
% Found (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K)) as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K))) x1) as proof of (P b)
% Found (((conj1 b) (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K))) x1) as proof of (P b)
% Found ((((conj (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) b) (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K))) x1) as proof of (P b)
% Found ((((conj (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) b) (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K))) x1) as proof of (P 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 eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% 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 x2:(P S)
% Instantiate: b:=S:(a->Prop)
% Found x2 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))):(((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) (fun (x:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x))))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))))
% Found eq_ref000:=(eq_ref00 K):((K Xx)->(K Xx))
% Found (eq_ref00 K) as proof of ((K Xx)->(T Xx))
% Found ((eq_ref0 Xx) K) as proof of ((K Xx)->(T Xx))
% Found (((eq_ref (a->Prop)) Xx) K) as proof of ((K Xx)->(T Xx))
% Found (((eq_ref (a->Prop)) Xx) K) as proof of ((K Xx)->(T Xx))
% Found (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K)) as proof of ((K Xx)->(T Xx))
% Found (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K)) as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K))) ((eq_ref (a->Prop)) b)) as proof of ((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))
% Found (((conj1 (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))))) (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K))) ((eq_ref (a->Prop)) b)) as proof of ((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))
% Found ((((conj (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))))) (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K))) ((eq_ref (a->Prop)) b)) as proof of ((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))
% Found ((((conj (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))))) (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K))) ((eq_ref (a->Prop)) b)) as proof of ((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))
% Found (x10 ((((conj (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))))) (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K))) ((eq_ref (a->Prop)) b))) as proof of (P b)
% Found ((x1 b) ((((conj (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))))) (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K))) ((eq_ref (a->Prop)) b))) as proof of (P b)
% Found (((x T) b) ((((conj (forall (Xx:(a->Prop)), ((T Xx)->(T Xx)))) (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (T S)) (S Xx))))))) (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) T))) ((eq_ref (a->Prop)) b))) as proof of (P b)
% Found (((x T) b) ((((conj (forall (Xx:(a->Prop)), ((T Xx)->(T Xx)))) (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (T S)) (S Xx))))))) (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) T))) ((eq_ref (a->Prop)) b))) as proof of (P b)
% Found x2:(P S)
% Instantiate: b:=S:(a->Prop)
% Found x2 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))):(((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) (fun (x:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x))))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found x2:(P S)
% Instantiate: f:=S:(a->Prop)
% Found x2 as proof of (P0 f)
% Found x2:(P S)
% Instantiate: f:=S:(a->Prop)
% Found x2 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x3)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x3)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x))))))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x3)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x3)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x))))))
% Found eq_ref00:=(eq_ref0 f):(((eq (a->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (a->Prop)) f) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))))
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))))
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))))
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))))
% Found eq_ref000:=(eq_ref00 K):((K Xx)->(K Xx))
% Found (eq_ref00 K) as proof of ((K Xx)->(T Xx))
% Found ((eq_ref0 Xx) K) as proof of ((K Xx)->(T Xx))
% Found (((eq_ref (a->Prop)) Xx) K) as proof of ((K Xx)->(T Xx))
% Found (((eq_ref (a->Prop)) Xx) K) as proof of ((K Xx)->(T Xx))
% Found (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K)) as proof of ((K Xx)->(T Xx))
% Found (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K)) as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K))) ((eq_ref (a->Prop)) f)) as proof of ((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) f) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))
% Found (((conj1 (((eq (a->Prop)) f) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))))) (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K))) ((eq_ref (a->Prop)) f)) as proof of ((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) f) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))
% Found ((((conj (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) f) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))))) (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K))) ((eq_ref (a->Prop)) f)) as proof of ((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) f) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))
