%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV011^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n118.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 13:33:33 EDT 2014

% Result   : Timeout 300.03s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV011^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n118.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 07:26:31 CDT 2014
% % CPUTime  : 300.03 
% 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 0x1243b00>, <kernel.Type object at 0x12432d8>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->(forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) of role conjecture named cTHM260_B_pme
% Conjecture to prove = (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->(forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->(forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))))']
% Parameter a:Type.
% Trying to prove (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->(forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref000:=(eq_ref00 P):((P Xq)->(P Xq))
% Found (eq_ref00 P) as proof of ((P Xq)->(P x0))
% Found ((eq_ref0 Xq) P) as proof of ((P Xq)->(P x0))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x0)
% Found x1:(P Xq)
% Instantiate: x0:=Xq:(a->Prop)
% Found (fun (x1:(P Xq))=> x1) as proof of (P x0)
% Found (fun (P:((a->Prop)->Prop)) (x1:(P Xq))=> x1) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop)) (x1:(P Xq))=> x1) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x0)
% Found x10:(P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of (P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion0000 (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion000 (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion00 Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion0 Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x10:(P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of (P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion_dep0000 (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion_dep000 (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion_dep00 Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref000:=(eq_ref00 P):((P Xq)->(P Xq))
% Found (eq_ref00 P) as proof of ((P Xq)->(P x2))
% Found ((eq_ref0 Xq) P) as proof of ((P Xq)->(P x2))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x2))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of ((P Xq)->(P x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x2))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x2))
% Found (((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x2))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x2))
% Found (((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x2)
% Found x3:(P Xq)
% Instantiate: x2:=Xq:(a->Prop)
% Found (fun (x3:(P Xq))=> x3) as proof of (P x2)
% Found (fun (P:((a->Prop)->Prop)) (x3:(P Xq))=> x3) as proof of ((P Xq)->(P x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop)) (x3:(P Xq))=> x3) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x2))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x2))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x2))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x2))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x2)
% Found x10:(P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of (P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion_dep0000 (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion_dep000 (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion_dep00 Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x10:(P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of (P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion0000 (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion000 (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion00 Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion0 Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref000:=(eq_ref00 P):((P Xq)->(P Xq))
% Found (eq_ref00 P) as proof of ((P Xq)->(P x0))
% Found ((eq_ref0 Xq) P) as proof of ((P Xq)->(P x0))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x0)
% Found x3:(P Xq)
% Instantiate: x0:=Xq:(a->Prop)
% Found (fun (x3:(P Xq))=> x3) as proof of (P x0)
% Found (fun (P:((a->Prop)->Prop)) (x3:(P Xq))=> x3) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop)) (x3:(P Xq))=> x3) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x2:(Xq Xx))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x1:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x2:(Xq Xx))=> (((eta_expansion a) Prop) Xq)) as proof of ((Xq Xx)->(((eq (a->Prop)) Xq) x0))
% Found (fun (x1:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x2:(Xq Xx))=> (((eta_expansion a) Prop) Xq)) as proof of ((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->(((eq (a->Prop)) Xq) x0)))
% Found (and_rect00 (fun (x1:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x2:(Xq Xx))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((and_rect0 (((eq (a->Prop)) Xq) x0)) (fun (x1:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x2:(Xq Xx))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((fun (P:Type) (x1:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P) x1) x00)) (((eq (a->Prop)) Xq) x0)) (fun (x1:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x2:(Xq Xx))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P:Type) (x1:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P) x1) x00)) (((eq (a->Prop)) Xq) x0)) (fun (x1:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x2:(Xq Xx))=> (((eta_expansion a) Prop) Xq)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x30:(P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion_dep0000 (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found ((eta_expansion_dep000 (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((eta_expansion_dep00 Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of ((P Xq)->(P x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x30:(P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion0000 (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found ((eta_expansion000 (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((eta_expansion00 Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion0 Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of ((P Xq)->(P x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eq_ref00:=(eq_ref0 x0):(((eq (a->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found (eq_sym000 ((eq_ref (a->Prop)) x0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eq_sym00 Xq) ((eq_ref (a->Prop)) x0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eq_sym0 x0) Xq) ((eq_ref (a->Prop)) x0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((((eq_sym (a->Prop)) x0) Xq) ((eq_ref (a->Prop)) x0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((((eq_sym (a->Prop)) x0) Xq) ((eq_ref (a->Prop)) x0))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x30:(P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion0000 (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion000 (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion00 Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion0 Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x30:(P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion_dep0000 (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion_dep000 (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion_dep00 Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))):(((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))
% Found (eq_ref0 (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) b)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x4))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x4))
% Found (((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x4))
% Found ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x4))
% Found ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x4))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x4))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x4))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x4))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x4))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x4))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eq_ref000:=(eq_ref00 P):((P Xq)->(P Xq))
% Found (eq_ref00 P) as proof of ((P Xq)->(P x4))
% Found ((eq_ref0 Xq) P) as proof of ((P Xq)->(P x4))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x4))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of ((P Xq)->(P x4))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x4)
% Found x5:(P Xq)
% Instantiate: x4:=Xq:(a->Prop)
% Found (fun (x5:(P Xq))=> x5) as proof of (P x4)
% Found (fun (P:((a->Prop)->Prop)) (x5:(P Xq))=> x5) as proof of ((P Xq)->(P x4))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop)) (x5:(P Xq))=> x5) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x4))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x4))
% Found (((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x4))
% Found ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x4))
% Found ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x4))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x4))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x4))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x4))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x4))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x4))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))=> (((eta_expansion a) Prop) Xq)) as proof of ((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->(((eq (a->Prop)) Xq) x0))
% Found (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))=> (((eta_expansion a) Prop) Xq)) as proof of (((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->(((eq (a->Prop)) Xq) x0)))
% Found (and_rect00 (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((and_rect0 (((eq (a->Prop)) Xq) x0)) (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((fun (P:Type) (x1:(((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->P)))=> (((((and_rect ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) P) x1) x)) (((eq (a->Prop)) Xq) x0)) (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P:Type) (x1:(((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->P)))=> (((((and_rect ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) P) x1) x)) (((eq (a->Prop)) Xq) x0)) (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))=> (((eta_expansion a) Prop) Xq)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x0)
% Found x30:(P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion0000 (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found ((eta_expansion000 (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((eta_expansion00 Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion0 Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of ((P Xq)->(P x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x5:(P Xq)
% Instantiate: x0:=Xq:(a->Prop)
% Found (fun (x5:(P Xq))=> x5) as proof of (P x0)
% Found (fun (P:((a->Prop)->Prop)) (x5:(P Xq))=> x5) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop)) (x5:(P Xq))=> x5) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref000:=(eq_ref00 P):((P Xq)->(P Xq))
% Found (eq_ref00 P) as proof of ((P Xq)->(P x0))
% Found ((eq_ref0 Xq) P) as proof of ((P Xq)->(P x0))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x0)
% Found x30:(P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion_dep0000 (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found ((eta_expansion_dep000 (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((eta_expansion_dep00 Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of ((P Xq)->(P x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x2))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x2))
% Found (((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x2))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x2))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref000:=(eq_ref00 P):((P Xq)->(P Xq))
% Found (eq_ref00 P) as proof of ((P Xq)->(P x2))
% Found ((eq_ref0 Xq) P) as proof of ((P Xq)->(P x2))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x2))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of ((P Xq)->(P x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x2)
% Found x5:(P Xq)
% Instantiate: x2:=Xq:(a->Prop)
% Found (fun (x5:(P Xq))=> x5) as proof of (P x2)
% Found (fun (P:((a->Prop)->Prop)) (x5:(P Xq))=> x5) as proof of ((P Xq)->(P x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop)) (x5:(P Xq))=> x5) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x2))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x2))
% Found (((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x2))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x2))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 (Xq x1)):(((eq Prop) (Xq x1)) (Xq x1))
% Found (eq_ref0 (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found ((eq_ref Prop) (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found ((eq_ref Prop) (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (Xq x1))) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (Xq x1))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found (functional_extensionality_dep0000 (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((functional_extensionality_dep000 x0) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((functional_extensionality_dep00 Xq) x0) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((((functional_extensionality_dep0 (fun (x3:a)=> Prop)) Xq) x0) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((((functional_extensionality_dep a) (fun (x3:a)=> Prop)) Xq) x0) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((((functional_extensionality_dep a) (fun (x3:a)=> Prop)) Xq) x0) (fun (x1:a)=> ((eq_ref Prop) (Xq x1))))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 (Xq x1)):(((eq Prop) (Xq x1)) (Xq x1))
% Found (eq_ref0 (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found ((eq_ref Prop) (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found ((eq_ref Prop) (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (Xq x1))) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (Xq x1))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found (functional_extensionality0000 (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((functional_extensionality000 x0) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((functional_extensionality00 Xq) x0) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((((functional_extensionality0 Prop) Xq) x0) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((((functional_extensionality a) Prop) Xq) x0) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((((functional_extensionality a) Prop) Xq) x0) (fun (x1:a)=> ((eq_ref Prop) (Xq x1))))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x30:(P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion0000 (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion000 (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion00 Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion0 Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x30:(P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion_dep0000 (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion_dep000 (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion_dep00 Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of (((eq (a->Prop)) Xq) x0)
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x4:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x3:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x4:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)) as proof of ((Xq Xx)->(((eq (a->Prop)) Xq) x2))
% Found (fun (x3:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x4:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)) as proof of ((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->(((eq (a->Prop)) Xq) x2)))
% Found (and_rect10 (fun (x3:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x4:(Xq Xx))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((and_rect1 (((eq (a->Prop)) Xq) x2)) (fun (x3:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x4:(Xq Xx))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((fun (P:Type) (x3:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P) x3) x00)) (((eq (a->Prop)) Xq) x2)) (fun (x3:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x4:(Xq Xx))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P) x3) x00)) (((eq (a->Prop)) Xq) x2)) (fun (x3:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x4:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))):(((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) (fun (x:(a->Prop))=> ((and ((and (forall (Xx0:a), ((x Xx0)->(forall (Xy:a), ((iff (x Xy)) ((R Xx0) Xy)))))) (x Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x))))))
% Found (eta_expansion_dep00 (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) b)
% Found ((eta_expansion_dep0 (fun (x3:(a->Prop))=> Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) b)
% Found eq_ref00:=(eq_ref0 x2):(((eq (a->Prop)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found (eq_sym000 ((eq_ref (a->Prop)) x2)) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eq_sym00 Xq) ((eq_ref (a->Prop)) x2)) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eq_sym0 x2) Xq) ((eq_ref (a->Prop)) x2)) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((((eq_sym (a->Prop)) x2) Xq) ((eq_ref (a->Prop)) x2)) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((((eq_sym (a->Prop)) x2) Xq) ((eq_ref (a->Prop)) x2))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x4:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x3:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x4:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)) as proof of ((Xq Xx)->(((eq (a->Prop)) Xq) x0))
% Found (fun (x3:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x4:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)) as proof of ((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->(((eq (a->Prop)) Xq) x0)))
% Found (and_rect10 (fun (x3:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x4:(Xq Xx))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((and_rect1 (((eq (a->Prop)) Xq) x0)) (fun (x3:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x4:(Xq Xx))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((fun (P:Type) (x3:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P) x3) x00)) (((eq (a->Prop)) Xq) x0)) (fun (x3:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x4:(Xq Xx))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P) x3) x00)) (((eq (a->Prop)) Xq) x0)) (fun (x3:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x4:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion0000 (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found ((eta_expansion000 (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((eta_expansion00 Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found ((((eta_expansion0 Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x4))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x4)
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion_dep0000 (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found ((eta_expansion_dep000 (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((eta_expansion_dep00 Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found ((((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x4))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x4)
% Found eq_ref00:=(eq_ref0 x0):(((eq (a->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found (eq_sym000 ((eq_ref (a->Prop)) x0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eq_sym00 Xq) ((eq_ref (a->Prop)) x0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eq_sym0 x0) Xq) ((eq_ref (a->Prop)) x0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((((eq_sym (a->Prop)) x0) Xq) ((eq_ref (a->Prop)) x0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((((eq_sym (a->Prop)) x0) Xq) ((eq_ref (a->Prop)) x0))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion_dep0000 (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion_dep000 (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion_dep00 Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion0000 (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion000 (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion00 Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion0 Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion_dep0000 (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found ((eta_expansion_dep000 (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((eta_expansion_dep00 Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion0000 (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found ((eta_expansion000 (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((eta_expansion00 Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion0 Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x3:(forall (Xx0:a), ((R Xx0) Xx0))) (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) as proof of ((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->(((eq (a->Prop)) Xq) x0))
% Found (fun (x3:(forall (Xx0:a), ((R Xx0) Xx0))) (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) as proof of ((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->(((eq (a->Prop)) Xq) x0)))
% Found (and_rect10 (fun (x3:(forall (Xx0:a), ((R Xx0) Xx0))) (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((and_rect1 (((eq (a->Prop)) Xq) x0)) (fun (x3:(forall (Xx0:a), ((R Xx0) Xx0))) (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((fun (P:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P) x3) x1)) (((eq (a->Prop)) Xq) x0)) (fun (x3:(forall (Xx0:a), ((R Xx0) Xx0))) (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P) x3) x1)) (((eq (a->Prop)) Xq) x0)) (fun (x3:(forall (Xx0:a), ((R Xx0) Xx0))) (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x3:(forall (Xx0:a), ((R Xx0) Xx0))) (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> (((eta_expansion a) Prop) Xq)) as proof of ((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->(((eq (a->Prop)) Xq) x2))
% Found (fun (x3:(forall (Xx0:a), ((R Xx0) Xx0))) (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> (((eta_expansion a) Prop) Xq)) as proof of ((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->(((eq (a->Prop)) Xq) x2)))
% Found (and_rect10 (fun (x3:(forall (Xx0:a), ((R Xx0) Xx0))) (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((and_rect1 (((eq (a->Prop)) Xq) x2)) (fun (x3:(forall (Xx0:a), ((R Xx0) Xx0))) (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((fun (P:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P) x3) x0)) (((eq (a->Prop)) Xq) x2)) (fun (x3:(forall (Xx0:a), ((R Xx0) Xx0))) (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P) x3) x0)) (((eq (a->Prop)) Xq) x2)) (fun (x3:(forall (Xx0:a), ((R Xx0) Xx0))) (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> (((eta_expansion a) Prop) Xq)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 (Xq x3)):(((eq Prop) (Xq x3)) (Xq x3))
% Found (eq_ref0 (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x2 x)))
% Found (functional_extensionality0000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((functional_extensionality000 x2) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((functional_extensionality00 Xq) x2) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((((functional_extensionality0 Prop) Xq) x2) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((((functional_extensionality a) Prop) Xq) x2) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((((functional_extensionality a) Prop) Xq) x2) (fun (x3:a)=> ((eq_ref Prop) (Xq x3))))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 (Xq x3)):(((eq Prop) (Xq x3)) (Xq x3))
% Found (eq_ref0 (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x2 x)))
% Found (functional_extensionality_dep0000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((functional_extensionality_dep000 x2) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((functional_extensionality_dep00 Xq) x2) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((((functional_extensionality_dep0 (fun (x5:a)=> Prop)) Xq) x2) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) Xq) x2) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) Xq) x2) (fun (x3:a)=> ((eq_ref Prop) (Xq x3))))) as proof of (((eq (a->Prop)) Xq) x2)
% Found iff_refl0:=(iff_refl (x2 Xy)):((iff (x2 Xy)) (x2 Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x2 Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (iff_refl (x2 Xy))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found eta_expansion000:=(eta_expansion00 (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))):(((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) (fun (x:(a->Prop))=> ((and ((and (forall (Xx0:a), ((x Xx0)->(forall (Xy:a), ((iff (x Xy)) ((R Xx0) Xy)))))) (x Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x))))))
% Found (eta_expansion00 (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) b)
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion0000 (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found ((eta_expansion000 (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((eta_expansion00 Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found ((((eta_expansion0 Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x4))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x4)
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion_dep0000 (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found ((eta_expansion_dep000 (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((eta_expansion_dep00 Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found ((((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x4))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x4)
% Found eq_ref00:=(eq_ref0 (Xq x3)):(((eq Prop) (Xq x3)) (Xq x3))
% Found (eq_ref0 (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found (functional_extensionality0000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((functional_extensionality000 x0) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((functional_extensionality00 Xq) x0) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((((functional_extensionality0 Prop) Xq) x0) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((((functional_extensionality a) Prop) Xq) x0) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((((functional_extensionality a) Prop) Xq) x0) (fun (x3:a)=> ((eq_ref Prop) (Xq x3))))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 (Xq x3)):(((eq Prop) (Xq x3)) (Xq x3))
% Found (eq_ref0 (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found (functional_extensionality_dep0000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((functional_extensionality_dep000 x0) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((functional_extensionality_dep00 Xq) x0) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((((functional_extensionality_dep0 (fun (x5:a)=> Prop)) Xq) x0) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) Xq) x0) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) Xq) x0) (fun (x3:a)=> ((eq_ref Prop) (Xq x3))))) as proof of (((eq (a->Prop)) Xq) x0)
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion_dep0000 (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion_dep000 (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion_dep00 Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion0000 (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion000 (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion00 Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion0 Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion0000 (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found ((eta_expansion000 (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((eta_expansion00 Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion0 Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion_dep0000 (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found ((eta_expansion_dep000 (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((eta_expansion_dep00 Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x6:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)) as proof of ((Xq Xx)->(((eq (a->Prop)) Xq) x4))
% Found (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)) as proof of ((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->(((eq (a->Prop)) Xq) x4)))
% Found (and_rect20 (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((and_rect2 (((eq (a->Prop)) Xq) x4)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((fun (P:Type) (x5:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P) x5) x00)) (((eq (a->Prop)) Xq) x4)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P:Type) (x5:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P) x5) x00)) (((eq (a->Prop)) Xq) x4)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)))) as proof of (((eq (a->Prop)) Xq) x4)
% Found eq_ref00:=(eq_ref0 x4):(((eq (a->Prop)) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq (a->Prop)) x4) Xq)
% Found ((eq_ref (a->Prop)) x4) as proof of (((eq (a->Prop)) x4) Xq)
% Found ((eq_ref (a->Prop)) x4) as proof of (((eq (a->Prop)) x4) Xq)
% Found ((eq_ref (a->Prop)) x4) as proof of (((eq (a->Prop)) x4) Xq)
% Found (eq_sym000 ((eq_ref (a->Prop)) x4)) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eq_sym00 Xq) ((eq_ref (a->Prop)) x4)) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eq_sym0 x4) Xq) ((eq_ref (a->Prop)) x4)) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((((eq_sym (a->Prop)) x4) Xq) ((eq_ref (a->Prop)) x4)) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((((eq_sym (a->Prop)) x4) Xq) ((eq_ref (a->Prop)) x4))) as proof of (((eq (a->Prop)) Xq) x4)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x6:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)) as proof of ((Xq Xx)->(((eq (a->Prop)) Xq) x0))
% Found (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)) as proof of ((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->(((eq (a->Prop)) Xq) x0)))
