%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV275^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n187.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 13:33:59 EDT 2014

% Result   : Timeout 300.10s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV275^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n187.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 08:42:06 CDT 2014
% % CPUTime  : 300.10 
% 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 0x261e9e0>, <kernel.Type object at 0x261e7e8>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (((ex ((a->Prop)->a)) (fun (Xc:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xc X))))))->((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz))))))))))) of role conjecture named cTHM550_pme
% Conjecture to prove = (((ex ((a->Prop)->a)) (fun (Xc:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xc X))))))->((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz))))))))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(((ex ((a->Prop)->a)) (fun (Xc:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xc X))))))->((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))))']
% Parameter a:Type.
% Trying to prove (((ex ((a->Prop)->a)) (fun (Xc:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xc X))))))->((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))))
% Found eta_expansion000:=(eta_expansion00 (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))):(((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) (fun (x:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x Xy) Xx))))->(((eq a) Xy) Xz))))))))))
% Found (eta_expansion00 (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) b)
% Found ((eta_expansion0 Prop) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) b)
% Found eq_ref00:=(eq_ref0 (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))):(((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz))))))))))
% Found (eq_ref0 (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) b)
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Found (fun (x00:((ex a) (fun (Xz:a)=> (X Xz))))=> x00) as proof of ((ex a) (fun (Xz:a)=> (X Xz)))
% Found (fun (x00:((ex a) (fun (Xz:a)=> (X Xz))))=> x00) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))))):(((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))))) (fun (x:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz))))))))))))
% Found (eta_expansion00 (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))))) b)
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Found (fun (x00:((ex a) (fun (Xz:a)=> (X Xz))))=> x00) as proof of ((ex a) (fun (Xz:a)=> (X Xz)))
% Found (fun (x00:((ex a) (fun (Xz:a)=> (X Xz))))=> x00) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Found (fun (x00:((ex a) (fun (Xz:a)=> (X Xz))))=> x00) as proof of ((ex a) (fun (Xz:a)=> (X Xz)))
% Found (fun (x00:((ex a) (fun (Xz:a)=> (X Xz))))=> x00) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz))))))))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz))))))))))))
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Found (fun (x00:((ex a) (fun (Xz:a)=> (X Xz))))=> x00) as proof of ((ex a) (fun (Xz:a)=> (X Xz)))
% Found (fun (x00:((ex a) (fun (Xz:a)=> (X Xz))))=> x00) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Found (fun (x00:((ex a) (fun (Xz:a)=> (X Xz))))=> x00) as proof of ((ex a) (fun (Xz:a)=> (X Xz)))
% Found (fun (x00:((ex a) (fun (Xz:a)=> (X Xz))))=> x00) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Found (fun (x00:((ex a) (fun (Xz:a)=> (X Xz))))=> x00) as proof of ((ex a) (fun (Xz:a)=> (X Xz)))
% Found (fun (x00:((ex a) (fun (Xz:a)=> (X Xz))))=> x00) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x1)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x1))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x1))
% Found (fun (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of ((P Xy)->(P x1))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of (((eq a) Xy) x1)
% Found x2:(P Xy)
% Instantiate: x1:=Xy:a
% Found (fun (x2:(P Xy))=> x2) as proof of (P x1)
% Found (fun (P:(a->Prop)) (x2:(P Xy))=> x2) as proof of ((P Xy)->(P x1))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop)) (x2:(P Xy))=> x2) as proof of (((eq a) Xy) x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x3)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found (fun (x01:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x3)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x1)
% Found x4:(P Xy)
% Instantiate: x3:=Xy:a
% Found (fun (x4:(P Xy))=> x4) as proof of (P x3)
% Found (fun (P:(a->Prop)) (x4:(P Xy))=> x4) as proof of ((P Xy)->(P x3))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop)) (x4:(P Xy))=> x4) as proof of (((eq a) Xy) x3)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x3))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x3))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x3))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x3))
% Found (fun (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of ((P Xy)->(P x3))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of (((eq a) Xy) x3)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x3))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x3))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x3))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x3))
% Found (fun (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of ((P Xy)->(P x3))
% Found (fun (x01:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))) (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of (((eq a) Xy) x3)
% Found x4:(P Xy)
% Instantiate: x3:=Xy:a
% Found (fun (x4:(P Xy))=> x4) as proof of (P x3)
% Found (fun (P:(a->Prop)) (x4:(P Xy))=> x4) as proof of ((P Xy)->(P x3))
% Found (fun (x01:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))) (P:(a->Prop)) (x4:(P Xy))=> x4) as proof of (((eq a) Xy) x3)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x3)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x1))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x1))
% Found (fun (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of ((P Xy)->(P x1))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of (((eq a) Xy) x1)
% Found x4:(P Xy)
% Instantiate: x1:=Xy:a
% Found (fun (x4:(P Xy))=> x4) as proof of (P x1)
% Found (fun (P:(a->Prop)) (x4:(P Xy))=> x4) as proof of ((P Xy)->(P x1))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop)) (x4:(P Xy))=> x4) as proof of (((eq a) Xy) x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x1)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x3))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x3))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x3))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x3))
% Found (fun (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of ((P Xy)->(P x3))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of (((eq a) Xy) x3)
% Found x4:(P Xy)
% Instantiate: x3:=Xy:a
% Found (fun (x4:(P Xy))=> x4) as proof of (P x3)
% Found (fun (P:(a->Prop)) (x4:(P Xy))=> x4) as proof of ((P Xy)->(P x3))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop)) (x4:(P Xy))=> x4) as proof of (((eq a) Xy) x3)
% Found x4:(P Xy)
% Instantiate: x1:=Xy:a
% Found (fun (x4:(P Xy))=> x4) as proof of (P x1)
% Found (fun (P:(a->Prop)) (x4:(P Xy))=> x4) as proof of ((P Xy)->(P x1))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop)) (x4:(P Xy))=> x4) as proof of (((eq a) Xy) x1)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x1))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x1))
% Found (fun (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of ((P Xy)->(P x1))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of (((eq a) Xy) x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found (fun (x3:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x1)
% Found (fun (x2:(X Xy)) (x3:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x1))
% Found (fun (x2:(X Xy)) (x3:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x1)))
% Found (and_rect00 (fun (x2:(X Xy)) (x3:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found ((and_rect0 (((eq a) Xy) x1)) (fun (x2:(X Xy)) (x3:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found (((fun (P:Type) (x2:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x2) x000)) (((eq a) Xy) x1)) (fun (x2:(X Xy)) (x3:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> (((fun (P:Type) (x2:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x2) x000)) (((eq a) Xy) x1)) (fun (x2:(X Xy)) (x3:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x1)
% Found x2:(X x1)
% Instantiate: x3:=x1:a
% Found x2 as proof of (X x3)
% Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq a) x1) Xy)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xy)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xy)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xy)
% Found (eq_sym000 ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found ((eq_sym00 Xy) ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found (((eq_sym0 x1) Xy) ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found ((((eq_sym a) x1) Xy) ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((((eq_sym a) x1) Xy) ((eq_ref a) x1))) as proof of (((eq a) Xy) x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x5)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x5)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x5)
% Found (fun (x01:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x5)