% Found ((((conj (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) f) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))))) (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K))) ((eq_ref (a->Prop)) f)) as proof of ((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) f) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))
% Found (x10 ((((conj (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) f) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))))) (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K))) ((eq_ref (a->Prop)) f))) as proof of (P f)
% Found ((x1 f) ((((conj (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) f) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))))) (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K))) ((eq_ref (a->Prop)) f))) as proof of (P f)
% Found (((x T) f) ((((conj (forall (Xx:(a->Prop)), ((T Xx)->(T Xx)))) (((eq (a->Prop)) f) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (T S)) (S Xx))))))) (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) T))) ((eq_ref (a->Prop)) f))) as proof of (P f)
% Found (((x T) f) ((((conj (forall (Xx:(a->Prop)), ((T Xx)->(T Xx)))) (((eq (a->Prop)) f) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (T S)) (S Xx))))))) (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) T))) ((eq_ref (a->Prop)) f))) as proof of (P f)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (a->Prop)) f) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))))
% Found ((eta_expansion0 Prop) f) as proof of (((eq (a->Prop)) f) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))))
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))))
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))))
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))))
% Found eq_ref000:=(eq_ref00 K):((K Xx)->(K Xx))
% Found (eq_ref00 K) as proof of ((K Xx)->(T Xx))
% Found ((eq_ref0 Xx) K) as proof of ((K Xx)->(T Xx))
% Found (((eq_ref (a->Prop)) Xx) K) as proof of ((K Xx)->(T Xx))
% Found (((eq_ref (a->Prop)) Xx) K) as proof of ((K Xx)->(T Xx))
% Found (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K)) as proof of ((K Xx)->(T Xx))
% Found (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K)) as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K))) (((eta_expansion a) Prop) f)) as proof of ((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) f) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))
% Found (((conj1 (((eq (a->Prop)) f) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))))) (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K))) (((eta_expansion a) Prop) f)) as proof of ((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) f) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))
% Found ((((conj (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) f) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))))) (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K))) (((eta_expansion a) Prop) f)) as proof of ((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) f) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))
% Found ((((conj (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) f) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))))) (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K))) (((eta_expansion a) Prop) f)) as proof of ((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) f) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx)))))))
% Found (x10 ((((conj (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) f) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))))) (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K))) (((eta_expansion a) Prop) f))) as proof of (P f)
% Found ((x1 f) ((((conj (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) f) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (K S)) (S Xx))))))) (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K))) (((eta_expansion a) Prop) f))) as proof of (P f)
% Found (((x T) f) ((((conj (forall (Xx:(a->Prop)), ((T Xx)->(T Xx)))) (((eq (a->Prop)) f) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (T S)) (S Xx))))))) (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) T))) (((eta_expansion a) Prop) f))) as proof of (P f)
% Found (((x T) f) ((((conj (forall (Xx:(a->Prop)), ((T Xx)->(T Xx)))) (((eq (a->Prop)) f) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (T S)) (S Xx))))))) (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) T))) (((eta_expansion a) Prop) f))) as proof of (P f)
% Found x2:(P S)
% Instantiate: f:=S:(a->Prop)
% Found x2 as proof of (P0 f)
% Found x2:(P S)
% Instantiate: f:=S:(a->Prop)
% Found x2 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x3)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x3)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x))))))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x3)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x3)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x))))))
% 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):(((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 eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% 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 (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found eq_ref000:=(eq_ref00 K):((K Xx)->(K Xx))
% Found (eq_ref00 K) as proof of ((K Xx)->(T Xx))
% Found ((eq_ref0 Xx) K) as proof of ((K Xx)->(T Xx))