% Found (and_rect20 (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((and_rect2 (((eq (a->Prop)) Xq) x0)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((fun (P:Type) (x5:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P) x5) x00)) (((eq (a->Prop)) Xq) x0)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P:Type) (x5:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P) x5) x00)) (((eq (a->Prop)) Xq) x0)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x6:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)) as proof of ((Xq Xx)->(((eq (a->Prop)) Xq) x2))
% Found (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)) as proof of ((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->(((eq (a->Prop)) Xq) x2)))
% Found (and_rect20 (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((and_rect2 (((eq (a->Prop)) Xq) x2)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((fun (P:Type) (x5:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P) x5) x00)) (((eq (a->Prop)) Xq) x2)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P:Type) (x5:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P) x5) x00)) (((eq (a->Prop)) Xq) x2)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 x0):(((eq (a->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found (eq_sym000 ((eq_ref (a->Prop)) x0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eq_sym00 Xq) ((eq_ref (a->Prop)) x0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eq_sym0 x0) Xq) ((eq_ref (a->Prop)) x0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((((eq_sym (a->Prop)) x0) Xq) ((eq_ref (a->Prop)) x0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((((eq_sym (a->Prop)) x0) Xq) ((eq_ref (a->Prop)) x0))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 x2):(((eq (a->Prop)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found (eq_sym000 ((eq_ref (a->Prop)) x2)) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eq_sym00 Xq) ((eq_ref (a->Prop)) x2)) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eq_sym0 x2) Xq) ((eq_ref (a->Prop)) x2)) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((((eq_sym (a->Prop)) x2) Xq) ((eq_ref (a->Prop)) x2)) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((((eq_sym (a->Prop)) x2) Xq) ((eq_ref (a->Prop)) x2))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 (Xq x5)):(((eq Prop) (Xq x5)) (Xq x5))
% Found (eq_ref0 (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x4 x)))
% Found (functional_extensionality0000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((functional_extensionality000 x4) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((functional_extensionality00 Xq) x4) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((((functional_extensionality0 Prop) Xq) x4) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((((functional_extensionality a) Prop) Xq) x4) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((((functional_extensionality a) Prop) Xq) x4) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of (((eq (a->Prop)) Xq) x4)
% Found eq_ref00:=(eq_ref0 (Xq x5)):(((eq Prop) (Xq x5)) (Xq x5))
% Found (eq_ref0 (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x4 x)))
% Found (functional_extensionality_dep0000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((functional_extensionality_dep000 x4) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((functional_extensionality_dep00 Xq) x4) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((((functional_extensionality_dep0 (fun (x7:a)=> Prop)) Xq) x4) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x4) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x4) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of (((eq (a->Prop)) Xq) x4)
% Found iff_refl0:=(iff_refl (x4 Xy)):((iff (x4 Xy)) (x4 Xy))
% Found (iff_refl (x4 Xy)) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x4 Xy)) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x4 Xy))) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x4 Xx0)) (Xy:a)=> (iff_refl (x4 Xy))) as proof of (forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eq_ref00:=(eq_ref0 (Xq x5)):(((eq Prop) (Xq x5)) (Xq x5))
% Found (eq_ref0 (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found (functional_extensionality0000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((functional_extensionality000 x0) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((functional_extensionality00 Xq) x0) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((((functional_extensionality0 Prop) Xq) x0) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((((functional_extensionality a) Prop) Xq) x0) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((((functional_extensionality a) Prop) Xq) x0) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 (Xq x5)):(((eq Prop) (Xq x5)) (Xq x5))
% Found (eq_ref0 (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found (functional_extensionality_dep0000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((functional_extensionality_dep000 x0) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((functional_extensionality_dep00 Xq) x0) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((((functional_extensionality_dep0 (fun (x7:a)=> Prop)) Xq) x0) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x0) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x0) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 (Xq x5)):(((eq Prop) (Xq x5)) (Xq x5))
% Found (eq_ref0 (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x2 x)))
% Found (functional_extensionality0000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((functional_extensionality000 x2) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((functional_extensionality00 Xq) x2) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((((functional_extensionality0 Prop) Xq) x2) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((((functional_extensionality a) Prop) Xq) x2) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((((functional_extensionality a) Prop) Xq) x2) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 (Xq x5)):(((eq Prop) (Xq x5)) (Xq x5))
% Found (eq_ref0 (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x2 x)))
% Found (functional_extensionality_dep0000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((functional_extensionality_dep000 x2) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((functional_extensionality_dep00 Xq) x2) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((((functional_extensionality_dep0 (fun (x7:a)=> Prop)) Xq) x2) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x2) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x2) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of (((eq (a->Prop)) Xq) x2)
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x2 Xy)):((iff (x2 Xy)) (x2 Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x2 Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (iff_refl (x2 Xy))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found x1:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x1:((R Xx0) Xy))=> x1) as proof of (x0 Xy)
% Found (fun (x1:((R Xx0) Xy))=> x1) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found x1:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x1:(x0 Xy))=> x1) as proof of ((R Xx0) Xy)
% Found (fun (x1:(x0 Xy))=> x1) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found ((conj20 (fun (x1:(x0 Xy))=> x1)) (fun (x1:((R Xx0) Xy))=> x1)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x1:(x0 Xy))=> x1)) (fun (x1:((R Xx0) Xy))=> x1)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x1:(x0 Xy))=> x1)) (fun (x1:((R Xx0) Xy))=> x1)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x1:(x0 Xy))=> x1)) (fun (x1:((R Xx0) Xy))=> x1))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x1:(x0 Xy))=> x1)) (fun (x1:((R Xx0) Xy))=> x1))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((eta_expansion0 Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((eta_expansion0 Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((eq_ref (a->Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((eq_ref (a->Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((eq_ref (a->Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_trans0000 (((eta_expansion a) Prop) Xq)) ((eq_ref (a->Prop)) b)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eq_trans000 x0) (((eta_expansion a) Prop) Xq)) ((eq_ref (a->Prop)) b)) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((((eq_trans00 (fun (x3:a)=> (Xq x3))) x0) (((eta_expansion a) Prop) Xq)) ((eq_ref (a->Prop)) (fun (x3:a)=> (Xq x3)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((((eq_trans0 Xq) (fun (x3:a)=> (Xq x3))) x0) (((eta_expansion a) Prop) Xq)) ((eq_ref (a->Prop)) (fun (x3:a)=> (Xq x3)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((((((eq_trans (a->Prop)) Xq) (fun (x3:a)=> (Xq x3))) x0) (((eta_expansion a) Prop) Xq)) ((eq_ref (a->Prop)) (fun (x3:a)=> (Xq x3)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((((((eq_trans (a->Prop)) Xq) (fun (x3:a)=> (Xq x3))) x0) (((eta_expansion a) Prop) Xq)) ((eq_ref (a->Prop)) (fun (x3:a)=> (Xq x3))))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x0)
% Found x1:(P Xq)
% Instantiate: x0:=Xq:(a->Prop)
% Found (fun (x1:(P Xq))=> x1) as proof of (P x0)
% Found (fun (P:((a->Prop)->Prop)) (x1:(P Xq))=> x1) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop)) (x1:(P Xq))=> x1) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop)) (x1:(P Xq))=> x1) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref000:=(eq_ref00 P):((P Xq)->(P Xq))
% Found (eq_ref00 P) as proof of ((P Xq)->(P x0))
% Found ((eq_ref0 Xq) P) as proof of ((P Xq)->(P x0))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 x0):(((eq (a->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found (eq_sym0000 ((eq_ref (a->Prop)) x0)) as proof of ((P Xq)->(P x0))
% Found (eq_sym0000 ((eq_ref (a->Prop)) x0)) as proof of ((P Xq)->(P x0))
% Found ((fun (x1:(((eq (a->Prop)) x0) Xq))=> ((eq_sym000 x1) P)) ((eq_ref (a->Prop)) x0)) as proof of ((P Xq)->(P x0))
% Found ((fun (x1:(((eq (a->Prop)) x0) Xq))=> (((eq_sym00 Xq) x1) P)) ((eq_ref (a->Prop)) x0)) as proof of ((P Xq)->(P x0))
% Found ((fun (x1:(((eq (a->Prop)) x0) Xq))=> ((((eq_sym0 x0) Xq) x1) P)) ((eq_ref (a->Prop)) x0)) as proof of ((P Xq)->(P x0))
% Found ((fun (x1:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x1) P)) ((eq_ref (a->Prop)) x0)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x1:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x1) P)) ((eq_ref (a->Prop)) x0))) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((fun (x1:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x1) P)) ((eq_ref (a->Prop)) x0))) as proof of (((eq (a->Prop)) Xq) x0)
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x2 Xy)):((iff (x2 Xy)) (x2 Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x2 Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (iff_refl (x2 Xy))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x10:(P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of (P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion_dep0000 (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion_dep000 (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion_dep00 Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x10:(P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of (P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion0000 (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion000 (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion00 Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion0 Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x10:(P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of (P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 x0):(((eq (a->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_sym0000 ((eq_ref (a->Prop)) x0)) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found ((eq_sym0000 ((eq_ref (a->Prop)) x0)) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((fun (x1:(((eq (a->Prop)) x0) Xq))=> ((eq_sym000 x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) ((eq_ref (a->Prop)) x0)) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((fun (x1:(((eq (a->Prop)) x0) Xq))=> (((eq_sym00 Xq) x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) ((eq_ref (a->Prop)) x0)) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((fun (x1:(((eq (a->Prop)) x0) Xq))=> ((((eq_sym0 x0) Xq) x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) ((eq_ref (a->Prop)) x0)) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((fun (x1:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) ((eq_ref (a->Prop)) x0)) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x1:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) ((eq_ref (a->Prop)) x0)) (fun (x10:(P Xq))=> x10))) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x1:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) ((eq_ref (a->Prop)) x0)) (fun (x10:(P Xq))=> x10))) as proof of (((eq (a->Prop)) Xq) x0)
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))) (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> (iff_refl (x0 Xy))) as proof of ((forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz)))->((iff (x0 Xy)) ((R Xx0) Xy)))
% Found (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))) (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> (iff_refl (x0 Xy))) as proof of (((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))->((forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz)))->((iff (x0 Xy)) ((R Xx0) Xy))))
% Found (and_rect00 (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))) (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((and_rect0 ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))) (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((fun (P:Type) (x1:(((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->P)))=> (((((and_rect ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) P) x1) x)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))) (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((fun (P:Type) (x1:(((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->P)))=> (((((and_rect ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) P) x1) x)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))) (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> (iff_refl (x0 Xy))))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((fun (P:Type) (x1:(((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->P)))=> (((((and_rect ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) P) x1) x)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))) (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> (iff_refl (x0 Xy))))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found ((eta_expansion0 Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found ((eta_expansion0 Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found ((eq_ref (a->Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found ((eq_ref (a->Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found ((eq_ref (a->Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found x3:(x2 Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x3:(x2 Xy))=> x3) as proof of ((R Xx0) Xy)
% Found (fun (x3:(x2 Xy))=> x3) as proof of ((x2 Xy)->((R Xx0) Xy))
% Found x3:((R Xx0) Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x3:((R Xx0) Xy))=> x3) as proof of (x2 Xy)
% Found (fun (x3:((R Xx0) Xy))=> x3) as proof of (((R Xx0) Xy)->(x2 Xy))
% Found ((conj20 (fun (x3:(x2 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x2 Xy))) (fun (x3:(x2 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x3:(x2 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x3:(x2 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x3:(x2 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found eq_ref00:=(eq_ref0 (Xq x1)):(((eq Prop) (Xq x1)) (Xq x1))
% Found (eq_ref0 (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found ((eq_ref Prop) (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found ((eq_ref Prop) (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (Xq x1))) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (Xq x1))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found (functional_extensionality_dep00000 (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of ((P Xq)->(P x0))
% Found (functional_extensionality_dep00000 (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((functional_extensionality_dep0000 x1) P)) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((functional_extensionality_dep000 x0) x1) P)) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((functional_extensionality_dep00 Xq) x0) x1) P)) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((((functional_extensionality_dep0 (fun (x3:a)=> Prop)) Xq) x0) x1) P)) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x3:a)=> Prop)) Xq) x0) x1) P)) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x3:a)=> Prop)) Xq) x0) x1) P)) (fun (x1:a)=> ((eq_ref Prop) (Xq x1))))) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x3:a)=> Prop)) Xq) x0) x1) P)) (fun (x1:a)=> ((eq_ref Prop) (Xq x1))))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 (Xq x1)):(((eq Prop) (Xq x1)) (Xq x1))
% Found (eq_ref0 (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found ((eq_ref Prop) (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found ((eq_ref Prop) (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (Xq x1))) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (Xq x1))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found (functional_extensionality00000 (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of ((P Xq)->(P x0))
% Found (functional_extensionality00000 (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((functional_extensionality0000 x1) P)) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((functional_extensionality000 x0) x1) P)) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((functional_extensionality00 Xq) x0) x1) P)) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((((functional_extensionality0 Prop) Xq) x0) x1) P)) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x1) P)) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x1) P)) (fun (x1:a)=> ((eq_ref Prop) (Xq x1))))) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x1) P)) (fun (x1:a)=> ((eq_ref Prop) (Xq x1))))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((eq_ref (a->Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((eq_ref (a->Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((eq_ref (a->Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((eta_expansion0 Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((eta_expansion0 Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found x10:(P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of (P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 (Xq x1)):(((eq Prop) (Xq x1)) (Xq x1))
% Found (eq_ref0 (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found ((eq_ref Prop) (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found ((eq_ref Prop) (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (Xq x1))) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (Xq x1))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found ((functional_extensionality_dep00000 (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found ((functional_extensionality_dep00000 (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((functional_extensionality_dep0000 x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((functional_extensionality_dep000 x0) x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((functional_extensionality_dep00 Xq) x0) x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((((functional_extensionality_dep0 (fun (x3:a)=> Prop)) Xq) x0) x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x3:a)=> Prop)) Xq) x0) x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x3:a)=> Prop)) Xq) x0) x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) (fun (x10:(P Xq))=> x10))) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x3:a)=> Prop)) Xq) x0) x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) (fun (x10:(P Xq))=> x10))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x10:(P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of (P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 (Xq x1)):(((eq Prop) (Xq x1)) (Xq x1))
% Found (eq_ref0 (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found ((eq_ref Prop) (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found ((eq_ref Prop) (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (Xq x1))) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (Xq x1))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found ((functional_extensionality00000 (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found ((functional_extensionality00000 (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((functional_extensionality0000 x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((functional_extensionality000 x0) x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((functional_extensionality00 Xq) x0) x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((((functional_extensionality0 Prop) Xq) x0) x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) (fun (x10:(P Xq))=> x10))) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) (fun (x10:(P Xq))=> x10))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref000:=(eq_ref00 P):((P Xq)->(P Xq))
% Found (eq_ref00 P) as proof of ((P Xq)->(P x2))
% Found ((eq_ref0 Xq) P) as proof of ((P Xq)->(P x2))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x2))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x2)
% Found x3:(P Xq)
% Instantiate: x2:=Xq:(a->Prop)
% Found (fun (x3:(P Xq))=> x3) as proof of (P x2)
% Found (fun (P:((a->Prop)->Prop)) (x3:(P Xq))=> x3) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop)) (x3:(P Xq))=> x3) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop)) (x3:(P Xq))=> x3) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x2))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x2))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x2))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x2))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x2))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x2))
% Found (((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x2))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x2))
% Found (((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x2)
% Found x3:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x3:(x0 Xy))=> x3) as proof of ((R Xx0) Xy)
% Found (fun (x3:(x0 Xy))=> x3) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found x3:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x3:((R Xx0) Xy))=> x3) as proof of (x0 Xy)
% Found (fun (x3:((R Xx0) Xy))=> x3) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found ((conj20 (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) 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) x2)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) x2)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) x2)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) x2)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) x2)
% Found ((eq_trans0000 ((eq_ref (a->Prop)) Xq)) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) b)) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eq_trans000 x2) ((eq_ref (a->Prop)) Xq)) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) b)) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((((eq_trans00 Xq) x2) ((eq_ref (a->Prop)) Xq)) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((((eq_trans0 Xq) Xq) x2) ((eq_ref (a->Prop)) Xq)) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((((((eq_trans (a->Prop)) Xq) Xq) x2) ((eq_ref (a->Prop)) Xq)) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((((((eq_trans (a->Prop)) Xq) Xq) x2) ((eq_ref (a->Prop)) Xq)) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x10:(P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of (P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion0000 (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion000 (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion00 Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion0 Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x10:(P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of (P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion_dep0000 (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion_dep000 (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion_dep00 Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref000:=(eq_ref00 P):((P Xq)->(P Xq))
% Found (eq_ref00 P) as proof of ((P Xq)->(P x0))
% Found ((eq_ref0 Xq) P) as proof of ((P Xq)->(P x0))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x0)
% Found x3:(P Xq)
% Instantiate: x0:=Xq:(a->Prop)
% Found (fun (x3:(P Xq))=> x3) as proof of (P x0)
% Found (fun (P:((a->Prop)->Prop)) (x3:(P Xq))=> x3) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop)) (x3:(P Xq))=> x3) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop)) (x3:(P Xq))=> x3) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 x2):(((eq (a->Prop)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found (eq_sym0000 ((eq_ref (a->Prop)) x2)) as proof of ((P Xq)->(P x2))
% Found (eq_sym0000 ((eq_ref (a->Prop)) x2)) as proof of ((P Xq)->(P x2))
% Found ((fun (x3:(((eq (a->Prop)) x2) Xq))=> ((eq_sym000 x3) P)) ((eq_ref (a->Prop)) x2)) as proof of ((P Xq)->(P x2))
% Found ((fun (x3:(((eq (a->Prop)) x2) Xq))=> (((eq_sym00 Xq) x3) P)) ((eq_ref (a->Prop)) x2)) as proof of ((P Xq)->(P x2))
% Found ((fun (x3:(((eq (a->Prop)) x2) Xq))=> ((((eq_sym0 x2) Xq) x3) P)) ((eq_ref (a->Prop)) x2)) as proof of ((P Xq)->(P x2))
% Found ((fun (x3:(((eq (a->Prop)) x2) Xq))=> (((((eq_sym (a->Prop)) x2) Xq) x3) P)) ((eq_ref (a->Prop)) x2)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x3:(((eq (a->Prop)) x2) Xq))=> (((((eq_sym (a->Prop)) x2) Xq) x3) P)) ((eq_ref (a->Prop)) x2))) as proof of ((P Xq)->(P x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((fun (x3:(((eq (a->Prop)) x2) Xq))=> (((((eq_sym (a->Prop)) x2) Xq) x3) P)) ((eq_ref (a->Prop)) x2))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) 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) x0)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_trans0000 (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) b)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eq_trans000 x0) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) b)) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((((eq_trans00 (fun (x5:a)=> (Xq x5))) x0) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (x5:a)=> (Xq x5)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((((eq_trans0 Xq) (fun (x5:a)=> (Xq x5))) x0) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (x5:a)=> (Xq x5)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((((((eq_trans (a->Prop)) Xq) (fun (x5:a)=> (Xq x5))) x0) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (x5:a)=> (Xq x5)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((((((eq_trans (a->Prop)) Xq) (fun (x5:a)=> (Xq x5))) x0) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (x5:a)=> (Xq x5))))) as proof of (((eq (a->Prop)) Xq) x0)
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eq_ref00:=(eq_ref0 x0):(((eq (a->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found (eq_sym0000 ((eq_ref (a->Prop)) x0)) as proof of ((P Xq)->(P x0))
% Found (eq_sym0000 ((eq_ref (a->Prop)) x0)) as proof of ((P Xq)->(P x0))
% Found ((fun (x3:(((eq (a->Prop)) x0) Xq))=> ((eq_sym000 x3) P)) ((eq_ref (a->Prop)) x0)) as proof of ((P Xq)->(P x0))
% Found ((fun (x3:(((eq (a->Prop)) x0) Xq))=> (((eq_sym00 Xq) x3) P)) ((eq_ref (a->Prop)) x0)) as proof of ((P Xq)->(P x0))
% Found ((fun (x3:(((eq (a->Prop)) x0) Xq))=> ((((eq_sym0 x0) Xq) x3) P)) ((eq_ref (a->Prop)) x0)) as proof of ((P Xq)->(P x0))
% Found ((fun (x3:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x3) P)) ((eq_ref (a->Prop)) x0)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x3:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x3) P)) ((eq_ref (a->Prop)) x0))) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((fun (x3:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x3) P)) ((eq_ref (a->Prop)) x0))) as proof of (((eq (a->Prop)) Xq) x0)
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x4 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x4 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x4 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x4 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x4 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x4 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x4 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x4 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))