% Found x3:(X x2)
% Instantiate: x1:=x2:a
% Found x3 as proof of (X x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found (fun (x01:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x3)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x5))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x5))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x5))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x5))
% Found (fun (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of ((P Xy)->(P x5))
% Found (fun (x01:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))) (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of (((eq a) Xy) x5)
% Found x6:(P Xy)
% Instantiate: x5:=Xy:a
% Found (fun (x6:(P Xy))=> x6) as proof of (P x5)
% Found (fun (P:(a->Prop)) (x6:(P Xy))=> x6) as proof of ((P Xy)->(P x5))
% Found (fun (x01:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))) (P:(a->Prop)) (x6:(P Xy))=> x6) as proof of (((eq a) Xy) x5)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found (fun (x3:(X x2))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x1)
% Found (fun (x2:a) (x3:(X x2))=> ((eq_ref a) Xy)) as proof of ((X x2)->(((eq a) Xy) x1))
% Found (fun (x2:a) (x3:(X x2))=> ((eq_ref a) Xy)) as proof of (forall (x:a), ((X x)->(((eq a) Xy) x1)))
% Found (ex_ind00 (fun (x2:a) (x3:(X x2))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found ((ex_ind0 (((eq a) Xy) x1)) (fun (x2:a) (x3:(X x2))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found (((fun (P:Prop) (x2:(forall (x:a), ((X x)->P)))=> (((((ex_ind a) (fun (Xz:a)=> (X Xz))) P) x2) x00)) (((eq a) Xy) x1)) (fun (x2:a) (x3:(X x2))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> (((fun (P:Prop) (x2:(forall (x:a), ((X x)->P)))=> (((((ex_ind a) (fun (Xz:a)=> (X Xz))) P) x2) x00)) (((eq a) Xy) x1)) (fun (x2:a) (x3:(X x2))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x1)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x3))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x3))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x3))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x3))
% Found (fun (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of ((P Xy)->(P x3))
% Found (fun (x01:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))) (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of (((eq a) Xy) x3)
% Found x6:(P Xy)
% Instantiate: x3:=Xy:a
% Found (fun (x6:(P Xy))=> x6) as proof of (P x3)
% Found (fun (P:(a->Prop)) (x6:(P Xy))=> x6) as proof of ((P Xy)->(P x3))
% Found (fun (x01:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))) (P:(a->Prop)) (x6:(P Xy))=> x6) as proof of (((eq a) Xy) x3)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x5)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x5)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x5)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x5)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x5)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x5)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x5)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x5)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found (fun (x3:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x2 X)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x1)
% Found (fun (x2:((a->Prop)->a)) (x3:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x2 X)))))=> ((eq_ref a) Xy)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x2 X))))->(((eq a) Xy) x1))
% Found (fun (x2:((a->Prop)->a)) (x3:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x2 X)))))=> ((eq_ref a) Xy)) as proof of (forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->(((eq a) Xy) x1)))
% Found (ex_ind00 (fun (x2:((a->Prop)->a)) (x3:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x2 X)))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found ((ex_ind0 (((eq a) Xy) x1)) (fun (x2:((a->Prop)->a)) (x3:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x2 X)))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found (((fun (P:Prop) (x2:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xc:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xc X)))))) P) x2) x)) (((eq a) Xy) x1)) (fun (x2:((a->Prop)->a)) (x3:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x2 X)))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> (((fun (P:Prop) (x2:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xc:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xc X)))))) P) x2) x)) (((eq a) Xy) x1)) (fun (x2:((a->Prop)->a)) (x3:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x2 X)))))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x3)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found (fun (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x3)
% Found (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x3))
% Found (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x3)))
% Found (and_rect00 (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found ((and_rect0 (((eq a) Xy) x3)) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found (((fun (P:Type) (x4:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x4) x000)) (((eq a) Xy) x3)) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> (((fun (P:Type) (x4:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x4) x000)) (((eq a) Xy) x3)) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x3)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x3)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found (fun (x5:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x3)
% Found (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))->(((eq a) Xy) x3))
% Found (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((X Xy)->((forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))->(((eq a) Xy) x3)))
% Found (and_rect00 (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found ((and_rect0 (((eq a) Xy) x3)) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found (((fun (P:Type) (x4:((X Xy)->((forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))) P) x4) x01)) (((eq a) Xy) x3)) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found (fun (x01:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))))=> (((fun (P:Type) (x4:((X Xy)->((forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))) P) x4) x01)) (((eq a) Xy) x3)) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x3)
% Found eq_ref00:=(eq_ref0 x3):(((eq a) x3) x3)
% Found (eq_ref0 x3) as proof of (((eq a) x3) Xy)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xy)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xy)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xy)
% Found (eq_sym000 ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found ((eq_sym00 Xy) ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found (((eq_sym0 x3) Xy) ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found ((((eq_sym a) x3) Xy) ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((((eq_sym a) x3) Xy) ((eq_ref a) x3))) as proof of (((eq a) Xy) x3)
% Found eq_ref00:=(eq_ref0 x3):(((eq a) x3) x3)
% Found (eq_ref0 x3) as proof of (((eq a) x3) Xy)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xy)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xy)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xy)
% Found (eq_sym000 ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found ((eq_sym00 Xy) ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found (((eq_sym0 x3) Xy) ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found ((((eq_sym a) x3) Xy) ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found (fun (x01:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))))=> ((((eq_sym a) x3) Xy) ((eq_ref a) x3))) as proof of (((eq a) Xy) x3)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x5))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x5))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x5))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x5))
% Found (fun (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of ((P Xy)->(P x5))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of (((eq a) Xy) x5)
% Found x6:(P Xy)
% Instantiate: x5:=Xy:a
% Found (fun (x6:(P Xy))=> x6) as proof of (P x5)
% Found (fun (P:(a->Prop)) (x6:(P Xy))=> x6) as proof of ((P Xy)->(P x5))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop)) (x6:(P Xy))=> x6) as proof of (((eq a) Xy) x5)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x5))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x5))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x5))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x5))
% Found (fun (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of ((P Xy)->(P x5))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of (((eq a) Xy) x5)
% Found x6:(P Xy)
% Instantiate: x5:=Xy:a
% Found (fun (x6:(P Xy))=> x6) as proof of (P x5)
% Found (fun (P:(a->Prop)) (x6:(P Xy))=> x6) as proof of ((P Xy)->(P x5))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop)) (x6:(P Xy))=> x6) as proof of (((eq a) Xy) x5)
% Found x4:(X x3)
% Instantiate: x5:=x3:a
% Found x4 as proof of (X x5)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found (fun (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x1)
% Found (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x1))
% Found (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x1)))