% Found (((eq_ref (a->Prop)) Xx) K) as proof of ((K Xx)->(T Xx))
% Found (((eq_ref (a->Prop)) Xx) K) as proof of ((K Xx)->(T Xx))
% Found (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K)) as proof of ((K Xx)->(T Xx))
% Found (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K)) as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 K):((K Xx)->(K Xx))
% Found (eq_ref00 K) as proof of ((K Xx)->(T Xx))
% Found ((eq_ref0 Xx) K) as proof of ((K Xx)->(T Xx))
% Found (((eq_ref (a->Prop)) Xx) K) as proof of ((K Xx)->(T Xx))
% Found (((eq_ref (a->Prop)) Xx) K) as proof of ((K Xx)->(T Xx))
% Found (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K)) as proof of ((K Xx)->(T Xx))
% Found (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K)) as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% 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 (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% 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 eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found eq_ref000:=(eq_ref00 K):((K Xx)->(K Xx))
% Found (eq_ref00 K) as proof of ((K Xx)->(T Xx))
% Found ((eq_ref0 Xx) K) as proof of ((K Xx)->(T Xx))
% Found (((eq_ref (a->Prop)) Xx) K) as proof of ((K Xx)->(T Xx))
% Found (((eq_ref (a->Prop)) Xx) K) as proof of ((K Xx)->(T Xx))
% Found (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K)) as proof of ((K Xx)->(T Xx))
% Found (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K)) as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 K):((K Xx)->(K Xx))
% Found (eq_ref00 K) as proof of ((K Xx)->(T Xx))
% Found ((eq_ref0 Xx) K) as proof of ((K Xx)->(T Xx))
% Found (((eq_ref (a->Prop)) Xx) K) as proof of ((K Xx)->(T Xx))
% Found (((eq_ref (a->Prop)) Xx) K) as proof of ((K Xx)->(T Xx))
% Found (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K)) as proof of ((K Xx)->(T Xx))
% Found (fun (Xx:(a->Prop))=> (((eq_ref (a->Prop)) Xx) K)) as proof of (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))
% 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 (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))):(((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (eq_ref0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) 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 (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))):(((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (eq_ref0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref (a->Prop)) b) as proof of (P b)
% Found ((eq_ref (a->Prop)) b) as proof of (P b)
% Found ((eq_ref (a->Prop)) b) as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))):(((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) (fun (x:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x))))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref (a->Prop)) b) as proof of (P b)
% Found ((eq_ref (a->Prop)) b) as proof of (P b)
% Found ((eq_ref (a->Prop)) b) as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))):(((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) (fun (x:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x))))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) 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 (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))):(((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (eq_ref0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) 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 (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))):(((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (eq_ref0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found x2:(P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Instantiate: b:=(fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))):(a->Prop)
% Found x2 as proof of (P0 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 eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))->(P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eq_ref0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))->(P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eq_ref0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))->(P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eq_ref0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))->(P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eq_ref0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found x2:(P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Instantiate: b:=(fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))):(a->Prop)
% Found x2 as proof of (P0 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 eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))->(P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eq_ref0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))->(P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eq_ref0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))->(P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eq_ref0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))->(P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eq_ref0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found eq_ref000:=(eq_ref00 P):((P (S x2))->(P (S x2)))
% Found (eq_ref00 P) as proof of (P0 (S x2))
% Found ((eq_ref0 (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% Found eq_ref000:=(eq_ref00 P):((P (S x2))->(P (S x2)))
% Found (eq_ref00 P) as proof of (P0 (S x2))
% Found ((eq_ref0 (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% Found eq_ref000:=(eq_ref00 P):((P (S x2))->(P (S x2)))