% Found x30:(P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion_dep0000 (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found ((eta_expansion_dep000 (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((eta_expansion_dep00 Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x30:(P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion0000 (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found ((eta_expansion000 (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((eta_expansion00 Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion0 Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found ((eq_ref (a->Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found ((eq_ref (a->Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found ((eq_ref (a->Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found x30:(P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 x2):(((eq (a->Prop)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_sym0000 ((eq_ref (a->Prop)) x2)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found ((eq_sym0000 ((eq_ref (a->Prop)) x2)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((fun (x3:(((eq (a->Prop)) x2) Xq))=> ((eq_sym000 x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) ((eq_ref (a->Prop)) x2)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((fun (x3:(((eq (a->Prop)) x2) Xq))=> (((eq_sym00 Xq) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) ((eq_ref (a->Prop)) x2)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((fun (x3:(((eq (a->Prop)) x2) Xq))=> ((((eq_sym0 x2) Xq) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) ((eq_ref (a->Prop)) x2)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((fun (x3:(((eq (a->Prop)) x2) Xq))=> (((((eq_sym (a->Prop)) x2) Xq) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) ((eq_ref (a->Prop)) x2)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x3:(((eq (a->Prop)) x2) Xq))=> (((((eq_sym (a->Prop)) x2) Xq) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) ((eq_ref (a->Prop)) x2)) (fun (x30:(P Xq))=> x30))) as proof of ((P Xq)->(P x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x3:(((eq (a->Prop)) x2) Xq))=> (((((eq_sym (a->Prop)) x2) Xq) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) ((eq_ref (a->Prop)) x2)) (fun (x30:(P Xq))=> x30))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found ((eta_expansion0 Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eq_ref00:=(eq_ref0 x0):(((eq (a->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found (eq_sym000 ((eq_ref (a->Prop)) x0)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((eq_sym00 Xq) ((eq_ref (a->Prop)) x0)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eq_sym0 x0) Xq) ((eq_ref (a->Prop)) x0)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((((eq_sym (a->Prop)) x0) Xq) ((eq_ref (a->Prop)) x0)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((((eq_sym (a->Prop)) x0) Xq) ((eq_ref (a->Prop)) x0)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((((eq_sym (a->Prop)) x0) Xq) ((eq_ref (a->Prop)) x0))) as proof of (((eq (a->Prop)) Xq) x0)
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found ((eta_expansion0 Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))) as proof of ((forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00)))->((iff (x0 Xy)) ((R Xx0) Xy)))
% Found (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))) as proof of ((forall (Xx00:a), ((R Xx00) Xx00))->((forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00)))->((iff (x0 Xy)) ((R Xx0) Xy))))
% Found (and_rect10 (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((and_rect1 ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((fun (P:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P) x3) x1)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((fun (P:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P) x3) x1)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((fun (P:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P) x3) x1)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x2 Xy)):((iff (x2 Xy)) (x2 Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x2 Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x2 Xy))) as proof of ((forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00)))->((iff (x2 Xy)) ((R Xx0) Xy)))
% Found (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x2 Xy))) as proof of ((forall (Xx00:a), ((R Xx00) Xx00))->((forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00)))->((iff (x2 Xy)) ((R Xx0) Xy))))
% Found (and_rect10 (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x2 Xy)))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found ((and_rect1 ((iff (x2 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x2 Xy)))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (((fun (P:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P) x3) x0)) ((iff (x2 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x2 Xy)))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((fun (P:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P) x3) x0)) ((iff (x2 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x2 Xy))))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (((fun (P:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P) x3) x0)) ((iff (x2 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x2 Xy))))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found x30:(P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion0000 (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion000 (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion00 Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion0 Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 (Xq x3)):(((eq Prop) (Xq x3)) (Xq x3))
% Found (eq_ref0 (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x2 x)))
% Found (functional_extensionality_dep00000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x2))
% Found (functional_extensionality_dep00000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((functional_extensionality_dep0000 x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> (((functional_extensionality_dep000 x2) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((functional_extensionality_dep00 Xq) x2) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> (((((functional_extensionality_dep0 (fun (x5:a)=> Prop)) Xq) x2) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) Xq) x2) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) Xq) x2) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3))))) as proof of ((P Xq)->(P x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) Xq) x2) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3))))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 (Xq x3)):(((eq Prop) (Xq x3)) (Xq x3))
% Found (eq_ref0 (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x2 x)))
% Found (functional_extensionality00000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x2))
% Found (functional_extensionality00000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((functional_extensionality0000 x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> (((functional_extensionality000 x2) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((functional_extensionality00 Xq) x2) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> (((((functional_extensionality0 Prop) Xq) x2) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality a) Prop) Xq) x2) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality a) Prop) Xq) x2) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3))))) as proof of ((P Xq)->(P x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality a) Prop) Xq) x2) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3))))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x30:(P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion_dep0000 (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion_dep000 (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion_dep00 Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((eq_ref (a->Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((eq_ref (a->Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((eq_ref (a->Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((eta_expansion0 Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found x30:(P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 x0):(((eq (a->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_sym0000 ((eq_ref (a->Prop)) x0)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found ((eq_sym0000 ((eq_ref (a->Prop)) x0)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((fun (x3:(((eq (a->Prop)) x0) Xq))=> ((eq_sym000 x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) ((eq_ref (a->Prop)) x0)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((fun (x3:(((eq (a->Prop)) x0) Xq))=> (((eq_sym00 Xq) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) ((eq_ref (a->Prop)) x0)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((fun (x3:(((eq (a->Prop)) x0) Xq))=> ((((eq_sym0 x0) Xq) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) ((eq_ref (a->Prop)) x0)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((fun (x3:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) ((eq_ref (a->Prop)) x0)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x3:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) ((eq_ref (a->Prop)) x0)) (fun (x30:(P Xq))=> x30))) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x3:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) ((eq_ref (a->Prop)) x0)) (fun (x30:(P Xq))=> x30))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((eta_expansion0 Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found ((eta_expansion0 Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found ((eq_ref (a->Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found ((eq_ref (a->Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found ((eq_ref (a->Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found ((eta_expansion0 Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found x5:((R Xx0) Xy)
% Instantiate: x4:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x4 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x4 Xy))
% Found x5:(x4 Xy)
% Instantiate: x4:=(R Xx0):(a->Prop)
% Found (fun (x5:(x4 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x4 Xy))=> x5) as proof of ((x4 Xy)->((R Xx0) Xy))
% Found ((conj20 (fun (x5:(x4 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x4 Xy))) (fun (x5:(x4 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x4 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x4 Xy))) (fun (x5:(x4 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x4 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x4 Xy))) (fun (x5:(x4 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x4 Xx0)) (Xy:a)=> ((((conj ((x4 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x4 Xy))) (fun (x5:(x4 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x4))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x4))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x4))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x4))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eq_ref000:=(eq_ref00 P):((P Xq)->(P Xq))
% Found (eq_ref00 P) as proof of ((P Xq)->(P x4))
% Found ((eq_ref0 Xq) P) as proof of ((P Xq)->(P x4))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x4))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x4)
% Found x5:(P Xq)
% Instantiate: x4:=Xq:(a->Prop)
% Found (fun (x5:(P Xq))=> x5) as proof of (P x4)
% Found (fun (P:((a->Prop)->Prop)) (x5:(P Xq))=> x5) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop)) (x5:(P Xq))=> x5) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop)) (x5:(P Xq))=> x5) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x4))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x4))
% Found (((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x4))
% Found ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x4))
% Found ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x4)
% Found x30:(P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 (Xq x3)):(((eq Prop) (Xq x3)) (Xq x3))
% Found (eq_ref0 (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x2 x)))
% Found ((functional_extensionality00000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found ((functional_extensionality00000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((functional_extensionality0000 x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> (((functional_extensionality000 x2) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((functional_extensionality00 Xq) x2) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> (((((functional_extensionality0 Prop) Xq) x2) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality a) Prop) Xq) x2) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality a) Prop) Xq) x2) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30))) as proof of ((P Xq)->(P x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality a) Prop) Xq) x2) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x30:(P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 (Xq x3)):(((eq Prop) (Xq x3)) (Xq x3))
% Found (eq_ref0 (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x2 x)))
% Found ((functional_extensionality_dep00000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found ((functional_extensionality_dep00000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((functional_extensionality_dep0000 x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> (((functional_extensionality_dep000 x2) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((functional_extensionality_dep00 Xq) x2) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> (((((functional_extensionality_dep0 (fun (x5:a)=> Prop)) Xq) x2) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) Xq) x2) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) Xq) x2) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30))) as proof of ((P Xq)->(P x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) Xq) x2) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x4))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x4))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x4))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x4))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x4))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x4))
% Found (((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x4))
% Found ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x4))
% Found ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x4)
% Found x3:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x3:(x0 Xy))=> x3) as proof of ((R Xx0) Xy)
% Found (fun (x3:(x0 Xy))=> x3) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found x3:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x3:((R Xx0) Xy))=> x3) as proof of (x0 Xy)
% Found (fun (x3:((R Xx0) Xy))=> x3) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found ((conj20 (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eq_ref00:=(eq_ref0 (Xq x3)):(((eq Prop) (Xq x3)) (Xq x3))
% Found (eq_ref0 (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found (functional_extensionality_dep00000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x0))
% Found (functional_extensionality_dep00000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((functional_extensionality_dep0000 x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((functional_extensionality_dep000 x0) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((functional_extensionality_dep00 Xq) x0) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((((functional_extensionality_dep0 (fun (x5:a)=> Prop)) Xq) x0) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) Xq) x0) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) Xq) x0) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3))))) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) Xq) x0) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3))))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 (Xq x3)):(((eq Prop) (Xq x3)) (Xq x3))
% Found (eq_ref0 (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found (functional_extensionality00000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x0))
% Found (functional_extensionality00000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((functional_extensionality0000 x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((functional_extensionality000 x0) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((functional_extensionality00 Xq) x0) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((((functional_extensionality0 Prop) Xq) x0) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3))))) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3))))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) 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) x4)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) x4)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) x4)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) x4)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) x4)
% Found ((eq_trans0000 (((eta_expansion a) Prop) Xq)) (((eta_expansion_dep a) (fun (x6:a)=> Prop)) b)) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eq_trans000 x4) (((eta_expansion a) Prop) Xq)) (((eta_expansion_dep a) (fun (x6:a)=> Prop)) b)) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((((eq_trans00 (fun (x7:a)=> (Xq x7))) x4) (((eta_expansion a) Prop) Xq)) (((eta_expansion_dep a) (fun (x6:a)=> Prop)) (fun (x7:a)=> (Xq x7)))) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((((eq_trans0 Xq) (fun (x7:a)=> (Xq x7))) x4) (((eta_expansion a) Prop) Xq)) (((eta_expansion_dep a) (fun (x6:a)=> Prop)) (fun (x7:a)=> (Xq x7)))) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((((((eq_trans (a->Prop)) Xq) (fun (x7:a)=> (Xq x7))) x4) (((eta_expansion a) Prop) Xq)) (((eta_expansion_dep a) (fun (x6:a)=> Prop)) (fun (x7:a)=> (Xq x7)))) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((((((eq_trans (a->Prop)) Xq) (fun (x7:a)=> (Xq x7))) x4) (((eta_expansion a) Prop) Xq)) (((eta_expansion_dep a) (fun (x6:a)=> Prop)) (fun (x7:a)=> (Xq x7))))) as proof of (((eq (a->Prop)) Xq) x4)
% Found x5:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x0 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found x5:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found ((conj20 (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x5:(x2 Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x5:(x2 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x2 Xy))=> x5) as proof of ((x2 Xy)->((R Xx0) Xy))
% Found x5:((R Xx0) Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x2 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x2 Xy))
% Found ((conj20 (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found x30:(P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion0000 (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found ((eta_expansion000 (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((eta_expansion00 Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion0 Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x30:(P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion_dep0000 (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found ((eta_expansion_dep000 (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((eta_expansion_dep00 Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref000:=(eq_ref00 P):((P Xq)->(P Xq))
% Found (eq_ref00 P) as proof of ((P Xq)->(P x0))
% Found ((eq_ref0 Xq) P) as proof of ((P Xq)->(P x0))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x0)
% Found x5:(P Xq)
% Instantiate: x0:=Xq:(a->Prop)
% Found (fun (x5:(P Xq))=> x5) as proof of (P x0)
% Found (fun (P:((a->Prop)->Prop)) (x5:(P Xq))=> x5) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop)) (x5:(P Xq))=> x5) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop)) (x5:(P Xq))=> x5) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 (Xq x1)):(((eq Prop) (Xq x1)) (Xq x1))
% Found (eq_ref0 (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found ((eq_ref Prop) (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found ((eq_ref Prop) (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (Xq x1))) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (Xq x1))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found (functional_extensionality_dep0000 (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((functional_extensionality_dep000 x0) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((functional_extensionality_dep00 Xq) x0) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((((functional_extensionality_dep0 (fun (x3:a)=> Prop)) Xq) x0) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((((functional_extensionality_dep a) (fun (x3:a)=> Prop)) Xq) x0) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((((functional_extensionality_dep a) (fun (x3:a)=> Prop)) Xq) x0) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((((functional_extensionality_dep a) (fun (x3:a)=> Prop)) Xq) x0) (fun (x1:a)=> ((eq_ref Prop) (Xq x1))))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 (Xq x1)):(((eq Prop) (Xq x1)) (Xq x1))
% Found (eq_ref0 (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found ((eq_ref Prop) (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found ((eq_ref Prop) (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (Xq x1))) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (Xq x1))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found (functional_extensionality0000 (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((functional_extensionality000 x0) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((functional_extensionality00 Xq) x0) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((((functional_extensionality0 Prop) Xq) x0) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((((functional_extensionality a) Prop) Xq) x0) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((((functional_extensionality a) Prop) Xq) x0) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((((functional_extensionality a) Prop) Xq) x0) (fun (x1:a)=> ((eq_ref Prop) (Xq x1))))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x2))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x2))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x2)
% Found x5:(P Xq)
% Instantiate: x2:=Xq:(a->Prop)
% Found (fun (x5:(P Xq))=> x5) as proof of (P x2)
% Found (fun (P:((a->Prop)->Prop)) (x5:(P Xq))=> x5) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop)) (x5:(P Xq))=> x5) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop)) (x5:(P Xq))=> x5) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x2))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x2))
% Found (((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref000:=(eq_ref00 P):((P Xq)->(P Xq))
% Found (eq_ref00 P) as proof of ((P Xq)->(P x2))
% Found ((eq_ref0 Xq) P) as proof of ((P Xq)->(P x2))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x2))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x2)
% Found x30:(P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 (Xq x3)):(((eq Prop) (Xq x3)) (Xq x3))
% Found (eq_ref0 (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found ((functional_extensionality00000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found ((functional_extensionality00000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((functional_extensionality0000 x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((functional_extensionality000 x0) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((functional_extensionality00 Xq) x0) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((((functional_extensionality0 Prop) Xq) x0) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30))) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x30:(P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 (Xq x3)):(((eq Prop) (Xq x3)) (Xq x3))
% Found (eq_ref0 (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found ((functional_extensionality_dep00000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found ((functional_extensionality_dep00000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((functional_extensionality_dep0000 x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((functional_extensionality_dep000 x0) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((functional_extensionality_dep00 Xq) x0) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((((functional_extensionality_dep0 (fun (x5:a)=> Prop)) Xq) x0) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) Xq) x0) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) Xq) x0) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30))) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) Xq) x0) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x2))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x2))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x2))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x2))
% Found (((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 x4):(((eq (a->Prop)) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq (a->Prop)) x4) Xq)
% Found ((eq_ref (a->Prop)) x4) as proof of (((eq (a->Prop)) x4) Xq)
% Found ((eq_ref (a->Prop)) x4) as proof of (((eq (a->Prop)) x4) Xq)
% Found ((eq_ref (a->Prop)) x4) as proof of (((eq (a->Prop)) x4) Xq)
% Found (eq_sym0000 ((eq_ref (a->Prop)) x4)) as proof of ((P Xq)->(P x4))
% Found (eq_sym0000 ((eq_ref (a->Prop)) x4)) as proof of ((P Xq)->(P x4))
% Found ((fun (x5:(((eq (a->Prop)) x4) Xq))=> ((eq_sym000 x5) P)) ((eq_ref (a->Prop)) x4)) as proof of ((P Xq)->(P x4))
% Found ((fun (x5:(((eq (a->Prop)) x4) Xq))=> (((eq_sym00 Xq) x5) P)) ((eq_ref (a->Prop)) x4)) as proof of ((P Xq)->(P x4))
% Found ((fun (x5:(((eq (a->Prop)) x4) Xq))=> ((((eq_sym0 x4) Xq) x5) P)) ((eq_ref (a->Prop)) x4)) as proof of ((P Xq)->(P x4))
% Found ((fun (x5:(((eq (a->Prop)) x4) Xq))=> (((((eq_sym (a->Prop)) x4) Xq) x5) P)) ((eq_ref (a->Prop)) x4)) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x5:(((eq (a->Prop)) x4) Xq))=> (((((eq_sym (a->Prop)) x4) Xq) x5) P)) ((eq_ref (a->Prop)) x4))) as proof of ((P Xq)->(P x4))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((fun (x5:(((eq (a->Prop)) x4) Xq))=> (((((eq_sym (a->Prop)) x4) Xq) x5) P)) ((eq_ref (a->Prop)) x4))) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) 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) x0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) x0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_trans0000 (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) b)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eq_trans000 x0) (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) b)) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((((eq_trans00 (fun (x7:a)=> (Xq x7))) x0) (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((((eq_trans0 Xq) (fun (x7:a)=> (Xq x7))) x0) (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((((((eq_trans (a->Prop)) Xq) (fun (x7:a)=> (Xq x7))) x0) (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((((((eq_trans (a->Prop)) Xq) (fun (x7:a)=> (Xq x7))) x0) (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7))))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) x2)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x2)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x2)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x2)
% 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) x2)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) x2)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x2)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x2)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x2)
% Found ((eq_trans0000 ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eq_trans000 x2) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((((eq_trans00 Xq) x2) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((((eq_trans0 Xq) Xq) x2) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((((((eq_trans (a->Prop)) Xq) Xq) x2) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((((((eq_trans (a->Prop)) Xq) Xq) x2) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x30:(P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion0000 (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion000 (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion00 Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion0 Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x30:(P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion_dep0000 (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion_dep000 (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion_dep00 Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P Xq)->(P x4)))) (fun (x30:(P Xq))=> x30))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq ((a->Prop)->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found ((eq_ref ((a->Prop)->Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found ((eq_ref ((a->Prop)->Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found ((eq_ref ((a->Prop)->Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 x2):(((eq (a->Prop)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found (eq_sym000 ((eq_ref (a->Prop)) x2)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found ((eq_sym00 Xq) ((eq_ref (a->Prop)) x2)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((eq_sym0 x2) Xq) ((eq_ref (a->Prop)) x2)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found ((((eq_sym (a->Prop)) x2) Xq) ((eq_ref (a->Prop)) x2)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found ((((eq_sym (a->Prop)) x2) Xq) ((eq_ref (a->Prop)) x2)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((((eq_sym (a->Prop)) x2) Xq) ((eq_ref (a->Prop)) x2))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 x0):(((eq (a->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found (eq_sym0000 ((eq_ref (a->Prop)) x0)) as proof of ((P Xq)->(P x0))
% Found (eq_sym0000 ((eq_ref (a->Prop)) x0)) as proof of ((P Xq)->(P x0))
% Found ((fun (x5:(((eq (a->Prop)) x0) Xq))=> ((eq_sym000 x5) P)) ((eq_ref (a->Prop)) x0)) as proof of ((P Xq)->(P x0))
% Found ((fun (x5:(((eq (a->Prop)) x0) Xq))=> (((eq_sym00 Xq) x5) P)) ((eq_ref (a->Prop)) x0)) as proof of ((P Xq)->(P x0))
% Found ((fun (x5:(((eq (a->Prop)) x0) Xq))=> ((((eq_sym0 x0) Xq) x5) P)) ((eq_ref (a->Prop)) x0)) as proof of ((P Xq)->(P x0))
% Found ((fun (x5:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x5) P)) ((eq_ref (a->Prop)) x0)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x5:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x5) P)) ((eq_ref (a->Prop)) x0))) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((fun (x5:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x5) P)) ((eq_ref (a->Prop)) x0))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 x2):(((eq (a->Prop)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found (eq_sym0000 ((eq_ref (a->Prop)) x2)) as proof of ((P Xq)->(P x2))
% Found (eq_sym0000 ((eq_ref (a->Prop)) x2)) as proof of ((P Xq)->(P x2))
% Found ((fun (x5:(((eq (a->Prop)) x2) Xq))=> ((eq_sym000 x5) P)) ((eq_ref (a->Prop)) x2)) as proof of ((P Xq)->(P x2))
% Found ((fun (x5:(((eq (a->Prop)) x2) Xq))=> (((eq_sym00 Xq) x5) P)) ((eq_ref (a->Prop)) x2)) as proof of ((P Xq)->(P x2))
% Found ((fun (x5:(((eq (a->Prop)) x2) Xq))=> ((((eq_sym0 x2) Xq) x5) P)) ((eq_ref (a->Prop)) x2)) as proof of ((P Xq)->(P x2))