% Found (and_rect00 (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found ((and_rect0 (((eq a) Xy) x1)) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found (((fun (P:Type) (x4:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x4) x000)) (((eq a) Xy) x1)) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> (((fun (P:Type) (x4:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x4) x000)) (((eq a) Xy) x1)) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x1)
% Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq a) x1) Xy)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xy)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xy)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xy)
% Found (eq_sym000 ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found ((eq_sym00 Xy) ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found (((eq_sym0 x1) Xy) ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found ((((eq_sym a) x1) Xy) ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((((eq_sym a) x1) Xy) ((eq_ref a) x1))) as proof of (((eq a) Xy) x1)
% Found x6:(P Xy)
% Instantiate: x3:=Xy:a
% Found (fun (x6:(P Xy))=> x6) as proof of (P x3)
% Found (fun (P:(a->Prop)) (x6:(P Xy))=> x6) as proof of ((P Xy)->(P x3))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop)) (x6:(P Xy))=> x6) as proof of (((eq a) Xy) x3)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x3))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x3))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x3))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x3))
% Found (fun (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of ((P Xy)->(P x3))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of (((eq a) Xy) x3)
% Found x6:(P Xy)
% Instantiate: x3:=Xy:a
% Found (fun (x6:(P Xy))=> x6) as proof of (P x3)
% Found (fun (P:(a->Prop)) (x6:(P Xy))=> x6) as proof of ((P Xy)->(P x3))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop)) (x6:(P Xy))=> x6) as proof of (((eq a) Xy) x3)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x3))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x3))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x3))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x3))
% Found (fun (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of ((P Xy)->(P x3))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of (((eq a) Xy) x3)
% Found x3:(X x2)
% Instantiate: x1:=x2:a
% Found x3 as proof of (X x1)
% Found x5:(X x4)
% Instantiate: x3:=x4:a
% Found x5 as proof of (X x3)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found (fun (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x3)
% Found (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x3))
% Found (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x3)))
% Found (and_rect00 (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found ((and_rect0 (((eq a) Xy) x3)) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found (((fun (P:Type) (x4:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x4) x000)) (((eq a) Xy) x3)) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> (((fun (P:Type) (x4:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x4) x000)) (((eq a) Xy) x3)) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x3)
% Found eq_ref00:=(eq_ref0 x3):(((eq a) x3) x3)
% Found (eq_ref0 x3) as proof of (((eq a) x3) Xy)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xy)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xy)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xy)
% Found (eq_sym000 ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found ((eq_sym00 Xy) ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found (((eq_sym0 x3) Xy) ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found ((((eq_sym a) x3) Xy) ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((((eq_sym a) x3) Xy) ((eq_ref a) x3))) as proof of (((eq a) Xy) x3)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x1))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x1))
% Found (fun (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of ((P Xy)->(P x1))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of (((eq a) Xy) x1)
% Found x6:(P Xy)
% Instantiate: x1:=Xy:a
% Found (fun (x6:(P Xy))=> x6) as proof of (P x1)
% Found (fun (P:(a->Prop)) (x6:(P Xy))=> x6) as proof of ((P Xy)->(P x1))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop)) (x6:(P Xy))=> x6) as proof of (((eq a) Xy) x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found (fun (x5:(X x4))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x3)
% Found (fun (x4:a) (x5:(X x4))=> ((eq_ref a) Xy)) as proof of ((X x4)->(((eq a) Xy) x3))
% Found (fun (x4:a) (x5:(X x4))=> ((eq_ref a) Xy)) as proof of (forall (x:a), ((X x)->(((eq a) Xy) x3)))
% Found (ex_ind10 (fun (x4:a) (x5:(X x4))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found ((ex_ind1 (((eq a) Xy) x3)) (fun (x4:a) (x5:(X x4))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found (((fun (P:Prop) (x4:(forall (x:a), ((X x)->P)))=> (((((ex_ind a) (fun (Xz:a)=> (X Xz))) P) x4) x00)) (((eq a) Xy) x3)) (fun (x4:a) (x5:(X x4))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found (fun (x01:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))))=> (((fun (P:Prop) (x4:(forall (x:a), ((X x)->P)))=> (((((ex_ind a) (fun (Xz:a)=> (X Xz))) P) x4) x00)) (((eq a) Xy) x3)) (fun (x4:a) (x5:(X x4))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x3)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x1))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x1))
% Found (fun (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of ((P Xy)->(P x1))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of (((eq a) Xy) x1)
% Found x6:(P Xy)
% Instantiate: x1:=Xy:a
% Found (fun (x6:(P Xy))=> x6) as proof of (P x1)
% Found (fun (P:(a->Prop)) (x6:(P Xy))=> x6) as proof of ((P Xy)->(P x1))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop)) (x6:(P Xy))=> x6) as proof of (((eq a) Xy) x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found (fun (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x1)
% Found (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x1))
% Found (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x1)))
% Found (and_rect00 (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found ((and_rect0 (((eq a) Xy) x1)) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found (((fun (P:Type) (x4:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x4) x000)) (((eq a) Xy) x1)) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> (((fun (P:Type) (x4:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x4) x000)) (((eq a) Xy) x1)) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x1)
% Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq a) x1) Xy)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xy)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xy)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xy)
% Found (eq_sym000 ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found ((eq_sym00 Xy) ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found (((eq_sym0 x1) Xy) ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found ((((eq_sym a) x1) Xy) ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((((eq_sym a) x1) Xy) ((eq_ref a) x1))) as proof of (((eq a) Xy) x1)
% Found x2:(X x1)
% Instantiate: x5:=x1:a
% Found x2 as proof of (X x5)
% Found x4:(X x3)
% Instantiate: x5:=x3:a
% Found x4 as proof of (X x5)
% Found x2:(X x1)
% Instantiate: x3:=x1:a
% Found x2 as proof of (X x3)
% Found x5:(X x4)
% Instantiate: x3:=x4:a
% Found x5 as proof of (X x3)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->(a->Prop))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz))))))))))
% Found ((eq_ref ((a->(a->Prop))->Prop)) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz))))))))))
% Found ((eq_ref ((a->(a->Prop))->Prop)) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz))))))))))
% Found ((eq_ref ((a->(a->Prop))->Prop)) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz))))))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq ((a->(a->Prop))->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found (fun (x5:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x4 X)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x3)
% Found (fun (x4:((a->Prop)->a)) (x5:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x4 X)))))=> ((eq_ref a) Xy)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x4 X))))->(((eq a) Xy) x3))
% Found (fun (x4:((a->Prop)->a)) (x5:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x4 X)))))=> ((eq_ref a) Xy)) as proof of (forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->(((eq a) Xy) x3)))
% Found (ex_ind10 (fun (x4:((a->Prop)->a)) (x5:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x4 X)))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found ((ex_ind1 (((eq a) Xy) x3)) (fun (x4:((a->Prop)->a)) (x5:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x4 X)))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found (((fun (P:Prop) (x4:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xc:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xc X)))))) P) x4) x)) (((eq a) Xy) x3)) (fun (x4:((a->Prop)->a)) (x5:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x4 X)))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> (((fun (P:Prop) (x4:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xc:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xc X)))))) P) x4) x)) (((eq a) Xy) x3)) (fun (x4:((a->Prop)->a)) (x5:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x4 X)))))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x3)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found (fun (x5:(X x4))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x3)