% Found (eq_ref00 P) as proof of (P0 (S x2))
% Found ((eq_ref0 (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% Found eq_ref000:=(eq_ref00 P):((P (S x2))->(P (S x2)))
% Found (eq_ref00 P) as proof of (P0 (S x2))
% Found ((eq_ref0 (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% Found eq_ref00:=(eq_ref0 (S x2)):(((eq Prop) (S x2)) (S x2))
% Found (eq_ref0 (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found eq_ref00:=(eq_ref0 (S x2)):(((eq Prop) (S x2)) (S x2))
% Found (eq_ref0 (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found eq_ref00:=(eq_ref0 (S x2)):(((eq Prop) (S x2)) (S x2))
% Found (eq_ref0 (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found eq_ref00:=(eq_ref0 (S x2)):(((eq Prop) (S x2)) (S x2))
% Found (eq_ref0 (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found eq_ref000:=(eq_ref00 P):((P (S x2))->(P (S x2)))
% Found (eq_ref00 P) as proof of (P0 (S x2))
% Found ((eq_ref0 (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% Found eq_ref000:=(eq_ref00 P):((P (S x2))->(P (S x2)))
% Found (eq_ref00 P) as proof of (P0 (S x2))
% Found ((eq_ref0 (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% Found eq_ref000:=(eq_ref00 P):((P (S x2))->(P (S x2)))
% Found (eq_ref00 P) as proof of (P0 (S x2))
% Found ((eq_ref0 (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% Found eq_ref000:=(eq_ref00 P):((P (S x2))->(P (S x2)))
% Found (eq_ref00 P) as proof of (P0 (S x2))
% Found ((eq_ref0 (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% Found eq_ref00:=(eq_ref0 (S x2)):(((eq Prop) (S x2)) (S x2))
% Found (eq_ref0 (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found eq_ref00:=(eq_ref0 (S x2)):(((eq Prop) (S x2)) (S x2))
% Found (eq_ref0 (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found eq_ref00:=(eq_ref0 (S x2)):(((eq Prop) (S x2)) (S x2))
% Found (eq_ref0 (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found eq_ref00:=(eq_ref0 (S x2)):(((eq Prop) (S x2)) (S x2))
% Found (eq_ref0 (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref (a->Prop)) b) P) as proof of (P0 b)
% Found (((eq_ref (a->Prop)) b) P) as proof of (P0 b)
% Found x2:(P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Instantiate: f:=(fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))):(a->Prop)
% Found x2 as proof of (P0 f)
% Found x2:(P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Instantiate: f:=(fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))):(a->Prop)
% Found x2 as proof of (P0 f)
% Found eta_expansion0000:=(eta_expansion000 P):((P b)->(P (fun (x:a)=> (b x))))
% Found (eta_expansion000 P) as proof of (P0 b)
% Found ((eta_expansion00 b) P) as proof of (P0 b)
% Found (((eta_expansion0 Prop) b) P) as proof of (P0 b)
% Found ((((eta_expansion a) Prop) b) P) as proof of (P0 b)
% Found ((((eta_expansion a) Prop) b) P) as proof of (P0 b)
% 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) b0)
% Found ((eta_expansion0 Prop) S) as proof of (((eq (a->Prop)) S) b0)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b0)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b0)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (S x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (S x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (S x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (S x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (S x)))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (S x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (S x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (S x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (S x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (S x)))
% Found x2:(P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Instantiate: f:=(fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))):(a->Prop)
% Found x2 as proof of (P0 f)
% Found x2:(P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Instantiate: f:=(fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))):(a->Prop)
% Found x2 as proof of (P0 f)
% Found x3:(P (S x2))
% Instantiate: b:=(S x2):Prop
% Found x3 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))):(((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found x3:(P (S x2))
% Instantiate: b:=(S x2):Prop
% Found x3 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))):(((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref (a->Prop)) b) P) as proof of (P0 b)
% Found (((eq_ref (a->Prop)) b) P) as proof of (P0 b)
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref (a->Prop)) b) P) as proof of (P0 b)
% Found (((eq_ref (a->Prop)) b) P) as proof of (P0 b)
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))->(P (fun (x:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x)))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (S x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (S x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (S x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (S x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (S x)))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (S x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (S x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (S x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (S x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (S x)))
% 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 (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))):(((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) (fun (x:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x))))))