% Found ((fun (x5:(((eq (a->Prop)) x2) Xq))=> (((((eq_sym (a->Prop)) x2) Xq) x5) P)) ((eq_ref (a->Prop)) x2)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x5:(((eq (a->Prop)) x2) Xq))=> (((((eq_sym (a->Prop)) x2) Xq) x5) P)) ((eq_ref (a->Prop)) x2))) as proof of ((P Xq)->(P x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((fun (x5:(((eq (a->Prop)) x2) Xq))=> (((((eq_sym (a->Prop)) x2) Xq) x5) P)) ((eq_ref (a->Prop)) x2))) as proof of (((eq (a->Prop)) Xq) x2)
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))) as proof of ((forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00)))->((iff (x0 Xy)) ((R Xx0) Xy)))
% Found (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))) as proof of ((forall (Xx00:a), ((R Xx00) Xx00))->((forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00)))->((iff (x0 Xy)) ((R Xx0) Xy))))
% Found (and_rect10 (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((and_rect1 ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((fun (P:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P) x3) x1)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((fun (P:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P) x3) x1)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((fun (P:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P) x3) x1)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 x4):(((eq (a->Prop)) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq (a->Prop)) x4) Xq)
% Found ((eq_ref (a->Prop)) x4) as proof of (((eq (a->Prop)) x4) Xq)
% Found ((eq_ref (a->Prop)) x4) as proof of (((eq (a->Prop)) x4) Xq)
% Found ((eq_ref (a->Prop)) x4) as proof of (((eq (a->Prop)) x4) Xq)
% Found ((eq_sym0000 ((eq_ref (a->Prop)) x4)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found ((eq_sym0000 ((eq_ref (a->Prop)) x4)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((fun (x5:(((eq (a->Prop)) x4) Xq))=> ((eq_sym000 x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x4)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((fun (x5:(((eq (a->Prop)) x4) Xq))=> (((eq_sym00 Xq) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x4)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((fun (x5:(((eq (a->Prop)) x4) Xq))=> ((((eq_sym0 x4) Xq) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x4)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((fun (x5:(((eq (a->Prop)) x4) Xq))=> (((((eq_sym (a->Prop)) x4) Xq) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x4)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x5:(((eq (a->Prop)) x4) Xq))=> (((((eq_sym (a->Prop)) x4) Xq) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x4)) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x4))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x5:(((eq (a->Prop)) x4) Xq))=> (((((eq_sym (a->Prop)) x4) Xq) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x4)) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x4)
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion_dep0000 (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found ((eta_expansion_dep000 (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((eta_expansion_dep00 Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found ((((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x4)
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion0000 (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found ((eta_expansion000 (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((eta_expansion00 Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found ((((eta_expansion0 Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x4)
% Found eq_ref00:=(eq_ref0 x0):(((eq (a->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found (eq_sym000 ((eq_ref (a->Prop)) x0)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((eq_sym00 Xq) ((eq_ref (a->Prop)) x0)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eq_sym0 x0) Xq) ((eq_ref (a->Prop)) x0)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((((eq_sym (a->Prop)) x0) Xq) ((eq_ref (a->Prop)) x0)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((((eq_sym (a->Prop)) x0) Xq) ((eq_ref (a->Prop)) x0)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((((eq_sym (a->Prop)) x0) Xq) ((eq_ref (a->Prop)) x0))) as proof of (((eq (a->Prop)) Xq) x0)
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found eq_ref00:=(eq_ref0 (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))):(((eq Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))
% Found (eq_ref0 (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))) as proof of (((eq Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))) b)
% Found ((eq_ref Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))) as proof of (((eq Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))) b)
% Found ((eq_ref Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))) as proof of (((eq Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))) b)
% Found ((eq_ref Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))) as proof of (((eq Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))) b)
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 x0):(((eq (a->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_sym0000 ((eq_ref (a->Prop)) x0)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found ((eq_sym0000 ((eq_ref (a->Prop)) x0)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((fun (x5:(((eq (a->Prop)) x0) Xq))=> ((eq_sym000 x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x0)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((fun (x5:(((eq (a->Prop)) x0) Xq))=> (((eq_sym00 Xq) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x0)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((fun (x5:(((eq (a->Prop)) x0) Xq))=> ((((eq_sym0 x0) Xq) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x0)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((fun (x5:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x0)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x5:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x0)) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x5:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x0)) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion0000 (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion000 (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion00 Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion0 Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 x2):(((eq (a->Prop)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_sym0000 ((eq_ref (a->Prop)) x2)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found ((eq_sym0000 ((eq_ref (a->Prop)) x2)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((fun (x5:(((eq (a->Prop)) x2) Xq))=> ((eq_sym000 x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x2)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((fun (x5:(((eq (a->Prop)) x2) Xq))=> (((eq_sym00 Xq) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x2)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((fun (x5:(((eq (a->Prop)) x2) Xq))=> ((((eq_sym0 x2) Xq) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x2)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((fun (x5:(((eq (a->Prop)) x2) Xq))=> (((((eq_sym (a->Prop)) x2) Xq) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x2)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x5:(((eq (a->Prop)) x2) Xq))=> (((((eq_sym (a->Prop)) x2) Xq) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x2)) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x5:(((eq (a->Prop)) x2) Xq))=> (((((eq_sym (a->Prop)) x2) Xq) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x2)) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion_dep0000 (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion_dep000 (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion_dep00 Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion_dep0000 (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found ((eta_expansion_dep000 (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((eta_expansion_dep00 Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion0000 (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found ((eta_expansion000 (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((eta_expansion00 Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion0 Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 (Xq x5)):(((eq Prop) (Xq x5)) (Xq x5))
% Found (eq_ref0 (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x4 x)))
% Found (functional_extensionality_dep00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x4))
% Found (functional_extensionality_dep00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x4))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((functional_extensionality_dep0000 x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x4))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> (((functional_extensionality_dep000 x4) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x4))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((((functional_extensionality_dep00 Xq) x4) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x4))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> (((((functional_extensionality_dep0 (fun (x7:a)=> Prop)) Xq) x4) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x4))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x4) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x4) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of ((P Xq)->(P x4))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x4) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of (((eq (a->Prop)) Xq) x4)
% Found eq_ref00:=(eq_ref0 (Xq x5)):(((eq Prop) (Xq x5)) (Xq x5))
% Found (eq_ref0 (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x4 x)))
% Found (functional_extensionality00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x4))
% Found (functional_extensionality00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x4))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((functional_extensionality0000 x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x4))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> (((functional_extensionality000 x4) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x4))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((((functional_extensionality00 Xq) x4) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x4))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> (((((functional_extensionality0 Prop) Xq) x4) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x4))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((((((functional_extensionality a) Prop) Xq) x4) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((((((functional_extensionality a) Prop) Xq) x4) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of ((P Xq)->(P x4))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((((((functional_extensionality a) Prop) Xq) x4) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of (((eq (a->Prop)) Xq) x4)
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 (Xq x5)):(((eq Prop) (Xq x5)) (Xq x5))
% Found (eq_ref0 (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x4 x)))
% Found ((functional_extensionality00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found ((functional_extensionality00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((functional_extensionality0000 x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> (((functional_extensionality000 x4) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((((functional_extensionality00 Xq) x4) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> (((((functional_extensionality0 Prop) Xq) x4) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((((((functional_extensionality a) Prop) Xq) x4) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((((((functional_extensionality a) Prop) Xq) x4) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x4))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((((((functional_extensionality a) Prop) Xq) x4) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x4)
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 (Xq x5)):(((eq Prop) (Xq x5)) (Xq x5))
% Found (eq_ref0 (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x4 x)))
% Found ((functional_extensionality_dep00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found ((functional_extensionality_dep00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((functional_extensionality_dep0000 x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> (((functional_extensionality_dep000 x4) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((((functional_extensionality_dep00 Xq) x4) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> (((((functional_extensionality_dep0 (fun (x7:a)=> Prop)) Xq) x4) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x4) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x4) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x4))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x4) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x4)
% Found eq_ref00:=(eq_ref0 (Xq x3)):(((eq Prop) (Xq x3)) (Xq x3))
% Found (eq_ref0 (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x2 x)))
% Found (functional_extensionality0000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found ((functional_extensionality000 x2) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((functional_extensionality00 Xq) x2) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found ((((functional_extensionality0 Prop) Xq) x2) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((((functional_extensionality a) Prop) Xq) x2) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((((functional_extensionality a) Prop) Xq) x2) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((((functional_extensionality a) Prop) Xq) x2) (fun (x3:a)=> ((eq_ref Prop) (Xq x3))))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 (Xq x3)):(((eq Prop) (Xq x3)) (Xq x3))
% Found (eq_ref0 (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x2 x)))
% Found (functional_extensionality_dep0000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found ((functional_extensionality_dep000 x2) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((functional_extensionality_dep00 Xq) x2) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found ((((functional_extensionality_dep0 (fun (x5:a)=> Prop)) Xq) x2) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) Xq) x2) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) Xq) x2) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) Xq) x2) (fun (x3:a)=> ((eq_ref Prop) (Xq x3))))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x5:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x0 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found x5:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found ((conj20 (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x5:((R Xx0) Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x2 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x2 Xy))
% Found x5:(x2 Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x5:(x2 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x2 Xy))=> x5) as proof of ((x2 Xy)->((R Xx0) Xy))
% Found ((conj20 (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq ((a->Prop)->Prop)) a0) (fun (x:(a->Prop))=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x3:(a->Prop))=> Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq ((a->Prop)->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found ((eq_ref ((a->Prop)->Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found ((eq_ref ((a->Prop)->Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found ((eq_ref ((a->Prop)->Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 (Xq x5)):(((eq Prop) (Xq x5)) (Xq x5))
% Found (eq_ref0 (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found (functional_extensionality00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x0))
% Found (functional_extensionality00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((functional_extensionality0000 x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((functional_extensionality000 x0) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((functional_extensionality00 Xq) x0) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((((functional_extensionality0 Prop) Xq) x0) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 (Xq x5)):(((eq Prop) (Xq x5)) (Xq x5))
% Found (eq_ref0 (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found (functional_extensionality_dep00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x0))
% Found (functional_extensionality_dep00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((functional_extensionality_dep0000 x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((functional_extensionality_dep000 x0) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((functional_extensionality_dep00 Xq) x0) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((((functional_extensionality_dep0 (fun (x7:a)=> Prop)) Xq) x0) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x0) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x0) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x0) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 (Xq x5)):(((eq Prop) (Xq x5)) (Xq x5))
% Found (eq_ref0 (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x2 x)))
% Found (functional_extensionality_dep00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x2))
% Found (functional_extensionality_dep00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((functional_extensionality_dep0000 x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> (((functional_extensionality_dep000 x2) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((functional_extensionality_dep00 Xq) x2) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> (((((functional_extensionality_dep0 (fun (x7:a)=> Prop)) Xq) x2) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x2) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x2) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of ((P Xq)->(P x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x2) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 (Xq x5)):(((eq Prop) (Xq x5)) (Xq x5))
% Found (eq_ref0 (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x2 x)))
% Found (functional_extensionality00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x2))
% Found (functional_extensionality00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((functional_extensionality0000 x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> (((functional_extensionality000 x2) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((functional_extensionality00 Xq) x2) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> (((((functional_extensionality0 Prop) Xq) x2) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality a) Prop) Xq) x2) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality a) Prop) Xq) x2) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of ((P Xq)->(P x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality a) Prop) Xq) x2) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion_dep0000 (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found ((eta_expansion_dep000 (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((eta_expansion_dep00 Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found ((((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x4)
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion0000 (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found ((eta_expansion000 (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((eta_expansion00 Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found ((((eta_expansion0 Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x4)
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 (Xq x5)):(((eq Prop) (Xq x5)) (Xq x5))
% Found (eq_ref0 (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found ((functional_extensionality_dep00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found ((functional_extensionality_dep00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((functional_extensionality_dep0000 x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((functional_extensionality_dep000 x0) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((functional_extensionality_dep00 Xq) x0) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((((functional_extensionality_dep0 (fun (x7:a)=> Prop)) Xq) x0) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x0) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x0) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x0) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 (Xq x5)):(((eq Prop) (Xq x5)) (Xq x5))
% Found (eq_ref0 (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found ((functional_extensionality00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found ((functional_extensionality00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((functional_extensionality0000 x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((functional_extensionality000 x0) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((functional_extensionality00 Xq) x0) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((((functional_extensionality0 Prop) Xq) x0) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 (Xq x5)):(((eq Prop) (Xq x5)) (Xq x5))
% Found (eq_ref0 (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x2 x)))
% Found ((functional_extensionality_dep00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found ((functional_extensionality_dep00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((functional_extensionality_dep0000 x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> (((functional_extensionality_dep000 x2) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((functional_extensionality_dep00 Xq) x2) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> (((((functional_extensionality_dep0 (fun (x7:a)=> Prop)) Xq) x2) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x2) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x2) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x2) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 (Xq x5)):(((eq Prop) (Xq x5)) (Xq x5))
% Found (eq_ref0 (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x2 x)))
% Found ((functional_extensionality00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found ((functional_extensionality00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((functional_extensionality0000 x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> (((functional_extensionality000 x2) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((functional_extensionality00 Xq) x2) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> (((((functional_extensionality0 Prop) Xq) x2) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality a) Prop) Xq) x2) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality a) Prop) Xq) x2) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality a) Prop) Xq) x2) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 (Xq x3)):(((eq Prop) (Xq x3)) (Xq x3))
% Found (eq_ref0 (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found (functional_extensionality0000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((functional_extensionality000 x0) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((functional_extensionality00 Xq) x0) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((((functional_extensionality0 Prop) Xq) x0) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((((functional_extensionality a) Prop) Xq) x0) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((((functional_extensionality a) Prop) Xq) x0) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((((functional_extensionality a) Prop) Xq) x0) (fun (x3:a)=> ((eq_ref Prop) (Xq x3))))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 (Xq x3)):(((eq Prop) (Xq x3)) (Xq x3))
% Found (eq_ref0 (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found (functional_extensionality_dep0000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((functional_extensionality_dep000 x0) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((functional_extensionality_dep00 Xq) x0) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((((functional_extensionality_dep0 (fun (x5:a)=> Prop)) Xq) x0) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) Xq) x0) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) Xq) x0) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) Xq) x0) (fun (x3:a)=> ((eq_ref Prop) (Xq x3))))) as proof of (((eq (a->Prop)) Xq) x0)
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion0000 (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion000 (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion00 Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion0 Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion_dep0000 (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found ((eta_expansion_dep000 (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((eta_expansion_dep00 Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found ((((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion_dep0000 (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found ((eta_expansion_dep000 (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((eta_expansion_dep00 Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion0000 (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found ((eta_expansion000 (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((eta_expansion00 Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found ((((eta_expansion0 Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x6:(a->Prop))=> ((P Xq)->(P x6)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))):(((eq Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))
% Found (eq_ref0 (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))) as proof of (((eq Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))) b)
% Found ((eq_ref Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))) as proof of (((eq Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))) b)
% Found ((eq_ref Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))) as proof of (((eq Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))) b)
% Found ((eq_ref Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))) as proof of (((eq Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))) b)
% Found eq_ref00:=(eq_ref0 x4):(((eq (a->Prop)) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq (a->Prop)) x4) Xq)
% Found ((eq_ref (a->Prop)) x4) as proof of (((eq (a->Prop)) x4) Xq)
% Found ((eq_ref (a->Prop)) x4) as proof of (((eq (a->Prop)) x4) Xq)
% Found ((eq_ref (a->Prop)) x4) as proof of (((eq (a->Prop)) x4) Xq)
% Found (eq_sym000 ((eq_ref (a->Prop)) x4)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found ((eq_sym00 Xq) ((eq_ref (a->Prop)) x4)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (((eq_sym0 x4) Xq) ((eq_ref (a->Prop)) x4)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found ((((eq_sym (a->Prop)) x4) Xq) ((eq_ref (a->Prop)) x4)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found ((((eq_sym (a->Prop)) x4) Xq) ((eq_ref (a->Prop)) x4)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((((eq_sym (a->Prop)) x4) Xq) ((eq_ref (a->Prop)) x4))) as proof of (((eq (a->Prop)) Xq) x4)
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eq_ref00:=(eq_ref0 (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))):(((eq Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))
% Found (eq_ref0 (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))) as proof of (((eq Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))) b)
% Found ((eq_ref Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))) as proof of (((eq Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))) b)
% Found ((eq_ref Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))) as proof of (((eq Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))) b)
% Found ((eq_ref Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))) as proof of (((eq Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))) b)
% Found eq_ref00:=(eq_ref0 x0):(((eq (a->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found (eq_sym000 ((eq_ref (a->Prop)) x0)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((eq_sym00 Xq) ((eq_ref (a->Prop)) x0)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eq_sym0 x0) Xq) ((eq_ref (a->Prop)) x0)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((((eq_sym (a->Prop)) x0) Xq) ((eq_ref (a->Prop)) x0)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((((eq_sym (a->Prop)) x0) Xq) ((eq_ref (a->Prop)) x0)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((((eq_sym (a->Prop)) x0) Xq) ((eq_ref (a->Prop)) x0))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 x2):(((eq (a->Prop)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found (eq_sym000 ((eq_ref (a->Prop)) x2)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found ((eq_sym00 Xq) ((eq_ref (a->Prop)) x2)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((eq_sym0 x2) Xq) ((eq_ref (a->Prop)) x2)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found ((((eq_sym (a->Prop)) x2) Xq) ((eq_ref (a->Prop)) x2)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found ((((eq_sym (a->Prop)) x2) Xq) ((eq_ref (a->Prop)) x2)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((((eq_sym (a->Prop)) x2) Xq) ((eq_ref (a->Prop)) x2))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq ((a->Prop)->Prop)) a0) (fun (x:(a->Prop))=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x5:(a->Prop))=> Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq ((a->Prop)->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found ((eq_ref ((a->Prop)->Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found ((eq_ref ((a->Prop)->Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found ((eq_ref ((a->Prop)->Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found x5:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found x5:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x0 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found ((conj20 (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found ex_intro0:=(ex_intro (a->Prop)):(forall (P:((a->Prop)->Prop)) (x:(a->Prop)), ((P x)->((ex (a->Prop)) P)))
% Instantiate: b:=(forall (P:((a->Prop)->Prop)) (x:(a->Prop)), ((P x)->((ex (a->Prop)) P))):Prop