% Found (fun (x4:a) (x5:(X x4))=> ((eq_ref a) Xy)) as proof of ((X x4)->(((eq a) Xy) x3))
% Found (fun (x4:a) (x5:(X x4))=> ((eq_ref a) Xy)) as proof of (forall (x:a), ((X x)->(((eq a) Xy) x3)))
% Found (ex_ind10 (fun (x4:a) (x5:(X x4))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found ((ex_ind1 (((eq a) Xy) x3)) (fun (x4:a) (x5:(X x4))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found (((fun (P:Prop) (x4:(forall (x:a), ((X x)->P)))=> (((((ex_ind a) (fun (Xz:a)=> (X Xz))) P) x4) x00)) (((eq a) Xy) x3)) (fun (x4:a) (x5:(X x4))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> (((fun (P:Prop) (x4:(forall (x:a), ((X x)->P)))=> (((((ex_ind a) (fun (Xz:a)=> (X Xz))) P) x4) x00)) (((eq a) Xy) x3)) (fun (x4:a) (x5:(X x4))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x3)
% Found x3:(X x2)
% Instantiate: x1:=x2:a
% Found x3 as proof of (X x1)
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: b:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P b)
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (eq_ref0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b)
% Found x5:(X x4)
% Instantiate: x1:=x4:a
% Found x5 as proof of (X x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x5)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x5)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x5)
% Found (fun (x7:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x5)
% Found (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))->(((eq a) Xy) x5))
% Found (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((X Xy)->((forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))->(((eq a) Xy) x5)))
% Found (and_rect00 (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x5)
% Found ((and_rect0 (((eq a) Xy) x5)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x5)
% Found (((fun (P:Type) (x6:((X Xy)->((forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))) P) x6) x01)) (((eq a) Xy) x5)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x5)
% Found (fun (x01:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))))=> (((fun (P:Type) (x6:((X Xy)->((forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))) P) x6) x01)) (((eq a) Xy) x5)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x5)
% Found eq_ref00:=(eq_ref0 x5):(((eq a) x5) x5)
% Found (eq_ref0 x5) as proof of (((eq a) x5) Xy)
% Found ((eq_ref a) x5) as proof of (((eq a) x5) Xy)
% Found ((eq_ref a) x5) as proof of (((eq a) x5) Xy)
% Found ((eq_ref a) x5) as proof of (((eq a) x5) Xy)
% Found (eq_sym000 ((eq_ref a) x5)) as proof of (((eq a) Xy) x5)
% Found ((eq_sym00 Xy) ((eq_ref a) x5)) as proof of (((eq a) Xy) x5)
% Found (((eq_sym0 x5) Xy) ((eq_ref a) x5)) as proof of (((eq a) Xy) x5)
% Found ((((eq_sym a) x5) Xy) ((eq_ref a) x5)) as proof of (((eq a) Xy) x5)
% Found (fun (x01:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))))=> ((((eq_sym a) x5) Xy) ((eq_ref a) x5))) as proof of (((eq a) Xy) x5)
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: f:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P f)
% 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)) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x0 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x))))))
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: f:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P f)
% 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)) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x0 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x))))))
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found (fun (x5:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x4 X)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x1)
% Found (fun (x4:((a->Prop)->a)) (x5:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x4 X)))))=> ((eq_ref a) Xy)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x4 X))))->(((eq a) Xy) x1))
% Found (fun (x4:((a->Prop)->a)) (x5:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x4 X)))))=> ((eq_ref a) Xy)) as proof of (forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->(((eq a) Xy) x1)))
% Found (ex_ind10 (fun (x4:((a->Prop)->a)) (x5:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x4 X)))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found ((ex_ind1 (((eq a) Xy) x1)) (fun (x4:((a->Prop)->a)) (x5:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x4 X)))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found (((fun (P:Prop) (x4:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xc:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xc X)))))) P) x4) x)) (((eq a) Xy) x1)) (fun (x4:((a->Prop)->a)) (x5:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x4 X)))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> (((fun (P:Prop) (x4:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xc:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xc X)))))) P) x4) x)) (((eq a) Xy) x1)) (fun (x4:((a->Prop)->a)) (x5:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x4 X)))))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x1)
% Found x5:(X x4)
% Instantiate: x3:=x4:a
% Found x5 as proof of (X x3)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found (fun (x5:(X x4))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x1)
% Found (fun (x4:a) (x5:(X x4))=> ((eq_ref a) Xy)) as proof of ((X x4)->(((eq a) Xy) x1))
% Found (fun (x4:a) (x5:(X x4))=> ((eq_ref a) Xy)) as proof of (forall (x:a), ((X x)->(((eq a) Xy) x1)))
% Found (ex_ind10 (fun (x4:a) (x5:(X x4))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found ((ex_ind1 (((eq a) Xy) x1)) (fun (x4:a) (x5:(X x4))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found (((fun (P:Prop) (x4:(forall (x:a), ((X x)->P)))=> (((((ex_ind a) (fun (Xz:a)=> (X Xz))) P) x4) x00)) (((eq a) Xy) x1)) (fun (x4:a) (x5:(X x4))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> (((fun (P:Prop) (x4:(forall (x:a), ((X x)->P)))=> (((((ex_ind a) (fun (Xz:a)=> (X Xz))) P) x4) x00)) (((eq a) Xy) x1)) (fun (x4:a) (x5:(X x4))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x1)
% Found x3:(X x2)
% Instantiate: x1:=x2:a
% Found (fun (x3:(X x2))=> x3) as proof of (X x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found (fun (x7:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x3)
% Found (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))->(((eq a) Xy) x3))
% Found (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((X Xy)->((forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))->(((eq a) Xy) x3)))
% Found (and_rect00 (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found ((and_rect0 (((eq a) Xy) x3)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found (((fun (P:Type) (x6:((X Xy)->((forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))) P) x6) x01)) (((eq a) Xy) x3)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found (fun (x01:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))))=> (((fun (P:Type) (x6:((X Xy)->((forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))) P) x6) x01)) (((eq a) Xy) x3)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x3)
% Found eq_ref00:=(eq_ref0 x3):(((eq a) x3) x3)
% Found (eq_ref0 x3) as proof of (((eq a) x3) Xy)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xy)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xy)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xy)
% Found (eq_sym000 ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found ((eq_sym00 Xy) ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found (((eq_sym0 x3) Xy) ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found ((((eq_sym a) x3) Xy) ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found (fun (x01:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))))=> ((((eq_sym a) x3) Xy) ((eq_ref a) x3))) as proof of (((eq a) Xy) x3)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> (X Xz))):(((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) (fun (x:a)=> (X x)))
% Found (eta_expansion_dep00 (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->(a->Prop))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz))))))))))
% Found ((eq_ref ((a->(a->Prop))->Prop)) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz))))))))))
% Found ((eq_ref ((a->(a->Prop))->Prop)) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz))))))))))
% Found ((eq_ref ((a->(a->Prop))->Prop)) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz))))))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq ((a->(a->Prop))->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x5)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x5)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x5)
% Found (fun (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x5)
% Found (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x5))
% Found (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x5)))
% Found (and_rect00 (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x5)
% Found ((and_rect0 (((eq a) Xy) x5)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x5)
% Found (((fun (P:Type) (x6:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x6) x000)) (((eq a) Xy) x5)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x5)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> (((fun (P:Type) (x6:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x6) x000)) (((eq a) Xy) x5)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x5)