% Found (eta_expansion00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) 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 eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))):(((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) (fun (x:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x))))))
% Found (eta_expansion00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) 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 eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))):(((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) (fun (x:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x))))))
% Found (eta_expansion00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) 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 eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))):(((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) (fun (x:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x))))))
% Found (eta_expansion00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found x3:(P (S x2))
% Instantiate: b:=(S x2):Prop
% Found x3 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))):(((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found x3:(P (S x2))
% Instantiate: b:=(S x2):Prop
% Found x3 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))):(((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2))))) b)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P b)->(P (fun (x:a)=> (b x))))
% Found (eta_expansion_dep000 P) as proof of (P0 b)
% Found ((eta_expansion_dep00 b) P) as proof of (P0 b)
% Found (((eta_expansion_dep0 (fun (x3:a)=> Prop)) b) P) as proof of (P0 b)
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) P) as proof of (P0 b)
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) P) as proof of (P0 b)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P b)->(P (fun (x:a)=> (b x))))
% Found (eta_expansion_dep000 P) as proof of (P0 b)
% Found ((eta_expansion_dep00 b) P) as proof of (P0 b)
% Found (((eta_expansion_dep0 (fun (x3:a)=> Prop)) b) P) as proof of (P0 b)
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) P) as proof of (P0 b)
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) P) as proof of (P0 b)
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))->(P (fun (x:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x)))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% 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 (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))):(((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) (fun (x:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x))))))
% Found (eta_expansion00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) 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 eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))):(((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) (fun (x:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x))))))
% Found (eta_expansion00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) 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 (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))):(((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (eq_ref0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) 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 eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))):(((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) (fun (x:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x))))))
% Found (eta_expansion00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) b)
% Found eta_expansion0000:=(eta_expansion000 P1):((P1 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))->(P1 (fun (x:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x)))))))
% Found (eta_expansion000 P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found eta_expansion0000:=(eta_expansion000 P1):((P1 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))->(P1 (fun (x:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x)))))))
% Found (eta_expansion000 P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found eta_expansion0000:=(eta_expansion000 P1):((P1 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))->(P1 (fun (x:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x)))))))
% Found (eta_expansion000 P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found eta_expansion0000:=(eta_expansion000 P1):((P1 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))->(P1 (fun (x:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x)))))))
% Found (eta_expansion000 P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% 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_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eta_expansion0000:=(eta_expansion000 P1):((P1 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))->(P1 (fun (x:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x)))))))
% Found (eta_expansion000 P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found eta_expansion0000:=(eta_expansion000 P1):((P1 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))->(P1 (fun (x:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x)))))))
% Found (eta_expansion000 P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found eta_expansion0000:=(eta_expansion000 P1):((P1 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))->(P1 (fun (x:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x)))))))