% Found ex_intro0 as proof of b
% Found eq_ref00:=(eq_ref0 (Xq x5)):(((eq Prop) (Xq x5)) (Xq x5))
% Found (eq_ref0 (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x4 x)))
% Found (functional_extensionality0000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found ((functional_extensionality000 x4) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (((functional_extensionality00 Xq) x4) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found ((((functional_extensionality0 Prop) Xq) x4) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (((((functional_extensionality a) Prop) Xq) x4) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (((((functional_extensionality a) Prop) Xq) x4) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((((functional_extensionality a) Prop) Xq) x4) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of (((eq (a->Prop)) Xq) x4)
% Found eq_ref00:=(eq_ref0 (Xq x5)):(((eq Prop) (Xq x5)) (Xq x5))
% Found (eq_ref0 (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x4 x)))
% Found (functional_extensionality_dep0000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found ((functional_extensionality_dep000 x4) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (((functional_extensionality_dep00 Xq) x4) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found ((((functional_extensionality_dep0 (fun (x7:a)=> Prop)) Xq) x4) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x4) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x4) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x4) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of (((eq (a->Prop)) Xq) x4)
% Found iff_refl0:=(iff_refl (x2 Xy)):((iff (x2 Xy)) (x2 Xy))
% Found (iff_refl (x2 Xy)) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found (iff_refl (x2 Xy)) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found (iff_refl (x2 Xy)) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found (fun (Xy:a)=> (iff_refl (x2 Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (iff_refl (x2 Xy))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found ex_intro0:=(ex_intro (a->Prop)):(forall (P:((a->Prop)->Prop)) (x:(a->Prop)), ((P x)->((ex (a->Prop)) P)))
% Instantiate: b:=(forall (P:((a->Prop)->Prop)) (x:(a->Prop)), ((P x)->((ex (a->Prop)) P))):Prop
% Found ex_intro0 as proof of b
% Found x1:(P0 Xq)
% Instantiate: x0:=Xq:(a->Prop)
% Found (fun (x1:(P0 Xq))=> x1) as proof of (P0 x0)
% Found (fun (P0:((a->Prop)->Prop)) (x1:(P0 Xq))=> x1) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x1:(P0 Xq))=> x1) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 (Xq x5)):(((eq Prop) (Xq x5)) (Xq x5))
% Found (eq_ref0 (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found (functional_extensionality_dep0000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((functional_extensionality_dep000 x0) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((functional_extensionality_dep00 Xq) x0) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((((functional_extensionality_dep0 (fun (x7:a)=> Prop)) Xq) x0) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x0) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x0) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x0) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 (Xq x5)):(((eq Prop) (Xq x5)) (Xq x5))
% Found (eq_ref0 (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found (functional_extensionality0000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((functional_extensionality000 x0) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((functional_extensionality00 Xq) x0) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((((functional_extensionality0 Prop) Xq) x0) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((((functional_extensionality a) Prop) Xq) x0) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((((functional_extensionality a) Prop) Xq) x0) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((((functional_extensionality a) Prop) Xq) x0) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 (Xq x5)):(((eq Prop) (Xq x5)) (Xq x5))
% Found (eq_ref0 (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x2 x)))
% Found (functional_extensionality_dep0000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found ((functional_extensionality_dep000 x2) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((functional_extensionality_dep00 Xq) x2) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found ((((functional_extensionality_dep0 (fun (x7:a)=> Prop)) Xq) x2) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x2) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x2) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x2) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 (Xq x5)):(((eq Prop) (Xq x5)) (Xq x5))
% Found (eq_ref0 (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x2 x)))
% Found (functional_extensionality0000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found ((functional_extensionality000 x2) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((functional_extensionality00 Xq) x2) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found ((((functional_extensionality0 Prop) Xq) x2) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((((functional_extensionality a) Prop) Xq) x2) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((((functional_extensionality a) Prop) Xq) x2) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((((functional_extensionality a) Prop) Xq) x2) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))):(((eq Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))
% Found (eq_ref0 (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))) as proof of (((eq Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))) b)
% Found ((eq_ref Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))) as proof of (((eq Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))) b)
% Found ((eq_ref Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))) as proof of (((eq Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))) b)
% Found ((eq_ref Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))) as proof of (((eq Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))) b)
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% 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) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found (((eq_trans000 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((((eq_trans00 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found (((((eq_trans0 (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((((((eq_trans Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% 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) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found (((eq_trans000 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((((eq_trans00 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found (((((eq_trans0 (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((((((eq_trans Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found eq_ref00:=(eq_ref0 (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))):(((eq Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))
% Found (eq_ref0 (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))) as proof of (((eq Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))) b)
% Found ((eq_ref Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))) as proof of (((eq Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))) b)
% Found ((eq_ref Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))) as proof of (((eq Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))) b)
% Found ((eq_ref Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))) as proof of (((eq Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))):(((eq Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))
% Found (eq_ref0 (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))) as proof of (((eq Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))) b)
% Found ((eq_ref Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))) as proof of (((eq Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))) b)
% Found ((eq_ref Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))) as proof of (((eq Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))) b)
% Found ((eq_ref Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))) as proof of (((eq Prop) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))) b)
% Found ex_intro0:=(ex_intro (a->Prop)):(forall (P:((a->Prop)->Prop)) (x:(a->Prop)), ((P x)->((ex (a->Prop)) P)))
% Instantiate: b:=(forall (P:((a->Prop)->Prop)) (x:(a->Prop)), ((P x)->((ex (a->Prop)) P))):Prop
% Found ex_intro0 as proof of b
% Found x10:(P0 (f x0))
% Found (fun (x10:(P0 (f x0)))=> x10) as proof of (P0 (f x0))
% Found (fun (x10:(P0 (f x0)))=> x10) as proof of (P1 (f x0))
% Found x10:(P0 (f x0))
% Found (fun (x10:(P0 (f x0)))=> x10) as proof of (P0 (f x0))
% Found (fun (x10:(P0 (f x0)))=> x10) as proof of (P1 (f x0))
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_trans0000 ((eq_ref (a->Prop)) Xq)) ((eq_ref (a->Prop)) b)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eq_trans000 x0) ((eq_ref (a->Prop)) Xq)) ((eq_ref (a->Prop)) b)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((((eq_trans00 Xq) x0) ((eq_ref (a->Prop)) Xq)) ((eq_ref (a->Prop)) Xq)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((((eq_trans0 Xq) Xq) x0) ((eq_ref (a->Prop)) Xq)) ((eq_ref (a->Prop)) Xq)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((((((eq_trans (a->Prop)) Xq) Xq) x0) ((eq_ref (a->Prop)) Xq)) ((eq_ref (a->Prop)) Xq)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((((((eq_trans (a->Prop)) Xq) Xq) x0) ((eq_ref (a->Prop)) Xq)) ((eq_ref (a->Prop)) Xq)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((((((eq_trans (a->Prop)) Xq) Xq) x0) ((eq_ref (a->Prop)) Xq)) ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b
% Found eq_ref00:=(eq_ref0 x0):(((eq (a->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found (eq_sym0000 ((eq_ref (a->Prop)) x0)) as proof of ((P Xq)->(P x0))
% Found (eq_sym0000 ((eq_ref (a->Prop)) x0)) as proof of ((P Xq)->(P x0))
% Found ((fun (x1:(((eq (a->Prop)) x0) Xq))=> ((eq_sym000 x1) P)) ((eq_ref (a->Prop)) x0)) as proof of ((P Xq)->(P x0))
% Found ((fun (x1:(((eq (a->Prop)) x0) Xq))=> (((eq_sym00 Xq) x1) P)) ((eq_ref (a->Prop)) x0)) as proof of ((P Xq)->(P x0))
% Found ((fun (x1:(((eq (a->Prop)) x0) Xq))=> ((((eq_sym0 x0) Xq) x1) P)) ((eq_ref (a->Prop)) x0)) as proof of ((P Xq)->(P x0))
% Found ((fun (x1:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x1) P)) ((eq_ref (a->Prop)) x0)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x1:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x1) P)) ((eq_ref (a->Prop)) x0))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x1:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x1) P)) ((eq_ref (a->Prop)) x0))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((fun (x1:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x1) P)) ((eq_ref (a->Prop)) x0))) as proof of (((eq (a->Prop)) Xq) x0)
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x4 Xy)):((iff (x4 Xy)) (x4 Xy))
% Found (iff_refl (x4 Xy)) as proof of ((and ((x4 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x4 Xy)))
% Found (iff_refl (x4 Xy)) as proof of ((and ((x4 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x4 Xy)))
% Found (iff_refl (x4 Xy)) as proof of ((and ((x4 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x4 Xy)))
% Found (fun (Xy:a)=> (iff_refl (x4 Xy))) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x4 Xx0)) (Xy:a)=> (iff_refl (x4 Xy))) as proof of (forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x3:(P0 Xq)
% Instantiate: x2:=Xq:(a->Prop)
% Found (fun (x3:(P0 Xq))=> x3) as proof of (P0 x2)
% Found (fun (P0:((a->Prop)->Prop)) (x3:(P0 Xq))=> x3) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x3:(P0 Xq))=> x3) as proof of (((eq (a->Prop)) Xq) x2)
% 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)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found (((eq_trans000 ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found ((((eq_trans00 ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found (((((eq_trans0 (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found ((((((eq_trans Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% 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)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found (((eq_trans000 ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found ((((eq_trans00 ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found (((((eq_trans0 (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found ((((((eq_trans Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x2 Xy)):((iff (x2 Xy)) (x2 Xy))
% Found (iff_refl (x2 Xy)) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found (iff_refl (x2 Xy)) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found (iff_refl (x2 Xy)) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found (fun (Xy:a)=> (iff_refl (x2 Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (iff_refl (x2 Xy))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found x3:(P0 Xq)
% Instantiate: x0:=Xq:(a->Prop)
% Found (fun (x3:(P0 Xq))=> x3) as proof of (P0 x0)
% Found (fun (P0:((a->Prop)->Prop)) (x3:(P0 Xq))=> x3) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x3:(P0 Xq))=> x3) as proof of (((eq (a->Prop)) Xq) x0)
% Found x10:(P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of (P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 x0):(((eq (a->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_sym0000 ((eq_ref (a->Prop)) x0)) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found ((eq_sym0000 ((eq_ref (a->Prop)) x0)) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((fun (x1:(((eq (a->Prop)) x0) Xq))=> ((eq_sym000 x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) ((eq_ref (a->Prop)) x0)) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((fun (x1:(((eq (a->Prop)) x0) Xq))=> (((eq_sym00 Xq) x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) ((eq_ref (a->Prop)) x0)) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((fun (x1:(((eq (a->Prop)) x0) Xq))=> ((((eq_sym0 x0) Xq) x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) ((eq_ref (a->Prop)) x0)) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((fun (x1:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) ((eq_ref (a->Prop)) x0)) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x1:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) ((eq_ref (a->Prop)) x0)) (fun (x10:(P Xq))=> x10))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x1:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) ((eq_ref (a->Prop)) x0)) (fun (x10:(P Xq))=> x10))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x1:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) ((eq_ref (a->Prop)) x0)) (fun (x10:(P Xq))=> x10))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b
% Found x1:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x1:((R Xx0) Xy))=> x1) as proof of (x0 Xy)
% Found (fun (x1:((R Xx0) Xy))=> x1) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found x1:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x1:(x0 Xy))=> x1) as proof of ((R Xx0) Xy)
% Found (fun (x1:(x0 Xy))=> x1) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found ((conj20 (fun (x1:(x0 Xy))=> x1)) (fun (x1:((R Xx0) Xy))=> x1)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x1:(x0 Xy))=> x1)) (fun (x1:((R Xx0) Xy))=> x1)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x1:(x0 Xy))=> x1)) (fun (x1:((R Xx0) Xy))=> x1)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x1:(x0 Xy))=> x1)) (fun (x1:((R Xx0) Xy))=> x1)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x1:(x0 Xy))=> x1)) (fun (x1:((R Xx0) Xy))=> x1))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x1:(x0 Xy))=> x1)) (fun (x1:((R Xx0) Xy))=> x1))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eq_ref00:=(eq_ref0 (Xq x1)):(((eq Prop) (Xq x1)) (Xq x1))
% Found (eq_ref0 (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found ((eq_ref Prop) (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found ((eq_ref Prop) (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (Xq x1))) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (Xq x1))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found (functional_extensionality00000 (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of ((P Xq)->(P x0))
% Found (functional_extensionality00000 (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((functional_extensionality0000 x1) P)) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((functional_extensionality000 x0) x1) P)) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((functional_extensionality00 Xq) x0) x1) P)) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((((functional_extensionality0 Prop) Xq) x0) x1) P)) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x1) P)) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x1) P)) (fun (x1:a)=> ((eq_ref Prop) (Xq x1))))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x1) P)) (fun (x1:a)=> ((eq_ref Prop) (Xq x1))))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x1) P)) (fun (x1:a)=> ((eq_ref Prop) (Xq x1))))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 (Xq x1)):(((eq Prop) (Xq x1)) (Xq x1))
% Found (eq_ref0 (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found ((eq_ref Prop) (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found ((eq_ref Prop) (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (Xq x1))) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (Xq x1))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found (functional_extensionality_dep00000 (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of ((P Xq)->(P x0))
% Found (functional_extensionality_dep00000 (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((functional_extensionality_dep0000 x1) P)) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((functional_extensionality_dep000 x0) x1) P)) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((functional_extensionality_dep00 Xq) x0) x1) P)) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((((functional_extensionality_dep0 (fun (x3:a)=> Prop)) Xq) x0) x1) P)) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x3:a)=> Prop)) Xq) x0) x1) P)) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x3:a)=> Prop)) Xq) x0) x1) P)) (fun (x1:a)=> ((eq_ref Prop) (Xq x1))))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x3:a)=> Prop)) Xq) x0) x1) P)) (fun (x1:a)=> ((eq_ref Prop) (Xq x1))))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x3:a)=> Prop)) Xq) x0) x1) P)) (fun (x1:a)=> ((eq_ref Prop) (Xq x1))))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x10:(P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of (P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 (Xq x1)):(((eq Prop) (Xq x1)) (Xq x1))
% Found (eq_ref0 (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found ((eq_ref Prop) (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found ((eq_ref Prop) (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (Xq x1))) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (Xq x1))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found ((functional_extensionality00000 (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found ((functional_extensionality00000 (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((functional_extensionality0000 x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((functional_extensionality000 x0) x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((functional_extensionality00 Xq) x0) x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((((functional_extensionality0 Prop) Xq) x0) x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) (fun (x10:(P Xq))=> x10))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) (fun (x10:(P Xq))=> x10))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) (fun (x10:(P Xq))=> x10))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x10:(P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of (P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 (Xq x1)):(((eq Prop) (Xq x1)) (Xq x1))
% Found (eq_ref0 (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found ((eq_ref Prop) (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found ((eq_ref Prop) (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (Xq x1))) as proof of (((eq Prop) (Xq x1)) (x0 x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (Xq x1))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found ((functional_extensionality_dep00000 (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found ((functional_extensionality_dep00000 (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((functional_extensionality_dep0000 x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((functional_extensionality_dep000 x0) x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((functional_extensionality_dep00 Xq) x0) x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((((functional_extensionality_dep0 (fun (x3:a)=> Prop)) Xq) x0) x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x3:a)=> Prop)) Xq) x0) x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x3:a)=> Prop)) Xq) x0) x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) (fun (x10:(P Xq))=> x10))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x3:a)=> Prop)) Xq) x0) x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) (fun (x10:(P Xq))=> x10))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x1:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x3:a)=> Prop)) Xq) x0) x1) (fun (x3:(a->Prop))=> ((P Xq)->(P x3))))) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) (fun (x10:(P Xq))=> x10))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x2:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x1:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x2:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)) as proof of ((Xq Xx)->(((eq (a->Prop)) Xq) x0))
% Found (fun (x1:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x2:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)) as proof of ((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->(((eq (a->Prop)) Xq) x0)))
% Found (and_rect00 (fun (x1:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x2:(Xq Xx))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((and_rect0 (((eq (a->Prop)) Xq) x0)) (fun (x1:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x2:(Xq Xx))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((fun (P0:Type) (x1:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x1) x00)) (((eq (a->Prop)) Xq) x0)) (fun (x1:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x2:(Xq Xx))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P0:Type) (x1:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x1) x00)) (((eq (a->Prop)) Xq) x0)) (fun (x1:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x2:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x3:(P0 Xq)
% Instantiate: x0:=Xq:(a->Prop)
% Found (fun (x3:(P0 Xq))=> x3) as proof of (P0 x0)
% Found (fun (P0:((a->Prop)->Prop)) (x3:(P0 Xq))=> x3) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x3:(P0 Xq))=> x3) as proof of (((eq (a->Prop)) Xq) x0)
% Found x30:(P0 (f x2))
% Found (fun (x30:(P0 (f x2)))=> x30) as proof of (P0 (f x2))
% Found (fun (x30:(P0 (f x2)))=> x30) as proof of (P1 (f x2))
% Found x30:(P0 (f x2))
% Found (fun (x30:(P0 (f x2)))=> x30) as proof of (P0 (f x2))
% Found (fun (x30:(P0 (f x2)))=> x30) as proof of (P1 (f x2))
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) 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) x2)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) x2)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) x2)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) x2)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) x2)
% Found ((eq_trans0000 (((eta_expansion a) Prop) Xq)) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) b)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((eq_trans000 x2) (((eta_expansion a) Prop) Xq)) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) b)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found ((((eq_trans00 (fun (x5:a)=> (Xq x5))) x2) (((eta_expansion a) Prop) Xq)) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (x5:a)=> (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((((eq_trans0 Xq) (fun (x5:a)=> (Xq x5))) x2) (((eta_expansion a) Prop) Xq)) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (x5:a)=> (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found ((((((eq_trans (a->Prop)) Xq) (fun (x5:a)=> (Xq x5))) x2) (((eta_expansion a) Prop) Xq)) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (x5:a)=> (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found ((((((eq_trans (a->Prop)) Xq) (fun (x5:a)=> (Xq x5))) x2) (((eta_expansion a) Prop) Xq)) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (x5:a)=> (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((((((eq_trans (a->Prop)) Xq) (fun (x5:a)=> (Xq x5))) x2) (((eta_expansion a) Prop) Xq)) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (x5:a)=> (Xq x5))))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_trans00000 (((eta_expansion a) Prop) Xq)) ((eq_ref (a->Prop)) b)) as proof of ((P Xq)->(P x0))
% Found ((eq_trans00000 (((eta_expansion a) Prop) Xq)) ((eq_ref (a->Prop)) b)) as proof of ((P Xq)->(P x0))
% Found (((fun (x1:(((eq (a->Prop)) Xq) b)) (x2:(((eq (a->Prop)) b) x0))=> (((eq_trans0000 x1) x2) P)) (((eta_expansion a) Prop) Xq)) ((eq_ref (a->Prop)) b)) as proof of ((P Xq)->(P x0))
% Found (((fun (x1:(((eq (a->Prop)) Xq) b)) (x2:(((eq (a->Prop)) b) x0))=> ((((eq_trans000 x0) x1) x2) P)) (((eta_expansion a) Prop) Xq)) ((eq_ref (a->Prop)) b)) as proof of ((P Xq)->(P x0))
% Found (((fun (x1:(((eq (a->Prop)) Xq) (fun (x3:a)=> (Xq x3)))) (x2:(((eq (a->Prop)) (fun (x3:a)=> (Xq x3))) x0))=> (((((eq_trans00 (fun (x3:a)=> (Xq x3))) x0) x1) x2) P)) (((eta_expansion a) Prop) Xq)) ((eq_ref (a->Prop)) (fun (x3:a)=> (Xq x3)))) as proof of ((P Xq)->(P x0))
% Found (((fun (x1:(((eq (a->Prop)) Xq) (fun (x3:a)=> (Xq x3)))) (x2:(((eq (a->Prop)) (fun (x3:a)=> (Xq x3))) x0))=> ((((((eq_trans0 Xq) (fun (x3:a)=> (Xq x3))) x0) x1) x2) P)) (((eta_expansion a) Prop) Xq)) ((eq_ref (a->Prop)) (fun (x3:a)=> (Xq x3)))) as proof of ((P Xq)->(P x0))
% Found (((fun (x1:(((eq (a->Prop)) Xq) (fun (x3:a)=> (Xq x3)))) (x2:(((eq (a->Prop)) (fun (x3:a)=> (Xq x3))) x0))=> (((((((eq_trans (a->Prop)) Xq) (fun (x3:a)=> (Xq x3))) x0) x1) x2) P)) (((eta_expansion a) Prop) Xq)) ((eq_ref (a->Prop)) (fun (x3:a)=> (Xq x3)))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x1:(((eq (a->Prop)) Xq) (fun (x3:a)=> (Xq x3)))) (x2:(((eq (a->Prop)) (fun (x3:a)=> (Xq x3))) x0))=> (((((((eq_trans (a->Prop)) Xq) (fun (x3:a)=> (Xq x3))) x0) x1) x2) P)) (((eta_expansion a) Prop) Xq)) ((eq_ref (a->Prop)) (fun (x3:a)=> (Xq x3))))) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x1:(((eq (a->Prop)) Xq) (fun (x3:a)=> (Xq x3)))) (x2:(((eq (a->Prop)) (fun (x3:a)=> (Xq x3))) x0))=> (((((((eq_trans (a->Prop)) Xq) (fun (x3:a)=> (Xq x3))) x0) x1) x2) P)) (((eta_expansion a) Prop) Xq)) ((eq_ref (a->Prop)) (fun (x3:a)=> (Xq x3))))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 x2):(((eq (a->Prop)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found (eq_sym0000 ((eq_ref (a->Prop)) x2)) as proof of ((P Xq)->(P x2))
% Found (eq_sym0000 ((eq_ref (a->Prop)) x2)) as proof of ((P Xq)->(P x2))
% Found ((fun (x3:(((eq (a->Prop)) x2) Xq))=> ((eq_sym000 x3) P)) ((eq_ref (a->Prop)) x2)) as proof of ((P Xq)->(P x2))
% Found ((fun (x3:(((eq (a->Prop)) x2) Xq))=> (((eq_sym00 Xq) x3) P)) ((eq_ref (a->Prop)) x2)) as proof of ((P Xq)->(P x2))
% Found ((fun (x3:(((eq (a->Prop)) x2) Xq))=> ((((eq_sym0 x2) Xq) x3) P)) ((eq_ref (a->Prop)) x2)) as proof of ((P Xq)->(P x2))
% Found ((fun (x3:(((eq (a->Prop)) x2) Xq))=> (((((eq_sym (a->Prop)) x2) Xq) x3) P)) ((eq_ref (a->Prop)) x2)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x3:(((eq (a->Prop)) x2) Xq))=> (((((eq_sym (a->Prop)) x2) Xq) x3) P)) ((eq_ref (a->Prop)) x2))) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x3:(((eq (a->Prop)) x2) Xq))=> (((((eq_sym (a->Prop)) x2) Xq) x3) P)) ((eq_ref (a->Prop)) x2))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((fun (x3:(((eq (a->Prop)) x2) Xq))=> (((((eq_sym (a->Prop)) x2) Xq) x3) P)) ((eq_ref (a->Prop)) x2))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) 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) x0)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_trans0000 (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) b)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eq_trans000 x0) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) b)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((((eq_trans00 (fun (x5:a)=> (Xq x5))) x0) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (x5:a)=> (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((((eq_trans0 Xq) (fun (x5:a)=> (Xq x5))) x0) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (x5:a)=> (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((((((eq_trans (a->Prop)) Xq) (fun (x5:a)=> (Xq x5))) x0) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (x5:a)=> (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((((((eq_trans (a->Prop)) Xq) (fun (x5:a)=> (Xq x5))) x0) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (x5:a)=> (Xq x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((((((eq_trans (a->Prop)) Xq) (fun (x5:a)=> (Xq x5))) x0) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (x5:a)=> (Xq x5))))) as proof of (((eq (a->Prop)) Xq) x0)
% Found iff_refl0:=(iff_refl (x2 Xy)):((iff (x2 Xy)) (x2 Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x2 Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (iff_refl (x2 Xy))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))):(((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found (eq_ref0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))):(((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found (eq_ref0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) b)