% Found eq_ref00:=(eq_ref0 x5):(((eq a) x5) x5)
% Found (eq_ref0 x5) as proof of (((eq a) x5) Xy)
% Found ((eq_ref a) x5) as proof of (((eq a) x5) Xy)
% Found ((eq_ref a) x5) as proof of (((eq a) x5) Xy)
% Found ((eq_ref a) x5) as proof of (((eq a) x5) Xy)
% Found (eq_sym000 ((eq_ref a) x5)) as proof of (((eq a) Xy) x5)
% Found ((eq_sym00 Xy) ((eq_ref a) x5)) as proof of (((eq a) Xy) x5)
% Found (((eq_sym0 x5) Xy) ((eq_ref a) x5)) as proof of (((eq a) Xy) x5)
% Found ((((eq_sym a) x5) Xy) ((eq_ref a) x5)) as proof of (((eq a) Xy) x5)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((((eq_sym a) x5) Xy) ((eq_ref a) x5))) as proof of (((eq a) Xy) x5)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x5)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x5)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x5)
% Found (fun (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x5)
% Found (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x5))
% Found (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x5)))
% Found (and_rect00 (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x5)
% Found ((and_rect0 (((eq a) Xy) x5)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x5)
% Found (((fun (P:Type) (x6:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x6) x000)) (((eq a) Xy) x5)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x5)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> (((fun (P:Type) (x6:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x6) x000)) (((eq a) Xy) x5)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x5)
% Found eq_ref00:=(eq_ref0 x5):(((eq a) x5) x5)
% Found (eq_ref0 x5) as proof of (((eq a) x5) Xy)
% Found ((eq_ref a) x5) as proof of (((eq a) x5) Xy)
% Found ((eq_ref a) x5) as proof of (((eq a) x5) Xy)
% Found ((eq_ref a) x5) as proof of (((eq a) x5) Xy)
% Found (eq_sym000 ((eq_ref a) x5)) as proof of (((eq a) Xy) x5)
% Found ((eq_sym00 Xy) ((eq_ref a) x5)) as proof of (((eq a) Xy) x5)
% Found (((eq_sym0 x5) Xy) ((eq_ref a) x5)) as proof of (((eq a) Xy) x5)
% Found ((((eq_sym a) x5) Xy) ((eq_ref a) x5)) as proof of (((eq a) Xy) x5)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((((eq_sym a) x5) Xy) ((eq_ref a) x5))) as proof of (((eq a) Xy) x5)
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: b:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P b)
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (eq_ref0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b)
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: b:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P b)
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (eq_ref0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) b)
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Found (fun (x00:((ex a) (fun (Xz:a)=> (X Xz))))=> x00) as proof of ((ex a) (fun (Xz:a)=> (X Xz)))
% Found (fun (x00:((ex a) (fun (Xz:a)=> (X Xz))))=> x00) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> (X Xz))):(((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) (fun (x:a)=> (X x)))
% Found (eta_expansion00 (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) 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 (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% 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_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 (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz))))))))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz))))))))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz))))))))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz))))))))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz))))))))))))
% Found x2:(X x1)
% Instantiate: x3:=x1:a
% Found x2 as proof of (X x3)
% Found x5:(X x4)
% Instantiate: x3:=x4:a
% Found x5 as proof of (X x3)
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: f:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P 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)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x0 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x0 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x0 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x0 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x0 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x))))))
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: f:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P 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)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x0 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x0 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x0 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x0 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x0 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x))))))
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found (fun (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x3)
% Found (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x3))
% Found (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x3)))
% Found (and_rect00 (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found ((and_rect0 (((eq a) Xy) x3)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found (((fun (P:Type) (x6:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x6) x000)) (((eq a) Xy) x3)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> (((fun (P:Type) (x6:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x6) x000)) (((eq a) Xy) x3)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x3)
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: f:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P 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)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x2 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x2 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x2 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x2 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x2 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x))))))
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: f:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P 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)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x2 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x2 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x2 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x2 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x2 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x))))))
% Found eq_ref00:=(eq_ref0 x3):(((eq a) x3) x3)
% Found (eq_ref0 x3) as proof of (((eq a) x3) Xy)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xy)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xy)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xy)
% Found (eq_sym000 ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found ((eq_sym00 Xy) ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found (((eq_sym0 x3) Xy) ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found ((((eq_sym a) x3) Xy) ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((((eq_sym a) x3) Xy) ((eq_ref a) x3))) as proof of (((eq a) Xy) x3)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found (fun (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x3)
% Found (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x3))
% Found (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x3)))
% Found (and_rect00 (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found ((and_rect0 (((eq a) Xy) x3)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found (((fun (P:Type) (x6:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x6) x000)) (((eq a) Xy) x3)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> (((fun (P:Type) (x6:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x6) x000)) (((eq a) Xy) x3)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x3)
% Found eq_ref00:=(eq_ref0 x3):(((eq a) x3) x3)
% Found (eq_ref0 x3) as proof of (((eq a) x3) Xy)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xy)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xy)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xy)
% Found (eq_sym000 ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found ((eq_sym00 Xy) ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found (((eq_sym0 x3) Xy) ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found ((((eq_sym a) x3) Xy) ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((((eq_sym a) x3) Xy) ((eq_ref a) x3))) as proof of (((eq a) Xy) x3)
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: b:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P b)
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (eq_ref0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b)
% Found x3:(X x2)
% Instantiate: x1:=x2:a
% Found x3 as proof of (X x1)
% Found x5:(X x4)
% Instantiate: x1:=x4:a
% Found x5 as proof of (X x1)
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> (X Xz))):(((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) (fun (Xz:a)=> (X Xz)))
% Found (eq_ref0 (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) 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 (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found x4:(X Xy)
% Instantiate: x3:=Xy:a
% Found x4 as proof of (X x3)
% Found (x50 x4) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy) x3)))