% Found (eta_expansion000 P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found eta_expansion0000:=(eta_expansion000 P1):((P1 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))->(P1 (fun (x:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x)))))))
% Found (eta_expansion000 P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))) P1) as proof of (P2 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found eq_ref000:=(eq_ref00 P):((P (S x2))->(P (S x2)))
% Found (eq_ref00 P) as proof of (P0 (S x2))
% Found ((eq_ref0 (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% Found eq_ref000:=(eq_ref00 P):((P (S x2))->(P (S x2)))
% Found (eq_ref00 P) as proof of (P0 (S x2))
% Found ((eq_ref0 (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% Found eq_ref00:=(eq_ref0 (S x2)):(((eq Prop) (S x2)) (S x2))
% Found (eq_ref0 (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found eq_ref00:=(eq_ref0 (S x2)):(((eq Prop) (S x2)) (S x2))
% Found (eq_ref0 (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found eq_ref00:=(eq_ref0 (S x2)):(((eq Prop) (S x2)) (S x2))
% Found (eq_ref0 (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found eq_ref00:=(eq_ref0 (S x2)):(((eq Prop) (S x2)) (S x2))
% Found (eq_ref0 (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found eq_ref000:=(eq_ref00 P1):((P1 (S x2))->(P1 (S x2)))
% Found (eq_ref00 P1) as proof of (P2 (S x2))
% Found ((eq_ref0 (S x2)) P1) as proof of (P2 (S x2))
% Found (((eq_ref Prop) (S x2)) P1) as proof of (P2 (S x2))
% Found (((eq_ref Prop) (S x2)) P1) as proof of (P2 (S x2))
% Found eq_ref000:=(eq_ref00 P1):((P1 (S x2))->(P1 (S x2)))
% Found (eq_ref00 P1) as proof of (P2 (S x2))
% Found ((eq_ref0 (S x2)) P1) as proof of (P2 (S x2))
% Found (((eq_ref Prop) (S x2)) P1) as proof of (P2 (S x2))
% Found (((eq_ref Prop) (S x2)) P1) as proof of (P2 (S x2))
% Found eq_ref000:=(eq_ref00 P1):((P1 (S x2))->(P1 (S x2)))
% Found (eq_ref00 P1) as proof of (P2 (S x2))
% Found ((eq_ref0 (S x2)) P1) as proof of (P2 (S x2))
% Found (((eq_ref Prop) (S x2)) P1) as proof of (P2 (S x2))
% Found (((eq_ref Prop) (S x2)) P1) as proof of (P2 (S x2))
% Found eq_ref000:=(eq_ref00 P1):((P1 (S x2))->(P1 (S x2)))
% Found (eq_ref00 P1) as proof of (P2 (S x2))
% Found ((eq_ref0 (S x2)) P1) as proof of (P2 (S x2))
% Found (((eq_ref Prop) (S x2)) P1) as proof of (P2 (S x2))
% Found (((eq_ref Prop) (S x2)) P1) as proof of (P2 (S x2))
% Found eq_ref00:=(eq_ref0 (S x2)):(((eq Prop) (S x2)) (S x2))
% Found (eq_ref0 (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found eq_ref00:=(eq_ref0 (S x2)):(((eq Prop) (S x2)) (S x2))
% Found (eq_ref0 (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found eq_ref00:=(eq_ref0 (S x2)):(((eq Prop) (S x2)) (S x2))
% Found (eq_ref0 (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found eq_ref00:=(eq_ref0 (S x2)):(((eq Prop) (S x2)) (S x2))
% Found (eq_ref0 (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found eq_ref000:=(eq_ref00 P):((P (S x2))->(P (S x2)))
% Found (eq_ref00 P) as proof of (P0 (S x2))
% Found ((eq_ref0 (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% Found eq_ref000:=(eq_ref00 P):((P (S x2))->(P (S x2)))
% Found (eq_ref00 P) as proof of (P0 (S x2))
% Found ((eq_ref0 (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% Found (((eq_ref Prop) (S x2)) P) as proof of (P0 (S x2))
% Found eq_ref00:=(eq_ref0 (S x2)):(((eq Prop) (S x2)) (S x2))
% Found (eq_ref0 (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found eq_ref00:=(eq_ref0 (S x2)):(((eq Prop) (S x2)) (S x2))
% Found (eq_ref0 (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found eq_ref00:=(eq_ref0 (S x2)):(((eq Prop) (S x2)) (S x2))
% Found (eq_ref0 (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found eq_ref00:=(eq_ref0 (S x2)):(((eq Prop) (S x2)) (S x2))
% Found (eq_ref0 (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found eq_ref000:=(eq_ref00 P1):((P1 (S x2))->(P1 (S x2)))
% Found (eq_ref00 P1) as proof of (P2 (S x2))
% Found ((eq_ref0 (S x2)) P1) as proof of (P2 (S x2))
% Found (((eq_ref Prop) (S x2)) P1) as proof of (P2 (S x2))
% Found (((eq_ref Prop) (S x2)) P1) as proof of (P2 (S x2))
% Found eq_ref000:=(eq_ref00 P1):((P1 (S x2))->(P1 (S x2)))
% Found (eq_ref00 P1) as proof of (P2 (S x2))
% Found ((eq_ref0 (S x2)) P1) as proof of (P2 (S x2))
% Found (((eq_ref Prop) (S x2)) P1) as proof of (P2 (S x2))
% Found (((eq_ref Prop) (S x2)) P1) as proof of (P2 (S x2))
% Found eq_ref000:=(eq_ref00 P1):((P1 (S x2))->(P1 (S x2)))
% Found (eq_ref00 P1) as proof of (P2 (S x2))
% Found ((eq_ref0 (S x2)) P1) as proof of (P2 (S x2))
% Found (((eq_ref Prop) (S x2)) P1) as proof of (P2 (S x2))
% Found (((eq_ref Prop) (S x2)) P1) as proof of (P2 (S x2))
% Found eq_ref000:=(eq_ref00 P1):((P1 (S x2))->(P1 (S x2)))
% Found (eq_ref00 P1) as proof of (P2 (S x2))
% Found ((eq_ref0 (S x2)) P1) as proof of (P2 (S x2))
% Found (((eq_ref Prop) (S x2)) P1) as proof of (P2 (S x2))
% Found (((eq_ref Prop) (S x2)) P1) as proof of (P2 (S x2))
% Found eq_ref00:=(eq_ref0 (S x2)):(((eq Prop) (S x2)) (S x2))
% Found (eq_ref0 (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 x2)))))
% Found eq_ref00:=(eq_ref0 (S x2)):(((eq Prop) (S x2)) (S x2))
% Found (eq_ref0 (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found ((eq_ref Prop) (S x2)) as proof of (((eq Prop) (S x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b
% EOF
%------------------------------------------------------------------------------