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found ((iff_sym0 (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x2 Xy)):((iff (x2 Xy)) (x2 Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x2 Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (iff_refl (x2 Xy))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found eq_ref00:=(eq_ref0 x0):(((eq (a->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found (eq_sym0000 ((eq_ref (a->Prop)) x0)) as proof of ((P Xq)->(P x0))
% Found (eq_sym0000 ((eq_ref (a->Prop)) x0)) as proof of ((P Xq)->(P x0))
% Found ((fun (x3:(((eq (a->Prop)) x0) Xq))=> ((eq_sym000 x3) P)) ((eq_ref (a->Prop)) x0)) as proof of ((P Xq)->(P x0))
% Found ((fun (x3:(((eq (a->Prop)) x0) Xq))=> (((eq_sym00 Xq) x3) P)) ((eq_ref (a->Prop)) x0)) as proof of ((P Xq)->(P x0))
% Found ((fun (x3:(((eq (a->Prop)) x0) Xq))=> ((((eq_sym0 x0) Xq) x3) P)) ((eq_ref (a->Prop)) x0)) as proof of ((P Xq)->(P x0))
% Found ((fun (x3:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x3) P)) ((eq_ref (a->Prop)) x0)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x3:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x3) P)) ((eq_ref (a->Prop)) x0))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x3:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x3) P)) ((eq_ref (a->Prop)) x0))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((fun (x3:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x3) P)) ((eq_ref (a->Prop)) x0))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) b)
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x4))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x4)) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))
% Found (((eq_trans000 ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x4)) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))
% Found ((((eq_trans00 ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))) as proof of (((eq Prop) (f x4)) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))
% Found (((((eq_trans0 (f x4)) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))) as proof of (((eq Prop) (f x4)) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))
% Found ((((((eq_trans Prop) (f x4)) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))) as proof of (((eq Prop) (f x4)) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) b)
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x4))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x4)) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))
% Found (((eq_trans000 ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x4)) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))
% Found ((((eq_trans00 ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))) as proof of (((eq Prop) (f x4)) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))
% Found (((((eq_trans0 (f x4)) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))) as proof of (((eq Prop) (f x4)) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))
% Found ((((((eq_trans Prop) (f x4)) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))) as proof of (((eq Prop) (f x4)) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))
% Found x5:(P0 Xq)
% Instantiate: x4:=Xq:(a->Prop)
% Found (fun (x5:(P0 Xq))=> x5) as proof of (P0 x4)
% Found (fun (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of (((eq (a->Prop)) Xq) x4)
% Found x10:(P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of (P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of (P0 Xq)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found (((eq_trans00000 (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq)) ((eq_ref (a->Prop)) b)) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((eq_trans00000 (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq)) ((eq_ref (a->Prop)) b)) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found ((((fun (x1:(((eq (a->Prop)) Xq) b)) (x2:(((eq (a->Prop)) b) x0))=> (((eq_trans0000 x1) x2) (fun (x4:(a->Prop))=> ((P Xq)->(P x4))))) (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq)) ((eq_ref (a->Prop)) b)) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found ((((fun (x1:(((eq (a->Prop)) Xq) b)) (x2:(((eq (a->Prop)) b) x0))=> ((((eq_trans000 x0) x1) x2) (fun (x4:(a->Prop))=> ((P Xq)->(P x4))))) (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq)) ((eq_ref (a->Prop)) b)) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found ((((fun (x1:(((eq (a->Prop)) Xq) (fun (x3:a)=> (Xq x3)))) (x2:(((eq (a->Prop)) (fun (x3:a)=> (Xq x3))) x0))=> (((((eq_trans00 (fun (x3:a)=> (Xq x3))) x0) x1) x2) (fun (x4:(a->Prop))=> ((P Xq)->(P x4))))) (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq)) ((eq_ref (a->Prop)) (fun (x3:a)=> (Xq x3)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found ((((fun (x1:(((eq (a->Prop)) Xq) (fun (x3:a)=> (Xq x3)))) (x2:(((eq (a->Prop)) (fun (x3:a)=> (Xq x3))) x0))=> ((((((eq_trans0 Xq) (fun (x3:a)=> (Xq x3))) x0) x1) x2) (fun (x4:(a->Prop))=> ((P Xq)->(P x4))))) (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq)) ((eq_ref (a->Prop)) (fun (x3:a)=> (Xq x3)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found ((((fun (x1:(((eq (a->Prop)) Xq) (fun (x3:a)=> (Xq x3)))) (x2:(((eq (a->Prop)) (fun (x3:a)=> (Xq x3))) x0))=> (((((((eq_trans (a->Prop)) Xq) (fun (x3:a)=> (Xq x3))) x0) x1) x2) (fun (x4:(a->Prop))=> ((P Xq)->(P x4))))) (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq)) ((eq_ref (a->Prop)) (fun (x3:a)=> (Xq x3)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((fun (x1:(((eq (a->Prop)) Xq) (fun (x3:a)=> (Xq x3)))) (x2:(((eq (a->Prop)) (fun (x3:a)=> (Xq x3))) x0))=> (((((((eq_trans (a->Prop)) Xq) (fun (x3:a)=> (Xq x3))) x0) x1) x2) (fun (x4:(a->Prop))=> ((P Xq)->(P x4))))) (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq)) ((eq_ref (a->Prop)) (fun (x3:a)=> (Xq x3)))) (fun (x10:(P Xq))=> x10))) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((fun (x1:(((eq (a->Prop)) Xq) (fun (x3:a)=> (Xq x3)))) (x2:(((eq (a->Prop)) (fun (x3:a)=> (Xq x3))) x0))=> (((((((eq_trans (a->Prop)) Xq) (fun (x3:a)=> (Xq x3))) x0) x1) x2) (fun (x4:(a->Prop))=> ((P Xq)->(P x4))))) (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq)) ((eq_ref (a->Prop)) (fun (x3:a)=> (Xq x3)))) (fun (x10:(P Xq))=> x10))) as proof of (((eq (a->Prop)) Xq) x0)
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x2 Xy)):((iff (x2 Xy)) (x2 Xy))
% Found (iff_refl (x2 Xy)) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found (iff_refl (x2 Xy)) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found (iff_refl (x2 Xy)) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found (fun (Xy:a)=> (iff_refl (x2 Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (iff_refl (x2 Xy))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x30:(P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 x2):(((eq (a->Prop)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_sym0000 ((eq_ref (a->Prop)) x2)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found ((eq_sym0000 ((eq_ref (a->Prop)) x2)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((fun (x3:(((eq (a->Prop)) x2) Xq))=> ((eq_sym000 x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) ((eq_ref (a->Prop)) x2)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((fun (x3:(((eq (a->Prop)) x2) Xq))=> (((eq_sym00 Xq) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) ((eq_ref (a->Prop)) x2)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((fun (x3:(((eq (a->Prop)) x2) Xq))=> ((((eq_sym0 x2) Xq) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) ((eq_ref (a->Prop)) x2)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((fun (x3:(((eq (a->Prop)) x2) Xq))=> (((((eq_sym (a->Prop)) x2) Xq) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) ((eq_ref (a->Prop)) x2)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x3:(((eq (a->Prop)) x2) Xq))=> (((((eq_sym (a->Prop)) x2) Xq) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) ((eq_ref (a->Prop)) x2)) (fun (x30:(P Xq))=> x30))) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x3:(((eq (a->Prop)) x2) Xq))=> (((((eq_sym (a->Prop)) x2) Xq) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) ((eq_ref (a->Prop)) x2)) (fun (x30:(P Xq))=> x30))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x3:(((eq (a->Prop)) x2) Xq))=> (((((eq_sym (a->Prop)) x2) Xq) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) ((eq_ref (a->Prop)) x2)) (fun (x30:(P Xq))=> x30))) as proof of (((eq (a->Prop)) Xq) x2)
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x5:(P0 Xq)
% Instantiate: x0:=Xq:(a->Prop)
% Found (fun (x5:(P0 Xq))=> x5) as proof of (P0 x0)
% Found (fun (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of (((eq (a->Prop)) Xq) x0)
% 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) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found (((eq_trans000 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((((eq_trans00 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found (((((eq_trans0 (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((((((eq_trans Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found (fun (x2:(forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz))))=> ((((((eq_trans Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found (fun (x1:((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))) (x2:(forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz))))=> ((((((eq_trans Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))) as proof of ((forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz)))->(((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))
% Found (fun (x1:((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))) (x2:(forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz))))=> ((((((eq_trans Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))) as proof of (((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))->((forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz)))->(((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))))
% Found (and_rect00 (fun (x1:((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))) (x2:(forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz))))=> ((((((eq_trans Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((and_rect0 (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (fun (x1:((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))) (x2:(forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz))))=> ((((((eq_trans Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found (((fun (P0:Type) (x1:(((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->P0)))=> (((((and_rect ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) P0) x1) x)) (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (fun (x1:((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))) (x2:(forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz))))=> ((((((eq_trans Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% 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) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found (((eq_trans000 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((((eq_trans00 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found (((((eq_trans0 (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((((((eq_trans Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found (fun (x2:(forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz))))=> ((((((eq_trans Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found (fun (x1:((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))) (x2:(forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz))))=> ((((((eq_trans Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))) as proof of ((forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz)))->(((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))
% Found (fun (x1:((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))) (x2:(forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz))))=> ((((((eq_trans Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))) as proof of (((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))->((forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz)))->(((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))))
% Found (and_rect00 (fun (x1:((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))) (x2:(forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz))))=> ((((((eq_trans Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((and_rect0 (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (fun (x1:((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))) (x2:(forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz))))=> ((((((eq_trans Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found (((fun (P0:Type) (x1:(((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->P0)))=> (((((and_rect ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) P0) x1) x)) (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (fun (x1:((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))) (x2:(forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz))))=> ((((((eq_trans Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found x5:(P0 Xq)
% Instantiate: x2:=Xq:(a->Prop)
% Found (fun (x5:(P0 Xq))=> x5) as proof of (P0 x2)
% Found (fun (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of (((eq (a->Prop)) Xq) x2)
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x3:(x2 Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x3:(x2 Xy))=> x3) as proof of ((R Xx0) Xy)
% Found (fun (x3:(x2 Xy))=> x3) as proof of ((x2 Xy)->((R Xx0) Xy))
% Found x3:((R Xx0) Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x3:((R Xx0) Xy))=> x3) as proof of (x2 Xy)
% Found (fun (x3:((R Xx0) Xy))=> x3) as proof of (((R Xx0) Xy)->(x2 Xy))
% Found ((conj20 (fun (x3:(x2 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found (((conj2 (((R Xx0) Xy)->(x2 Xy))) (fun (x3:(x2 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x3:(x2 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x3:(x2 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found (fun (Xy:a)=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x3:(x2 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x3:(x2 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))=> (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))=> (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) as proof of ((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->(((eq (a->Prop)) Xq) x0))
% Found (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))=> (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) as proof of (((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->(((eq (a->Prop)) Xq) x0)))
% Found (and_rect00 (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))=> (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((and_rect0 (((eq (a->Prop)) Xq) x0)) (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))=> (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((fun (P0:Type) (x1:(((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->P0)))=> (((((and_rect ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) P0) x1) x)) (((eq (a->Prop)) Xq) x0)) (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))=> (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P0:Type) (x1:(((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->P0)))=> (((((and_rect ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) P0) x1) x)) (((eq (a->Prop)) Xq) x0)) (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))=> (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 (Xq x3)):(((eq Prop) (Xq x3)) (Xq x3))
% Found (eq_ref0 (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x2 x)))
% Found (functional_extensionality_dep00000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x2))
% Found (functional_extensionality_dep00000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((functional_extensionality_dep0000 x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> (((functional_extensionality_dep000 x2) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((functional_extensionality_dep00 Xq) x2) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> (((((functional_extensionality_dep0 (fun (x5:a)=> Prop)) Xq) x2) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) Xq) x2) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) Xq) x2) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3))))) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) Xq) x2) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3))))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) Xq) x2) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3))))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 (Xq x3)):(((eq Prop) (Xq x3)) (Xq x3))
% Found (eq_ref0 (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x2 x)))
% Found (functional_extensionality00000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x2))
% Found (functional_extensionality00000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((functional_extensionality0000 x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> (((functional_extensionality000 x2) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((functional_extensionality00 Xq) x2) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> (((((functional_extensionality0 Prop) Xq) x2) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality a) Prop) Xq) x2) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality a) Prop) Xq) x2) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3))))) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality a) Prop) Xq) x2) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3))))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality a) Prop) Xq) x2) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3))))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x30:(P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 x0):(((eq (a->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_sym0000 ((eq_ref (a->Prop)) x0)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found ((eq_sym0000 ((eq_ref (a->Prop)) x0)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((fun (x3:(((eq (a->Prop)) x0) Xq))=> ((eq_sym000 x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) ((eq_ref (a->Prop)) x0)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((fun (x3:(((eq (a->Prop)) x0) Xq))=> (((eq_sym00 Xq) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) ((eq_ref (a->Prop)) x0)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((fun (x3:(((eq (a->Prop)) x0) Xq))=> ((((eq_sym0 x0) Xq) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) ((eq_ref (a->Prop)) x0)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((fun (x3:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) ((eq_ref (a->Prop)) x0)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x3:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) ((eq_ref (a->Prop)) x0)) (fun (x30:(P Xq))=> x30))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x3:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) ((eq_ref (a->Prop)) x0)) (fun (x30:(P Xq))=> x30))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x3:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) ((eq_ref (a->Prop)) x0)) (fun (x30:(P Xq))=> x30))) as proof of (((eq (a->Prop)) Xq) x0)
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x30:(P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 (Xq x3)):(((eq Prop) (Xq x3)) (Xq x3))
% Found (eq_ref0 (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x2 x)))
% Found ((functional_extensionality00000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found ((functional_extensionality00000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((functional_extensionality0000 x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> (((functional_extensionality000 x2) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((functional_extensionality00 Xq) x2) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> (((((functional_extensionality0 Prop) Xq) x2) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality a) Prop) Xq) x2) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality a) Prop) Xq) x2) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30))) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality a) Prop) Xq) x2) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality a) Prop) Xq) x2) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x30:(P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 (Xq x3)):(((eq Prop) (Xq x3)) (Xq x3))
% Found (eq_ref0 (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (((eq Prop) (Xq x3)) (x2 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x2 x)))
% Found ((functional_extensionality_dep00000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found ((functional_extensionality_dep00000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((functional_extensionality_dep0000 x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> (((functional_extensionality_dep000 x2) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((functional_extensionality_dep00 Xq) x2) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> (((((functional_extensionality_dep0 (fun (x5:a)=> Prop)) Xq) x2) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) Xq) x2) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) Xq) x2) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30))) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) Xq) x2) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) Xq) x2) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30))) as proof of (((eq (a->Prop)) Xq) x2)
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x3:(P0 Xq)
% Instantiate: x0:=Xq:(a->Prop)
% Found (fun (x3:(P0 Xq))=> x3) as proof of (P0 x0)
% Found (fun (P0:((a->Prop)->Prop)) (x3:(P0 Xq))=> x3) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x3:(P0 Xq))=> x3) as proof of (((eq (a->Prop)) Xq) x0)
% Found x3:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x3:((R Xx0) Xy))=> x3) as proof of (x0 Xy)
% Found (fun (x3:((R Xx0) Xy))=> x3) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found x3:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x3:(x0 Xy))=> x3) as proof of ((R Xx0) Xy)
% Found (fun (x3:(x0 Xy))=> x3) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found ((conj20 (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x50:(P0 (f x4))
% Found (fun (x50:(P0 (f x4)))=> x50) as proof of (P0 (f x4))
% Found (fun (x50:(P0 (f x4)))=> x50) as proof of (P1 (f x4))
% Found x50:(P0 (f x4))
% Found (fun (x50:(P0 (f x4)))=> x50) as proof of (P0 (f x4))
% Found (fun (x50:(P0 (f x4)))=> x50) as proof of (P1 (f x4))
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x4:(Xq Xx))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x3:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x4:(Xq Xx))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) as proof of ((Xq Xx)->(((eq (a->Prop)) Xq) x2))
% Found (fun (x3:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x4:(Xq Xx))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) as proof of ((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->(((eq (a->Prop)) Xq) x2)))
% Found (and_rect10 (fun (x3:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x4:(Xq Xx))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((and_rect1 (((eq (a->Prop)) Xq) x2)) (fun (x3:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x4:(Xq Xx))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((fun (P0:Type) (x3:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x3) x00)) (((eq (a->Prop)) Xq) x2)) (fun (x3:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x4:(Xq Xx))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P0:Type) (x3:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x3) x00)) (((eq (a->Prop)) Xq) x2)) (fun (x3:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x4:(Xq Xx))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x5:(P0 Xq)
% Instantiate: x0:=Xq:(a->Prop)
% Found (fun (x5:(P0 Xq))=> x5) as proof of (P0 x0)
% Found (fun (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of (((eq (a->Prop)) Xq) x0)
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eq_ref00:=(eq_ref0 (Xq x3)):(((eq Prop) (Xq x3)) (Xq x3))
% Found (eq_ref0 (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found (functional_extensionality00000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x0))
% Found (functional_extensionality00000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((functional_extensionality0000 x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((functional_extensionality000 x0) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((functional_extensionality00 Xq) x0) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((((functional_extensionality0 Prop) Xq) x0) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3))))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3))))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3))))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x5:(P0 Xq)
% Instantiate: x0:=Xq:(a->Prop)
% Found (fun (x5:(P0 Xq))=> x5) as proof of (P0 x0)
% Found (fun (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 (Xq x3)):(((eq Prop) (Xq x3)) (Xq x3))
% Found (eq_ref0 (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found (functional_extensionality_dep00000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x0))
% Found (functional_extensionality_dep00000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((functional_extensionality_dep0000 x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((functional_extensionality_dep000 x0) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((functional_extensionality_dep00 Xq) x0) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((((functional_extensionality_dep0 (fun (x5:a)=> Prop)) Xq) x0) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) Xq) x0) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) Xq) x0) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3))))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) Xq) x0) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3))))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) Xq) x0) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (Xq x3))))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x5:(P0 Xq)
% Instantiate: x2:=Xq:(a->Prop)
% Found (fun (x5:(P0 Xq))=> x5) as proof of (P0 x2)
% Found (fun (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) 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) x2)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) x2)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x2)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x2)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x2)
% Found ((eq_trans00000 ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x2))
% Found ((eq_trans00000 ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x2))
% Found (((fun (x3:(((eq (a->Prop)) Xq) b)) (x4:(((eq (a->Prop)) b) x2))=> (((eq_trans0000 x3) x4) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x2))
% Found (((fun (x3:(((eq (a->Prop)) Xq) b)) (x4:(((eq (a->Prop)) b) x2))=> ((((eq_trans000 x2) x3) x4) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x2))
% Found (((fun (x3:(((eq (a->Prop)) Xq) Xq)) (x4:(((eq (a->Prop)) Xq) x2))=> (((((eq_trans00 Xq) x2) x3) x4) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) as proof of ((P Xq)->(P x2))
% Found (((fun (x3:(((eq (a->Prop)) Xq) Xq)) (x4:(((eq (a->Prop)) Xq) x2))=> ((((((eq_trans0 Xq) Xq) x2) x3) x4) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) as proof of ((P Xq)->(P x2))
% Found (((fun (x3:(((eq (a->Prop)) Xq) Xq)) (x4:(((eq (a->Prop)) Xq) x2))=> (((((((eq_trans (a->Prop)) Xq) Xq) x2) x3) x4) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x3:(((eq (a->Prop)) Xq) Xq)) (x4:(((eq (a->Prop)) Xq) x2))=> (((((((eq_trans (a->Prop)) Xq) Xq) x2) x3) x4) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq))) as proof of ((P Xq)->(P x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x3:(((eq (a->Prop)) Xq) Xq)) (x4:(((eq (a->Prop)) Xq) x2))=> (((((((eq_trans (a->Prop)) Xq) Xq) x2) x3) x4) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) x4)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x4)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x4)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x4)
% 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) x4)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) x4)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x4)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x4)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x4)
% Found ((eq_trans0000 (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (((eq_trans000 x4) (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found ((((eq_trans00 (fun (x7:a)=> (Xq x7))) x4) (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (((((eq_trans0 Xq) (fun (x7:a)=> (Xq x7))) x4) (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found ((((((eq_trans (a->Prop)) Xq) (fun (x7:a)=> (Xq x7))) x4) (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found ((((((eq_trans (a->Prop)) Xq) (fun (x7:a)=> (Xq x7))) x4) (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((((((eq_trans (a->Prop)) Xq) (fun (x7:a)=> (Xq x7))) x4) (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7))))) as proof of (((eq (a->Prop)) Xq) x4)
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x30:(P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 (Xq x3)):(((eq Prop) (Xq x3)) (Xq x3))
% Found (eq_ref0 (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found ((functional_extensionality00000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found ((functional_extensionality00000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((functional_extensionality0000 x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((functional_extensionality000 x0) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((functional_extensionality00 Xq) x0) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((((functional_extensionality0 Prop) Xq) x0) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x30:(P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 (Xq x3)):(((eq Prop) (Xq x3)) (Xq x3))