% Found ((x5 x3) x4) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy) x3)))
% Found ((x5 x3) x4) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy) x3)))
% Found (fun (x5:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((x5 x3) x4)) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy) x3)))
% Found (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((x5 x3) x4)) as proof of ((forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))->((ex a) (fun (Xt:a)=> (((eq a) Xy) x3))))
% Found (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((x5 x3) x4)) as proof of ((X Xy)->((forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))->((ex a) (fun (Xt:a)=> (((eq a) Xy) x3)))))
% Found (and_rect00 (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((x5 x3) x4))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy) x3)))
% Found ((and_rect0 ((ex a) (fun (Xt:a)=> (((eq a) Xy) x3)))) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((x5 x3) x4))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy) x3)))
% Found (((fun (P:Type) (x4:((X Xy)->((forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))) P) x4) x01)) ((ex a) (fun (Xt:a)=> (((eq a) Xy) x3)))) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((x5 x3) x4))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy) x3)))
% Found (((fun (P:Type) (x4:((X Xy)->((forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))) P) x4) x01)) ((ex a) (fun (Xt:a)=> (((eq a) Xy) x3)))) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((x5 x3) x4))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy) x3)))
% Found (x10 (((fun (P:Type) (x4:((X Xy)->((forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))) P) x4) x01)) ((ex a) (fun (Xt:a)=> (((eq a) Xy) x3)))) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((x5 x3) x4)))) as proof of (((eq a) Xy) x3)
% Found ((x1 (fun (x4:a)=> (((eq a) Xy) x3))) (((fun (P:Type) (x4:((X Xy)->((forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))) P) x4) x01)) ((ex a) (fun (Xt:a)=> (((eq a) Xy) x3)))) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((x5 x3) x4)))) as proof of (((eq a) Xy) x3)
% Found (fun (x01:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))))=> ((x1 (fun (x4:a)=> (((eq a) Xy) x3))) (((fun (P:Type) (x4:((X Xy)->((forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))) P) x4) x01)) ((ex a) (fun (Xt:a)=> (((eq a) Xy) x3)))) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((x5 x3) x4))))) as proof of (((eq a) Xy) x3)
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))->(P0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eq_ref0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))->(P0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eq_ref0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found (fun (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x1)
% Found (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x1))
% Found (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x1)))
% Found (and_rect00 (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found ((and_rect0 (((eq a) Xy) x1)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found (((fun (P:Type) (x6:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x6) x000)) (((eq a) Xy) x1)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> (((fun (P:Type) (x6:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x6) x000)) (((eq a) Xy) x1)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x1)))
% Found ((eq_ref a) Xy) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x1)))
% Found ((eq_ref a) Xy) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x1)))
% Found ((eq_ref a) Xy) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x1)))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found (fun (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x1)
% Found (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x1))
% Found (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x1)))
% Found (and_rect00 (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found ((and_rect0 (((eq a) Xy) x1)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found (((fun (P:Type) (x6:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x6) x000)) (((eq a) Xy) x1)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> (((fun (P:Type) (x6:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x6) x000)) (((eq a) Xy) x1)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x1)
% Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq a) x1) Xy)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xy)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xy)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xy)
% Found (eq_sym000 ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found ((eq_sym00 Xy) ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found (((eq_sym0 x1) Xy) ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found ((((eq_sym a) x1) Xy) ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((((eq_sym a) x1) Xy) ((eq_ref a) x1))) as proof of (((eq a) Xy) x1)
% Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq a) x1) Xy)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xy)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xy)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xy)
% Found (eq_sym000 ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found ((eq_sym00 Xy) ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found (((eq_sym0 x1) Xy) ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found ((((eq_sym a) x1) Xy) ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((((eq_sym a) x1) Xy) ((eq_ref a) x1))) as proof of (((eq a) Xy) x1)
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: b:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:a)=> ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))):(((eq (a->Prop)) (fun (Xt:a)=> ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))) (fun (x:a)=> ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))
% Found (eta_expansion_dep00 (fun (Xt:a)=> ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))) b)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) (fun (Xt:a)=> ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xt:a)=> ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xt:a)=> ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xt:a)=> ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))) b)
% Found eq_ref00:=(eq_ref0 (fun (Xt:a)=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))))):(((eq (a->Prop)) (fun (Xt:a)=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))))) (fun (Xt:a)=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))))
% Found (eq_ref0 (fun (Xt:a)=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))))) b)
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: f:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P 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)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x0 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x0 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x0 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x0 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x0 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x))))))
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: f:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P 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)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x0 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x0 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x0 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x0 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x0 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x))))))
% Found x5:(X x4)
% Instantiate: x3:=x4:a
% Found (fun (x5:(X x4))=> x5) as proof of (X x3)
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:a)=> (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))):(((eq (a->Prop)) (fun (Xt:a)=> (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))) (fun (x:a)=> (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found (eta_expansion00 (fun (Xt:a)=> (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xt:a)=> (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))) b)
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (eq_ref0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> (X Xz)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> (X Xz)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> (X Xz)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> (X Xz)))
% 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 (Xz:a)=> (X Xz)))
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> (X Xz)))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> (X Xz)))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> (X Xz)))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> (X Xz)))
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (eq_ref0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b)
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: f:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P f)
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: f:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P 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) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))
% 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) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))
% Found x2:(P Xy)
% Instantiate: x1:=Xy:a