% Found (eq_ref0 (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found ((eq_ref Prop) (Xq x3)) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (((eq Prop) (Xq x3)) (x0 x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (Xq x3))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found ((functional_extensionality_dep00000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found ((functional_extensionality_dep00000 (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((functional_extensionality_dep0000 x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((functional_extensionality_dep000 x0) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((functional_extensionality_dep00 Xq) x0) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((((functional_extensionality_dep0 (fun (x5:a)=> Prop)) Xq) x0) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) Xq) x0) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) Xq) x0) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) Xq) x0) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x3:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) Xq) x0) x3) (fun (x5:(a->Prop))=> ((P Xq)->(P x5))))) (fun (x3:a)=> ((eq_ref Prop) (Xq x3)))) (fun (x30:(P Xq))=> x30))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x1:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x1:(x0 Xy))=> x1) as proof of ((R Xx0) Xy)
% Found (fun (x1:(x0 Xy))=> x1) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found x1:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x1:((R Xx0) Xy))=> x1) as proof of (x0 Xy)
% Found (fun (x1:((R Xx0) Xy))=> x1) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found ((conj20 (fun (x1:(x0 Xy))=> x1)) (fun (x1:((R Xx0) Xy))=> x1)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x1:(x0 Xy))=> x1)) (fun (x1:((R Xx0) Xy))=> x1)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x1:(x0 Xy))=> x1)) (fun (x1:((R Xx0) Xy))=> x1)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x1:(x0 Xy))=> x1)) (fun (x1:((R Xx0) Xy))=> x1))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x1:(x0 Xy))=> x1)) (fun (x1:((R Xx0) Xy))=> x1))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x4:(Xq Xx))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x3:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x4:(Xq Xx))=> (((eta_expansion a) Prop) Xq)) as proof of ((Xq Xx)->(((eq (a->Prop)) Xq) x0))
% Found (fun (x3:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x4:(Xq Xx))=> (((eta_expansion a) Prop) Xq)) as proof of ((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->(((eq (a->Prop)) Xq) x0)))
% Found (and_rect10 (fun (x3:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x4:(Xq Xx))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((and_rect1 (((eq (a->Prop)) Xq) x0)) (fun (x3:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x4:(Xq Xx))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((fun (P0:Type) (x3:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x3) x00)) (((eq (a->Prop)) Xq) x0)) (fun (x3:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x4:(Xq Xx))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P0:Type) (x3:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x3) x00)) (((eq (a->Prop)) Xq) x0)) (fun (x3:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x4:(Xq Xx))=> (((eta_expansion a) Prop) Xq)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x1:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x1:(x0 Xy))=> x1) as proof of ((R Xx0) Xy)
% Found (fun (x1:(x0 Xy))=> x1) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found x1:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x1:((R Xx0) Xy))=> x1) as proof of (x0 Xy)
% Found (fun (x1:((R Xx0) Xy))=> x1) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found ((conj20 (fun (x1:(x0 Xy))=> x1)) (fun (x1:((R Xx0) Xy))=> x1)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x1:(x0 Xy))=> x1)) (fun (x1:((R Xx0) Xy))=> x1)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x1:(x0 Xy))=> x1)) (fun (x1:((R Xx0) Xy))=> x1)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x1:(x0 Xy))=> x1)) (fun (x1:((R Xx0) Xy))=> x1))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x1:(x0 Xy))=> x1)) (fun (x1:((R Xx0) Xy))=> x1))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) 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) x0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) x0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_trans00000 ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x0))
% Found ((eq_trans00000 ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x0))
% Found (((fun (x3:(((eq (a->Prop)) Xq) b)) (x4:(((eq (a->Prop)) b) x0))=> (((eq_trans0000 x3) x4) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x0))
% Found (((fun (x3:(((eq (a->Prop)) Xq) b)) (x4:(((eq (a->Prop)) b) x0))=> ((((eq_trans000 x0) x3) x4) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x0))
% Found (((fun (x3:(((eq (a->Prop)) Xq) Xq)) (x4:(((eq (a->Prop)) Xq) x0))=> (((((eq_trans00 Xq) x0) x3) x4) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) as proof of ((P Xq)->(P x0))
% Found (((fun (x3:(((eq (a->Prop)) Xq) Xq)) (x4:(((eq (a->Prop)) Xq) x0))=> ((((((eq_trans0 Xq) Xq) x0) x3) x4) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) as proof of ((P Xq)->(P x0))
% Found (((fun (x3:(((eq (a->Prop)) Xq) Xq)) (x4:(((eq (a->Prop)) Xq) x0))=> (((((((eq_trans (a->Prop)) Xq) Xq) x0) x3) x4) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x3:(((eq (a->Prop)) Xq) Xq)) (x4:(((eq (a->Prop)) Xq) x0))=> (((((((eq_trans (a->Prop)) Xq) Xq) x0) x3) x4) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq))) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x3:(((eq (a->Prop)) Xq) Xq)) (x4:(((eq (a->Prop)) Xq) x0))=> (((((((eq_trans (a->Prop)) Xq) Xq) x0) x3) x4) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))):(((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found (eq_ref0 ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))):(((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found (eq_ref0 ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) 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) x0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) x0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_trans0000 (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((eq_trans000 x0) (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((((eq_trans00 (fun (x7:a)=> (Xq x7))) x0) (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (((((eq_trans0 Xq) (fun (x7:a)=> (Xq x7))) x0) (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((((((eq_trans (a->Prop)) Xq) (fun (x7:a)=> (Xq x7))) x0) (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found ((((((eq_trans (a->Prop)) Xq) (fun (x7:a)=> (Xq x7))) x0) (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7)))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((((((eq_trans (a->Prop)) Xq) (fun (x7:a)=> (Xq x7))) x0) (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7))))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 x4):(((eq (a->Prop)) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq (a->Prop)) x4) Xq)
% Found ((eq_ref (a->Prop)) x4) as proof of (((eq (a->Prop)) x4) Xq)
% Found ((eq_ref (a->Prop)) x4) as proof of (((eq (a->Prop)) x4) Xq)
% Found ((eq_ref (a->Prop)) x4) as proof of (((eq (a->Prop)) x4) Xq)
% Found (eq_sym0000 ((eq_ref (a->Prop)) x4)) as proof of ((P Xq)->(P x4))
% Found (eq_sym0000 ((eq_ref (a->Prop)) x4)) as proof of ((P Xq)->(P x4))
% Found ((fun (x5:(((eq (a->Prop)) x4) Xq))=> ((eq_sym000 x5) P)) ((eq_ref (a->Prop)) x4)) as proof of ((P Xq)->(P x4))
% Found ((fun (x5:(((eq (a->Prop)) x4) Xq))=> (((eq_sym00 Xq) x5) P)) ((eq_ref (a->Prop)) x4)) as proof of ((P Xq)->(P x4))
% Found ((fun (x5:(((eq (a->Prop)) x4) Xq))=> ((((eq_sym0 x4) Xq) x5) P)) ((eq_ref (a->Prop)) x4)) as proof of ((P Xq)->(P x4))
% Found ((fun (x5:(((eq (a->Prop)) x4) Xq))=> (((((eq_sym (a->Prop)) x4) Xq) x5) P)) ((eq_ref (a->Prop)) x4)) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x5:(((eq (a->Prop)) x4) Xq))=> (((((eq_sym (a->Prop)) x4) Xq) x5) P)) ((eq_ref (a->Prop)) x4))) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x5:(((eq (a->Prop)) x4) Xq))=> (((((eq_sym (a->Prop)) x4) Xq) x5) P)) ((eq_ref (a->Prop)) x4))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((fun (x5:(((eq (a->Prop)) x4) Xq))=> (((((eq_sym (a->Prop)) x4) Xq) x5) P)) ((eq_ref (a->Prop)) x4))) as proof of (((eq (a->Prop)) Xq) x4)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) 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) x2)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) x2)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x2)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x2)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x2)
% Found ((eq_trans0000 ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((eq_trans000 x2) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found ((((eq_trans00 Xq) x2) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (((((eq_trans0 Xq) Xq) x2) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found ((((((eq_trans (a->Prop)) Xq) Xq) x2) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found ((((((eq_trans (a->Prop)) Xq) Xq) x2) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((((((eq_trans (a->Prop)) Xq) Xq) x2) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x30:(P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) 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) x2)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) x2)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x2)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x2)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x2)
% Found (((eq_trans00000 ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((eq_trans00000 ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found ((((fun (x3:(((eq (a->Prop)) Xq) b)) (x4:(((eq (a->Prop)) b) x2))=> (((eq_trans0000 x3) x4) (fun (x6:(a->Prop))=> ((P Xq)->(P x6))))) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found ((((fun (x3:(((eq (a->Prop)) Xq) b)) (x4:(((eq (a->Prop)) b) x2))=> ((((eq_trans000 x2) x3) x4) (fun (x6:(a->Prop))=> ((P Xq)->(P x6))))) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found ((((fun (x3:(((eq (a->Prop)) Xq) Xq)) (x4:(((eq (a->Prop)) Xq) x2))=> (((((eq_trans00 Xq) x2) x3) x4) (fun (x6:(a->Prop))=> ((P Xq)->(P x6))))) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found ((((fun (x3:(((eq (a->Prop)) Xq) Xq)) (x4:(((eq (a->Prop)) Xq) x2))=> ((((((eq_trans0 Xq) Xq) x2) x3) x4) (fun (x6:(a->Prop))=> ((P Xq)->(P x6))))) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found ((((fun (x3:(((eq (a->Prop)) Xq) Xq)) (x4:(((eq (a->Prop)) Xq) x2))=> (((((((eq_trans (a->Prop)) Xq) Xq) x2) x3) x4) (fun (x6:(a->Prop))=> ((P Xq)->(P x6))))) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((((fun (x3:(((eq (a->Prop)) Xq) Xq)) (x4:(((eq (a->Prop)) Xq) x2))=> (((((((eq_trans (a->Prop)) Xq) Xq) x2) x3) x4) (fun (x6:(a->Prop))=> ((P Xq)->(P x6))))) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) (fun (x30:(P Xq))=> x30))) as proof of ((P Xq)->(P x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((fun (x3:(((eq (a->Prop)) Xq) Xq)) (x4:(((eq (a->Prop)) Xq) x2))=> (((((((eq_trans (a->Prop)) Xq) Xq) x2) x3) x4) (fun (x6:(a->Prop))=> ((P Xq)->(P x6))))) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) (fun (x30:(P Xq))=> x30))) as proof of (((eq (a->Prop)) Xq) x2)
% 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_sym010 ((eq_ref Prop) (f x0))) as proof of (((eq Prop) b) (f x0))
% Found ((eq_sym01 b) ((eq_ref Prop) (f x0))) as proof of (((eq Prop) b) (f x0))
% Found (((eq_sym0 (f x0)) b) ((eq_ref Prop) (f x0))) as proof of (((eq Prop) b) (f x0))
% Found (((eq_sym0 (f x0)) b) ((eq_ref Prop) (f x0))) as proof of (((eq Prop) b) (f x0))
% Found ((eq_trans0000 ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (((eq_sym0 (f x0)) b) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0))
% Found (((eq_trans000 (f x0)) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (((eq_sym0 (f x0)) b) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0))
% Found ((((eq_trans00 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0)) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (((eq_sym0 (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0))
% Found (((((eq_trans0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0)) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (((eq_sym0 (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0))
% Found ((((((eq_trans Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0)) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (((eq_sym0 (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0))
% Found ((((((eq_trans Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0)) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (((eq_sym0 (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0))
% Found (eq_sym000 ((((((eq_trans Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0)) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (((eq_sym0 (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_sym00 (f x0)) ((((((eq_trans Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0)) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (((eq_sym0 (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found (((eq_sym0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0)) ((((((eq_trans Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0)) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (((eq_sym0 (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((((eq_sym Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0)) ((((((eq_trans Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0)) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) ((((eq_sym Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found iff_refl0:=(iff_refl (x4 Xy)):((iff (x4 Xy)) (x4 Xy))
% Found (iff_refl (x4 Xy)) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x4 Xy)) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x4 Xy))) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x4 Xx0)) (Xy:a)=> (iff_refl (x4 Xy))) as proof of (forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x4 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x4 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x4 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((and ((x4 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x4 Xy)))
% Found ((iff_sym0 (x4 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((and ((x4 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x4 Xy)))
% Found (((iff_sym ((R Xx0) Xy)) (x4 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((and ((x4 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x4 Xy)))
% Found (((iff_sym ((R Xx0) Xy)) (x4 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((and ((x4 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x4 Xy)))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x4 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x4 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x4 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))
% Found eq_ref00:=(eq_ref0 x0):(((eq (a->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found (eq_sym0000 ((eq_ref (a->Prop)) x0)) as proof of ((P Xq)->(P x0))
% Found (eq_sym0000 ((eq_ref (a->Prop)) x0)) as proof of ((P Xq)->(P x0))
% Found ((fun (x5:(((eq (a->Prop)) x0) Xq))=> ((eq_sym000 x5) P)) ((eq_ref (a->Prop)) x0)) as proof of ((P Xq)->(P x0))
% Found ((fun (x5:(((eq (a->Prop)) x0) Xq))=> (((eq_sym00 Xq) x5) P)) ((eq_ref (a->Prop)) x0)) as proof of ((P Xq)->(P x0))
% Found ((fun (x5:(((eq (a->Prop)) x0) Xq))=> ((((eq_sym0 x0) Xq) x5) P)) ((eq_ref (a->Prop)) x0)) as proof of ((P Xq)->(P x0))
% Found ((fun (x5:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x5) P)) ((eq_ref (a->Prop)) x0)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x5:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x5) P)) ((eq_ref (a->Prop)) x0))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x5:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x5) P)) ((eq_ref (a->Prop)) x0))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((fun (x5:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x5) P)) ((eq_ref (a->Prop)) x0))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 x2):(((eq (a->Prop)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found (eq_sym0000 ((eq_ref (a->Prop)) x2)) as proof of ((P Xq)->(P x2))
% Found (eq_sym0000 ((eq_ref (a->Prop)) x2)) as proof of ((P Xq)->(P x2))
% Found ((fun (x5:(((eq (a->Prop)) x2) Xq))=> ((eq_sym000 x5) P)) ((eq_ref (a->Prop)) x2)) as proof of ((P Xq)->(P x2))
% Found ((fun (x5:(((eq (a->Prop)) x2) Xq))=> (((eq_sym00 Xq) x5) P)) ((eq_ref (a->Prop)) x2)) as proof of ((P Xq)->(P x2))
% Found ((fun (x5:(((eq (a->Prop)) x2) Xq))=> ((((eq_sym0 x2) Xq) x5) P)) ((eq_ref (a->Prop)) x2)) as proof of ((P Xq)->(P x2))
% Found ((fun (x5:(((eq (a->Prop)) x2) Xq))=> (((((eq_sym (a->Prop)) x2) Xq) x5) P)) ((eq_ref (a->Prop)) x2)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x5:(((eq (a->Prop)) x2) Xq))=> (((((eq_sym (a->Prop)) x2) Xq) x5) P)) ((eq_ref (a->Prop)) x2))) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x5:(((eq (a->Prop)) x2) Xq))=> (((((eq_sym (a->Prop)) x2) Xq) x5) P)) ((eq_ref (a->Prop)) x2))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((fun (x5:(((eq (a->Prop)) x2) Xq))=> (((((eq_sym (a->Prop)) x2) Xq) x5) P)) ((eq_ref (a->Prop)) x2))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x4:(Xq Xx))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x3:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x4:(Xq Xx))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) as proof of ((Xq Xx)->(((eq (a->Prop)) Xq) x0))
% Found (fun (x3:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x4:(Xq Xx))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) as proof of ((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->(((eq (a->Prop)) Xq) x0)))
% Found (and_rect10 (fun (x3:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x4:(Xq Xx))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((and_rect1 (((eq (a->Prop)) Xq) x0)) (fun (x3:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x4:(Xq Xx))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((fun (P0:Type) (x3:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x3) x00)) (((eq (a->Prop)) Xq) x0)) (fun (x3:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x4:(Xq Xx))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P0:Type) (x3:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x3) x00)) (((eq (a->Prop)) Xq) x0)) (fun (x3:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x4:(Xq Xx))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found iff_refl0:=(iff_refl (x4 Xy)):((iff (x4 Xy)) (x4 Xy))
% Found (iff_refl (x4 Xy)) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x4 Xy)) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x4 Xy))) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x4 Xx0)) (Xy:a)=> (iff_refl (x4 Xy))) as proof of (forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x30:(P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) 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) x0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) x0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x0)
% Found (((eq_trans00000 ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((eq_trans00000 ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found ((((fun (x3:(((eq (a->Prop)) Xq) b)) (x4:(((eq (a->Prop)) b) x0))=> (((eq_trans0000 x3) x4) (fun (x6:(a->Prop))=> ((P Xq)->(P x6))))) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found ((((fun (x3:(((eq (a->Prop)) Xq) b)) (x4:(((eq (a->Prop)) b) x0))=> ((((eq_trans000 x0) x3) x4) (fun (x6:(a->Prop))=> ((P Xq)->(P x6))))) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found ((((fun (x3:(((eq (a->Prop)) Xq) Xq)) (x4:(((eq (a->Prop)) Xq) x0))=> (((((eq_trans00 Xq) x0) x3) x4) (fun (x6:(a->Prop))=> ((P Xq)->(P x6))))) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found ((((fun (x3:(((eq (a->Prop)) Xq) Xq)) (x4:(((eq (a->Prop)) Xq) x0))=> ((((((eq_trans0 Xq) Xq) x0) x3) x4) (fun (x6:(a->Prop))=> ((P Xq)->(P x6))))) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found ((((fun (x3:(((eq (a->Prop)) Xq) Xq)) (x4:(((eq (a->Prop)) Xq) x0))=> (((((((eq_trans (a->Prop)) Xq) Xq) x0) x3) x4) (fun (x6:(a->Prop))=> ((P Xq)->(P x6))))) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((fun (x3:(((eq (a->Prop)) Xq) Xq)) (x4:(((eq (a->Prop)) Xq) x0))=> (((((((eq_trans (a->Prop)) Xq) Xq) x0) x3) x4) (fun (x6:(a->Prop))=> ((P Xq)->(P x6))))) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) (fun (x30:(P Xq))=> x30))) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((fun (x3:(((eq (a->Prop)) Xq) Xq)) (x4:(((eq (a->Prop)) Xq) x0))=> (((((((eq_trans (a->Prop)) Xq) Xq) x0) x3) x4) (fun (x6:(a->Prop))=> ((P Xq)->(P x6))))) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) (fun (x30:(P Xq))=> x30))) as proof of (((eq (a->Prop)) Xq) x0)
% 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)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found (((eq_trans000 ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found ((((eq_trans00 ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found (((((eq_trans0 (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found ((((((eq_trans Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found (fun (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> ((((((eq_trans Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))))) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found (fun (x3:(forall (Xx0:a), ((R Xx0) Xx0))) (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> ((((((eq_trans Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))))) as proof of ((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->(((eq Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))))
% Found (fun (x3:(forall (Xx0:a), ((R Xx0) Xx0))) (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> ((((((eq_trans Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))))) as proof of ((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->(((eq Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))))
% Found (and_rect10 (fun (x3:(forall (Xx0:a), ((R Xx0) Xx0))) (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> ((((((eq_trans Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))))) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found ((and_rect1 (((eq Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))) (fun (x3:(forall (Xx0:a), ((R Xx0) Xx0))) (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> ((((((eq_trans Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))))) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found (((fun (P0:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P0)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P0) x3) x0)) (((eq Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))) (fun (x3:(forall (Xx0:a), ((R Xx0) Xx0))) (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> ((((((eq_trans Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))))) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% 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)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found (((eq_trans000 ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found ((((eq_trans00 ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found (((((eq_trans0 (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found ((((((eq_trans Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found (fun (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> ((((((eq_trans Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))))) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found (fun (x3:(forall (Xx0:a), ((R Xx0) Xx0))) (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> ((((((eq_trans Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))))) as proof of ((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->(((eq Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))))
% Found (fun (x3:(forall (Xx0:a), ((R Xx0) Xx0))) (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> ((((((eq_trans Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))))) as proof of ((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->(((eq Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))))
% Found (and_rect10 (fun (x3:(forall (Xx0:a), ((R Xx0) Xx0))) (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> ((((((eq_trans Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))))) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found ((and_rect1 (((eq Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))) (fun (x3:(forall (Xx0:a), ((R Xx0) Xx0))) (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> ((((((eq_trans Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))))) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found (((fun (P0:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P0)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P0) x3) x0)) (((eq Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))) (fun (x3:(forall (Xx0:a), ((R Xx0) Xx0))) (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> ((((((eq_trans Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))))) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x2 Xy)):((iff (x2 Xy)) (x2 Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x2 Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (iff_refl (x2 Xy))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 x4):(((eq (a->Prop)) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq (a->Prop)) x4) Xq)
% Found ((eq_ref (a->Prop)) x4) as proof of (((eq (a->Prop)) x4) Xq)
% Found ((eq_ref (a->Prop)) x4) as proof of (((eq (a->Prop)) x4) Xq)
% Found ((eq_ref (a->Prop)) x4) as proof of (((eq (a->Prop)) x4) Xq)
% Found ((eq_sym0000 ((eq_ref (a->Prop)) x4)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found ((eq_sym0000 ((eq_ref (a->Prop)) x4)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((fun (x5:(((eq (a->Prop)) x4) Xq))=> ((eq_sym000 x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x4)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((fun (x5:(((eq (a->Prop)) x4) Xq))=> (((eq_sym00 Xq) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x4)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((fun (x5:(((eq (a->Prop)) x4) Xq))=> ((((eq_sym0 x4) Xq) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x4)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((fun (x5:(((eq (a->Prop)) x4) Xq))=> (((((eq_sym (a->Prop)) x4) Xq) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x4)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x5:(((eq (a->Prop)) x4) Xq))=> (((((eq_sym (a->Prop)) x4) Xq) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x4)) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x5:(((eq (a->Prop)) x4) Xq))=> (((((eq_sym (a->Prop)) x4) Xq) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x4)) (fun (x50:(P Xq))=> x50))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x5:(((eq (a->Prop)) x4) Xq))=> (((((eq_sym (a->Prop)) x4) Xq) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x4)) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x4)
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found ((iff_sym0 (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x2 Xy)):((iff (x2 Xy)) (x2 Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x2 Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (iff_refl (x2 Xy))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x2 Xy)):((iff (x2 Xy)) (x2 Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x2 Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (iff_refl (x2 Xy))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x3:(forall (Xx0:a), ((R Xx0) Xx0))) (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) as proof of ((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->(((eq (a->Prop)) Xq) x0))
% Found (fun (x3:(forall (Xx0:a), ((R Xx0) Xx0))) (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) as proof of ((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->(((eq (a->Prop)) Xq) x0)))
% Found (and_rect10 (fun (x3:(forall (Xx0:a), ((R Xx0) Xx0))) (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((and_rect1 (((eq (a->Prop)) Xq) x0)) (fun (x3:(forall (Xx0:a), ((R Xx0) Xx0))) (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((fun (P0:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P0)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P0) x3) x1)) (((eq (a->Prop)) Xq) x0)) (fun (x3:(forall (Xx0:a), ((R Xx0) Xx0))) (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P0:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P0)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P0) x3) x1)) (((eq (a->Prop)) Xq) x0)) (fun (x3:(forall (Xx0:a), ((R Xx0) Xx0))) (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x4:(forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x3:(forall (Xx:a), ((R Xx) Xx))) (x4:(forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))=> (((eta_expansion a) Prop) Xq)) as proof of ((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))->(((eq (a->Prop)) Xq) x0))
% Found (fun (x3:(forall (Xx:a), ((R Xx) Xx))) (x4:(forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))=> (((eta_expansion a) Prop) Xq)) as proof of ((forall (Xx:a), ((R Xx) Xx))->((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))->(((eq (a->Prop)) Xq) x0)))
% Found (and_rect10 (fun (x3:(forall (Xx:a), ((R Xx) Xx))) (x4:(forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((and_rect1 (((eq (a->Prop)) Xq) x0)) (fun (x3:(forall (Xx:a), ((R Xx) Xx))) (x4:(forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((fun (P0:Type) (x3:((forall (Xx:a), ((R Xx) Xx))->((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))->P0)))=> (((((and_rect (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) P0) x3) x1)) (((eq (a->Prop)) Xq) x0)) (fun (x3:(forall (Xx:a), ((R Xx) Xx))) (x4:(forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P0:Type) (x3:((forall (Xx:a), ((R Xx) Xx))->((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))->P0)))=> (((((and_rect (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) P0) x3) x1)) (((eq (a->Prop)) Xq) x0)) (fun (x3:(forall (Xx:a), ((R Xx) Xx))) (x4:(forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))=> (((eta_expansion a) Prop) Xq)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x3:(forall (Xx0:a), ((R Xx0) Xx0))) (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> (((eta_expansion a) Prop) Xq)) as proof of ((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->(((eq (a->Prop)) Xq) x2))