% Found (fun (x2:(P Xy))=> x2) as proof of (P x1)
% Found (fun (P:(a->Prop)) (x2:(P Xy))=> x2) as proof of ((P Xy)->(P x1))
% Found (fun (P:(a->Prop)) (x2:(P Xy))=> x2) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x1)))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop)) (x2:(P Xy))=> x2) as proof of (((eq a) Xy) x1)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x1))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x1))
% Found (fun (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of ((P Xy)->(P x1))
% Found (fun (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x1)))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of (((eq a) Xy) x1)
% Found x4:(X Xy)
% Instantiate: x3:=Xy:a
% Found x4 as proof of (X x3)
% Found (x50 x4) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy) x3)))
% Found ((x5 x3) x4) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy) x3)))
% Found ((x5 x3) x4) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy) x3)))
% Found (fun (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((x5 x3) x4)) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy) x3)))
% Found (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((x5 x3) x4)) as proof of ((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->((ex a) (fun (Xt:a)=> (((eq a) Xy) x3))))
% Found (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((x5 x3) x4)) as proof of ((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->((ex a) (fun (Xt:a)=> (((eq a) Xy) x3)))))
% Found (and_rect00 (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((x5 x3) x4))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy) x3)))
% Found ((and_rect0 ((ex a) (fun (Xt:a)=> (((eq a) Xy) x3)))) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((x5 x3) x4))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy) x3)))
% Found (((fun (P:Type) (x4:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x4) x000)) ((ex a) (fun (Xt:a)=> (((eq a) Xy) x3)))) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((x5 x3) x4))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy) x3)))
% Found (((fun (P:Type) (x4:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x4) x000)) ((ex a) (fun (Xt:a)=> (((eq a) Xy) x3)))) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((x5 x3) x4))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy) x3)))
% Found (x20 (((fun (P:Type) (x4:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x4) x000)) ((ex a) (fun (Xt:a)=> (((eq a) Xy) x3)))) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((x5 x3) x4)))) as proof of (((eq a) Xy) x3)
% Found ((x2 (fun (x4:a)=> (((eq a) Xy) x3))) (((fun (P:Type) (x4:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x4) x000)) ((ex a) (fun (Xt:a)=> (((eq a) Xy) x3)))) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((x5 x3) x4)))) as proof of (((eq a) Xy) x3)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((x2 (fun (x4:a)=> (((eq a) Xy) x3))) (((fun (P:Type) (x4:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x4) x000)) ((ex a) (fun (Xt:a)=> (((eq a) Xy) x3)))) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((x5 x3) x4))))) as proof of (((eq a) Xy) x3)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))->(P0 (fun (x:a)=> ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x0 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x)))))))
% Found (eta_expansion_dep000 P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eta_expansion_dep00 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))->(P0 (fun (x:a)=> ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x0 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x)))))))
% Found (eta_expansion_dep000 P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eta_expansion_dep00 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))->(P0 (fun (x:a)=> ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x0 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x)))))))
% Found (eta_expansion_dep000 P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eta_expansion_dep00 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))->(P0 (fun (x:a)=> ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x0 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x)))))))
% Found (eta_expansion_dep000 P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eta_expansion_dep00 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Found (fun (x00:((ex a) (fun (Xz:a)=> (X Xz))))=> x00) as proof of ((ex a) (fun (Xz:a)=> (X Xz)))
% Found (fun (x00:((ex a) (fun (Xz:a)=> (X Xz))))=> x00) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> (X Xz))):(((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) (fun (x:a)=> (X x)))
% Found (eta_expansion00 (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: b:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:a)=> ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))):(((eq (a->Prop)) (fun (Xt:a)=> ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))) (fun (x:a)=> ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))))
% Found (eta_expansion_dep00 (fun (Xt:a)=> ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))) b)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) (fun (Xt:a)=> ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xt:a)=> ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xt:a)=> ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xt:a)=> ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))) b)
% Found eq_ref00:=(eq_ref0 (X x1)):(((eq Prop) (X x1)) (X x1))
% Found (eq_ref0 (X x1)) as proof of (((eq Prop) (X x1)) b)
% Found ((eq_ref Prop) (X x1)) as proof of (((eq Prop) (X x1)) b)
% Found ((eq_ref Prop) (X x1)) as proof of (((eq Prop) (X x1)) b)
% Found ((eq_ref Prop) (X x1)) as proof of (((eq Prop) (X x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found eq_ref00:=(eq_ref0 (X x1)):(((eq Prop) (X x1)) (X x1))
% Found (eq_ref0 (X x1)) as proof of (((eq Prop) (X x1)) b)
% Found ((eq_ref Prop) (X x1)) as proof of (((eq Prop) (X x1)) b)
% Found ((eq_ref Prop) (X x1)) as proof of (((eq Prop) (X x1)) b)
% Found ((eq_ref Prop) (X x1)) as proof of (((eq Prop) (X x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found eq_ref00:=(eq_ref0 (X x1)):(((eq Prop) (X x1)) (X x1))
% Found (eq_ref0 (X x1)) as proof of (((eq Prop) (X x1)) b)
% Found ((eq_ref Prop) (X x1)) as proof of (((eq Prop) (X x1)) b)
% Found ((eq_ref Prop) (X x1)) as proof of (((eq Prop) (X x1)) b)
% Found ((eq_ref Prop) (X x1)) as proof of (((eq Prop) (X x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found eq_ref00:=(eq_ref0 (X x1)):(((eq Prop) (X x1)) (X x1))
% Found (eq_ref0 (X x1)) as proof of (((eq Prop) (X x1)) b)
% Found ((eq_ref Prop) (X x1)) as proof of (((eq Prop) (X x1)) b)
% Found ((eq_ref Prop) (X x1)) as proof of (((eq Prop) (X x1)) b)
% Found ((eq_ref Prop) (X x1)) as proof of (((eq Prop) (X x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% 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)) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))))
% 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)) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))))
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: b:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) (fun (x:a)=> ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x2 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x))))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) b)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) b)
% Found x4:(X Xy)
% Instantiate: x1:=Xy:a
% Found x4 as proof of (X x1)
% Found (x50 x4) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy) x1)))
% Found ((x5 x1) x4) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy) x1)))
% Found ((x5 x1) x4) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy) x1)))
% Found (fun (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((x5 x1) x4)) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy) x1)))
% Found (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((x5 x1) x4)) as proof of ((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->((ex a) (fun (Xt:a)=> (((eq a) Xy) x1))))
% Found (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((x5 x1) x4)) as proof of ((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->((ex a) (fun (Xt:a)=> (((eq a) Xy) x1)))))
% Found (and_rect00 (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((x5 x1) x4))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy) x1)))
% Found ((and_rect0 ((ex a) (fun (Xt:a)=> (((eq a) Xy) x1)))) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((x5 x1) x4))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy) x1)))
% Found (((fun (P:Type) (x4:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x4) x000)) ((ex a) (fun (Xt:a)=> (((eq a) Xy) x1)))) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((x5 x1) x4))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy) x1)))
% Found (((fun (P:Type) (x4:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x4) x000)) ((ex a) (fun (Xt:a)=> (((eq a) Xy) x1)))) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((x5 x1) x4))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy) x1)))
% Found (x30 (((fun (P:Type) (x4:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x4) x000)) ((ex a) (fun (Xt:a)=> (((eq a) Xy) x1)))) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((x5 x1) x4)))) as proof of (((eq a) Xy) x1)
% Found ((x3 (fun (x4:a)=> (((eq a) Xy) x1))) (((fun (P:Type) (x4:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x4) x000)) ((ex a) (fun (Xt:a)=> (((eq a) Xy) x1)))) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((x5 x1) x4)))) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((x3 (fun (x4:a)=> (((eq a) Xy) x1))) (((fun (P:Type) (x4:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x4) x000)) ((ex a) (fun (Xt:a)=> (((eq a) Xy) x1)))) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((x5 x1) x4))))) as proof of (((eq a) Xy) x1)
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: f:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P f)
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: f:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P 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) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))))
% 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) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))))
% 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) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% 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) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: b:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:a)=> (forall (x30:a), ((X x30)->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))))):(((eq (a->Prop)) (fun (Xt:a)=> (forall (x30:a), ((X x30)->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))))) (fun (x:a)=> (forall (x30:a), ((X x30)->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))))
% Found (eta_expansion_dep00 (fun (Xt:a)=> (forall (x30:a), ((X x30)->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (forall (x30:a), ((X x30)->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))))) b)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) (fun (Xt:a)=> (forall (x30:a), ((X x30)->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (forall (x30:a), ((X x30)->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))))) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xt:a)=> (forall (x30:a), ((X x30)->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (forall (x30:a), ((X x30)->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))))) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xt:a)=> (forall (x30:a), ((X x30)->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (forall (x30:a), ((X x30)->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))))) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xt:a)=> (forall (x30:a), ((X x30)->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (forall (x30:a), ((X x30)->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))))) b)
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: f:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P f)
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: f:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P f)
% Found x20:(P0 (X x1))
% Found (fun (x20:(P0 (X x1)))=> x20) as proof of (P0 (X x1))
% Found (fun (x20:(P0 (X x1)))=> x20) as proof of (P1 (X x1))
% Found x20:(P0 (X x1))
% Found (fun (x20:(P0 (X x1)))=> x20) as proof of (P0 (X x1))
% Found (fun (x20:(P0 (X x1)))=> x20) as proof of (P1 (X x1))
% Found x20:(P0 (X x1))
% Found (fun (x20:(P0 (X x1)))=> x20) as proof of (P0 (X x1))
% Found (fun (x20:(P0 (X x1)))=> x20) as proof of (P1 (X x1))
% Found x20:(P0 (X x1))
% Found (fun (x20:(P0 (X x1)))=> x20) as proof of (P0 (X x1))
% Found (fun (x20:(P0 (X x1)))=> x20) as proof of (P1 (X x1))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((and ((and (X x5)) (forall (Xx:a), ((X Xx)->((x2 x5) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x5)))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((and (X x5)) (forall (Xx:a), ((X Xx)->((x2 x5) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x5)))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((and (X x5)) (forall (Xx:a), ((X Xx)->((x2 x5) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x5)))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((and ((and (X x5)) (forall (Xx:a), ((X Xx)->((x2 x5) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x5)))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x2 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x))))))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((and ((and (X x5)) (forall (Xx:a), ((X Xx)->((x2 x5) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x5)))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((and (X x5)) (forall (Xx:a), ((X Xx)->((x2 x5) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x5)))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((and (X x5)) (forall (Xx:a), ((X Xx)->((x2 x5) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x5)))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((and ((and (X x5)) (forall (Xx:a), ((X Xx)->((x2 x5) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x5)))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x2 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x))))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found (((eq_trans000 (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found ((((eq_trans00 (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))))) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq Prop) (f x0)) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found (((((eq_trans0 (f x0)) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))))) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq Prop) (f x0)) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found ((((((eq_trans Prop) (f x0)) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))))) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq Prop) (f x0)) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found x5:(X x4)
% Instantiate: x3:=x4:a
% Found (fun (x5:(X x4))=> x5) as proof of (X x3)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found (((eq_trans000 (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found ((((eq_trans00 (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))))) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq Prop) (f x0)) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found (((((eq_trans0 (f x0)) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))))) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq Prop) (f x0)) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found ((((((eq_trans Prop) (f x0)) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))))) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq Prop) (f x0)) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: b:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:a)=> ((X x3)->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))):(((eq (a->Prop)) (fun (Xt:a)=> ((X x3)->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))) (fun (x:a)=> ((X x3)->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found (eta_expansion_dep00 (fun (Xt:a)=> ((X x3)->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((X x3)->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))) b)
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) (fun (Xt:a)=> ((X x3)->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((X x3)->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xt:a)=> ((X x3)->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((X x3)->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xt:a)=> ((X x3)->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((X x3)->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xt:a)=> ((X x3)->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((X x3)->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))))) b)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))->(P0 (fun (x:a)=> ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x2 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x)))))))
% Found (eta_expansion_dep000 P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eta_expansion_dep00 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (((eta_expansion_dep0 (fun (x4:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))->(P0 (fun (x:a)=> ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x2 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x)))))))
% Found (eta_expansion_dep000 P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eta_expansion_dep00 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (((eta_expansion_dep0 (fun (x4:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: f:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P f)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x3)))
% Found ((eq_ref a) Xy) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x3)))
% Found ((eq_ref a) Xy) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x3)))
% Found ((eq_ref a) Xy) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x3)))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x3)
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: f:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P f)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x3)))
% Found ((eq_ref a) Xy) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x3)))
% Found ((eq_ref a) Xy) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x3)))
% Found ((eq_ref a) Xy) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x3)))
% Found (fun (x01:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x3)
% 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)) (forall (x30:a), ((X x30)->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% EOF
%------------------------------------------------------------------------------