% Found (fun (x3:(forall (Xx0:a), ((R Xx0) Xx0))) (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> (((eta_expansion a) Prop) Xq)) as proof of ((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->(((eq (a->Prop)) Xq) x2)))
% Found (and_rect10 (fun (x3:(forall (Xx0:a), ((R Xx0) Xx0))) (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((and_rect1 (((eq (a->Prop)) Xq) x2)) (fun (x3:(forall (Xx0:a), ((R Xx0) Xx0))) (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((fun (P0:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P0)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P0) x3) x0)) (((eq (a->Prop)) Xq) x2)) (fun (x3:(forall (Xx0:a), ((R Xx0) Xx0))) (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P0:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P0)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P0) x3) x0)) (((eq (a->Prop)) Xq) x2)) (fun (x3:(forall (Xx0:a), ((R Xx0) Xx0))) (x4:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> (((eta_expansion a) Prop) Xq)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x5:((R Xx0) Xy)
% Instantiate: x4:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x4 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x4 Xy))
% Found x5:(x4 Xy)
% Instantiate: x4:=(R Xx0):(a->Prop)
% Found (fun (x5:(x4 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x4 Xy))=> x5) as proof of ((x4 Xy)->((R Xx0) Xy))
% Found ((conj20 (fun (x5:(x4 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((and ((x4 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x4 Xy)))
% Found (((conj2 (((R Xx0) Xy)->(x4 Xy))) (fun (x5:(x4 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((and ((x4 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x4 Xy)))
% Found ((((conj ((x4 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x4 Xy))) (fun (x5:(x4 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((and ((x4 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x4 Xy)))
% Found ((((conj ((x4 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x4 Xy))) (fun (x5:(x4 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((and ((x4 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x4 Xy)))
% Found (fun (Xy:a)=> ((((conj ((x4 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x4 Xy))) (fun (x5:(x4 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x4 Xx0)) (Xy:a)=> ((((conj ((x4 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x4 Xy))) (fun (x5:(x4 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 x0):(((eq (a->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_sym0000 ((eq_ref (a->Prop)) x0)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found ((eq_sym0000 ((eq_ref (a->Prop)) x0)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((fun (x5:(((eq (a->Prop)) x0) Xq))=> ((eq_sym000 x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x0)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((fun (x5:(((eq (a->Prop)) x0) Xq))=> (((eq_sym00 Xq) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x0)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((fun (x5:(((eq (a->Prop)) x0) Xq))=> ((((eq_sym0 x0) Xq) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x0)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((fun (x5:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x0)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x5:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x0)) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x5:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x0)) (fun (x50:(P Xq))=> x50))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x5:(((eq (a->Prop)) x0) Xq))=> (((((eq_sym (a->Prop)) x0) Xq) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x0)) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 x2):(((eq (a->Prop)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_sym0000 ((eq_ref (a->Prop)) x2)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found ((eq_sym0000 ((eq_ref (a->Prop)) x2)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((fun (x5:(((eq (a->Prop)) x2) Xq))=> ((eq_sym000 x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x2)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((fun (x5:(((eq (a->Prop)) x2) Xq))=> (((eq_sym00 Xq) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x2)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((fun (x5:(((eq (a->Prop)) x2) Xq))=> ((((eq_sym0 x2) Xq) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x2)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((fun (x5:(((eq (a->Prop)) x2) Xq))=> (((((eq_sym (a->Prop)) x2) Xq) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x2)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x5:(((eq (a->Prop)) x2) Xq))=> (((((eq_sym (a->Prop)) x2) Xq) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x2)) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x5:(((eq (a->Prop)) x2) Xq))=> (((((eq_sym (a->Prop)) x2) Xq) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x2)) (fun (x50:(P Xq))=> x50))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x5:(((eq (a->Prop)) x2) Xq))=> (((((eq_sym (a->Prop)) x2) Xq) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) ((eq_ref (a->Prop)) x2)) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x2)
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found eq_ref00:=(eq_ref0 (Xq x5)):(((eq Prop) (Xq x5)) (Xq x5))
% Found (eq_ref0 (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x4 x)))
% Found (functional_extensionality_dep00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x4))
% Found (functional_extensionality_dep00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x4))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((functional_extensionality_dep0000 x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x4))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> (((functional_extensionality_dep000 x4) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x4))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((((functional_extensionality_dep00 Xq) x4) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x4))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> (((((functional_extensionality_dep0 (fun (x7:a)=> Prop)) Xq) x4) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x4))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x4) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x4) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x4) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x4) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of (((eq (a->Prop)) Xq) x4)
% Found eq_ref00:=(eq_ref0 (Xq x5)):(((eq Prop) (Xq x5)) (Xq x5))
% Found (eq_ref0 (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x4 x)))
% Found (functional_extensionality00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x4))
% Found (functional_extensionality00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x4))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((functional_extensionality0000 x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x4))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> (((functional_extensionality000 x4) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x4))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((((functional_extensionality00 Xq) x4) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x4))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> (((((functional_extensionality0 Prop) Xq) x4) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x4))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((((((functional_extensionality a) Prop) Xq) x4) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((((((functional_extensionality a) Prop) Xq) x4) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((((((functional_extensionality a) Prop) Xq) x4) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((((((functional_extensionality a) Prop) Xq) x4) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of (((eq (a->Prop)) Xq) x4)
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x3:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x3:((R Xx0) Xy))=> x3) as proof of (x0 Xy)
% Found (fun (x3:((R Xx0) Xy))=> x3) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found x3:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x3:(x0 Xy))=> x3) as proof of ((R Xx0) Xy)
% Found (fun (x3:(x0 Xy))=> x3) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found ((conj20 (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x2 Xy)):((iff (x2 Xy)) (x2 Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x2 Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (iff_refl (x2 Xy))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found x5:(P0 Xq)
% Instantiate: x0:=Xq:(a->Prop)
% Found (fun (x5:(P0 Xq))=> x5) as proof of (P0 x0)
% Found (fun (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of (((eq (a->Prop)) Xq) x0)
% Found iff_refl0:=(iff_refl (x2 Xy)):((iff (x2 Xy)) (x2 Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x2 Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (iff_refl (x2 Xy))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x5:(P0 Xq)
% Instantiate: x0:=Xq:(a->Prop)
% Found (fun (x5:(P0 Xq))=> x5) as proof of (P0 x0)
% Found (fun (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of (((eq (a->Prop)) Xq) x0)
% Found x5:(P0 Xq)
% Instantiate: x2:=Xq:(a->Prop)
% Found (fun (x5:(P0 Xq))=> x5) as proof of (P0 x2)
% Found (fun (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of (((eq (a->Prop)) Xq) x2)
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 (Xq x5)):(((eq Prop) (Xq x5)) (Xq x5))
% Found (eq_ref0 (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x4 x)))
% Found ((functional_extensionality_dep00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found ((functional_extensionality_dep00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((functional_extensionality_dep0000 x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> (((functional_extensionality_dep000 x4) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((((functional_extensionality_dep00 Xq) x4) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> (((((functional_extensionality_dep0 (fun (x7:a)=> Prop)) Xq) x4) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x4) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x4) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x4) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x4) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x4)
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 (Xq x5)):(((eq Prop) (Xq x5)) (Xq x5))
% Found (eq_ref0 (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (((eq Prop) (Xq x5)) (x4 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x4 x)))
% Found ((functional_extensionality00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found ((functional_extensionality00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((functional_extensionality0000 x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> (((functional_extensionality000 x4) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((((functional_extensionality00 Xq) x4) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> (((((functional_extensionality0 Prop) Xq) x4) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((((((functional_extensionality a) Prop) Xq) x4) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((((((functional_extensionality a) Prop) Xq) x4) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((((((functional_extensionality a) Prop) Xq) x4) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x4 x))))=> ((((((functional_extensionality a) Prop) Xq) x4) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x4)
% Found x5:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found x5:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x0 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found ((conj20 (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x5:(x2 Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x5:(x2 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x2 Xy))=> x5) as proof of ((x2 Xy)->((R Xx0) Xy))
% Found x5:((R Xx0) Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x2 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x2 Xy))
% Found ((conj20 (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found (((conj2 (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found (fun (Xy:a)=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x2 Xy)):((iff (x2 Xy)) (x2 Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x2 Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (iff_refl (x2 Xy))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) 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) x4)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) x4)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x4)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x4)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x4)
% Found ((eq_trans00000 (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x4))
% Found ((eq_trans00000 (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x4))
% Found (((fun (x5:(((eq (a->Prop)) Xq) b)) (x6:(((eq (a->Prop)) b) x4))=> (((eq_trans0000 x5) x6) P)) (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x4))
% Found (((fun (x5:(((eq (a->Prop)) Xq) b)) (x6:(((eq (a->Prop)) b) x4))=> ((((eq_trans000 x4) x5) x6) P)) (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x4))
% Found (((fun (x5:(((eq (a->Prop)) Xq) (fun (x7:a)=> (Xq x7)))) (x6:(((eq (a->Prop)) (fun (x7:a)=> (Xq x7))) x4))=> (((((eq_trans00 (fun (x7:a)=> (Xq x7))) x4) x5) x6) P)) (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7)))) as proof of ((P Xq)->(P x4))
% Found (((fun (x5:(((eq (a->Prop)) Xq) (fun (x7:a)=> (Xq x7)))) (x6:(((eq (a->Prop)) (fun (x7:a)=> (Xq x7))) x4))=> ((((((eq_trans0 Xq) (fun (x7:a)=> (Xq x7))) x4) x5) x6) P)) (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7)))) as proof of ((P Xq)->(P x4))
% Found (((fun (x5:(((eq (a->Prop)) Xq) (fun (x7:a)=> (Xq x7)))) (x6:(((eq (a->Prop)) (fun (x7:a)=> (Xq x7))) x4))=> (((((((eq_trans (a->Prop)) Xq) (fun (x7:a)=> (Xq x7))) x4) x5) x6) P)) (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7)))) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x5:(((eq (a->Prop)) Xq) (fun (x7:a)=> (Xq x7)))) (x6:(((eq (a->Prop)) (fun (x7:a)=> (Xq x7))) x4))=> (((((((eq_trans (a->Prop)) Xq) (fun (x7:a)=> (Xq x7))) x4) x5) x6) P)) (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7))))) as proof of ((P Xq)->(P x4))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x5:(((eq (a->Prop)) Xq) (fun (x7:a)=> (Xq x7)))) (x6:(((eq (a->Prop)) (fun (x7:a)=> (Xq x7))) x4))=> (((((((eq_trans (a->Prop)) Xq) (fun (x7:a)=> (Xq x7))) x4) x5) x6) P)) (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7))))) as proof of (((eq (a->Prop)) Xq) x4)
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))) (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> (iff_refl (x0 Xy))) as proof of ((forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz)))->((iff (x0 Xy)) ((R Xx0) Xy)))
% Found (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))) (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> (iff_refl (x0 Xy))) as proof of (((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))->((forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz)))->((iff (x0 Xy)) ((R Xx0) Xy))))
% Found (and_rect00 (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))) (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((and_rect0 ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))) (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((fun (P0:Type) (x1:(((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->P0)))=> (((((and_rect ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) P0) x1) x)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))) (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((fun (P0:Type) (x1:(((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->P0)))=> (((((and_rect ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) P0) x1) x)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))) (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> (iff_refl (x0 Xy))))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((fun (P0:Type) (x1:(((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->P0)))=> (((((and_rect ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) P0) x1) x)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))) (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> (iff_refl (x0 Xy))))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eq_ref00:=(eq_ref0 (Xq x5)):(((eq Prop) (Xq x5)) (Xq x5))
% Found (eq_ref0 (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found (functional_extensionality_dep00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x0))
% Found (functional_extensionality_dep00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((functional_extensionality_dep0000 x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((functional_extensionality_dep000 x0) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((functional_extensionality_dep00 Xq) x0) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((((functional_extensionality_dep0 (fun (x7:a)=> Prop)) Xq) x0) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x0) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x0) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x0) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x0) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 (Xq x5)):(((eq Prop) (Xq x5)) (Xq x5))
% Found (eq_ref0 (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found (functional_extensionality00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x0))
% Found (functional_extensionality00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((functional_extensionality0000 x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((functional_extensionality000 x0) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((functional_extensionality00 Xq) x0) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((((functional_extensionality0 Prop) Xq) x0) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x0))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 (Xq x5)):(((eq Prop) (Xq x5)) (Xq x5))
% Found (eq_ref0 (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x2 x)))
% Found (functional_extensionality_dep00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x2))
% Found (functional_extensionality_dep00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((functional_extensionality_dep0000 x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> (((functional_extensionality_dep000 x2) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((functional_extensionality_dep00 Xq) x2) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> (((((functional_extensionality_dep0 (fun (x7:a)=> Prop)) Xq) x2) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x2) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x2) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x2) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x2) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 (Xq x5)):(((eq Prop) (Xq x5)) (Xq x5))
% Found (eq_ref0 (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x2 x)))
% Found (functional_extensionality00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x2))
% Found (functional_extensionality00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((functional_extensionality0000 x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> (((functional_extensionality000 x2) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((functional_extensionality00 Xq) x2) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> (((((functional_extensionality0 Prop) Xq) x2) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x2))
% Found ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality a) Prop) Xq) x2) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality a) Prop) Xq) x2) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality a) Prop) Xq) x2) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality a) Prop) Xq) x2) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (Xq x5))))) as proof of (((eq (a->Prop)) Xq) x2)
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq)) as proof of ((Xq Xx)->(((eq (a->Prop)) Xq) x4))
% Found (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq)) as proof of ((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->(((eq (a->Prop)) Xq) x4)))
% Found (and_rect20 (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((and_rect2 (((eq (a->Prop)) Xq) x4)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((fun (P0:Type) (x5:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x5) x00)) (((eq (a->Prop)) Xq) x4)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P0:Type) (x5:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x5) x00)) (((eq (a->Prop)) Xq) x4)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq)))) as proof of (((eq (a->Prop)) Xq) x4)
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x2 Xy)):((iff (x2 Xy)) (x2 Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x2 Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (iff_refl (x2 Xy))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))) (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> (iff_refl (x0 Xy))) as proof of ((forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz)))->((iff (x0 Xy)) ((R Xx0) Xy)))
% Found (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))) (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> (iff_refl (x0 Xy))) as proof of (((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))->((forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz)))->((iff (x0 Xy)) ((R Xx0) Xy))))
% Found (and_rect00 (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))) (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((and_rect0 ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))) (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((fun (P0:Type) (x1:(((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->P0)))=> (((((and_rect ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) P0) x1) x)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))) (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((fun (P0:Type) (x1:(((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->P0)))=> (((((and_rect ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) P0) x1) x)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))) (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> (iff_refl (x0 Xy))))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((fun (P0:Type) (x1:(((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->P0)))=> (((((and_rect ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) P0) x1) x)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))) (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> (iff_refl (x0 Xy))))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 (Xq x5)):(((eq Prop) (Xq x5)) (Xq x5))
% Found (eq_ref0 (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found ((functional_extensionality_dep00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found ((functional_extensionality_dep00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((functional_extensionality_dep0000 x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((functional_extensionality_dep000 x0) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((functional_extensionality_dep00 Xq) x0) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((((functional_extensionality_dep0 (fun (x7:a)=> Prop)) Xq) x0) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x0) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x0) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x0) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x0) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 (Xq x5)):(((eq Prop) (Xq x5)) (Xq x5))
% Found (eq_ref0 (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (((eq Prop) (Xq x5)) (x0 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found ((functional_extensionality00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found ((functional_extensionality00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((functional_extensionality0000 x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((functional_extensionality000 x0) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((functional_extensionality00 Xq) x0) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> (((((functional_extensionality0 Prop) Xq) x0) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x0 x))))=> ((((((functional_extensionality a) Prop) Xq) x0) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 (Xq x5)):(((eq Prop) (Xq x5)) (Xq x5))
% Found (eq_ref0 (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x2 x)))
% Found ((functional_extensionality_dep00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found ((functional_extensionality_dep00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((functional_extensionality_dep0000 x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> (((functional_extensionality_dep000 x2) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((functional_extensionality_dep00 Xq) x2) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> (((((functional_extensionality_dep0 (fun (x7:a)=> Prop)) Xq) x2) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x2) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x2) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x2) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xq) x2) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 (Xq x5)):(((eq Prop) (Xq x5)) (Xq x5))
% Found (eq_ref0 (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found ((eq_ref Prop) (Xq x5)) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (((eq Prop) (Xq x5)) (x2 x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (Xq x5))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x2 x)))
% Found ((functional_extensionality00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found ((functional_extensionality00000 (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((functional_extensionality0000 x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> (((functional_extensionality000 x2) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((functional_extensionality00 Xq) x2) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> (((((functional_extensionality0 Prop) Xq) x2) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality a) Prop) Xq) x2) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality a) Prop) Xq) x2) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality a) Prop) Xq) x2) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x5:(forall (x:a), (((eq Prop) (Xq x)) (x2 x))))=> ((((((functional_extensionality a) Prop) Xq) x2) x5) (fun (x7:(a->Prop))=> ((P Xq)->(P x7))))) (fun (x5:a)=> ((eq_ref Prop) (Xq x5)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x2)
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x3:((R Xx0) Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x3:((R Xx0) Xy))=> x3) as proof of (x2 Xy)
% Found (fun (x3:((R Xx0) Xy))=> x3) as proof of (((R Xx0) Xy)->(x2 Xy))
% Found x3:(x2 Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x3:(x2 Xy))=> x3) as proof of ((R Xx0) Xy)
% Found (fun (x3:(x2 Xy))=> x3) as proof of ((x2 Xy)->((R Xx0) Xy))
% Found ((conj20 (fun (x3:(x2 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x2 Xy))) (fun (x3:(x2 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x3:(x2 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x3:(x2 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x3:(x2 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x2 Xy)):((iff (x2 Xy)) (x2 Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x2 Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (iff_refl (x2 Xy))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) 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) x0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) x0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_trans00000 (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x0))
% Found ((eq_trans00000 (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x0))
% Found (((fun (x5:(((eq (a->Prop)) Xq) b)) (x6:(((eq (a->Prop)) b) x0))=> (((eq_trans0000 x5) x6) P)) (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x0))
% Found (((fun (x5:(((eq (a->Prop)) Xq) b)) (x6:(((eq (a->Prop)) b) x0))=> ((((eq_trans000 x0) x5) x6) P)) (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x0))
% Found (((fun (x5:(((eq (a->Prop)) Xq) (fun (x7:a)=> (Xq x7)))) (x6:(((eq (a->Prop)) (fun (x7:a)=> (Xq x7))) x0))=> (((((eq_trans00 (fun (x7:a)=> (Xq x7))) x0) x5) x6) P)) (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7)))) as proof of ((P Xq)->(P x0))
% Found (((fun (x5:(((eq (a->Prop)) Xq) (fun (x7:a)=> (Xq x7)))) (x6:(((eq (a->Prop)) (fun (x7:a)=> (Xq x7))) x0))=> ((((((eq_trans0 Xq) (fun (x7:a)=> (Xq x7))) x0) x5) x6) P)) (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7)))) as proof of ((P Xq)->(P x0))
% Found (((fun (x5:(((eq (a->Prop)) Xq) (fun (x7:a)=> (Xq x7)))) (x6:(((eq (a->Prop)) (fun (x7:a)=> (Xq x7))) x0))=> (((((((eq_trans (a->Prop)) Xq) (fun (x7:a)=> (Xq x7))) x0) x5) x6) P)) (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7)))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x5:(((eq (a->Prop)) Xq) (fun (x7:a)=> (Xq x7)))) (x6:(((eq (a->Prop)) (fun (x7:a)=> (Xq x7))) x0))=> (((((((eq_trans (a->Prop)) Xq) (fun (x7:a)=> (Xq x7))) x0) x5) x6) P)) (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7))))) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x5:(((eq (a->Prop)) Xq) (fun (x7:a)=> (Xq x7)))) (x6:(((eq (a->Prop)) (fun (x7:a)=> (Xq x7))) x0))=> (((((((eq_trans (a->Prop)) Xq) (fun (x7:a)=> (Xq x7))) x0) x5) x6) P)) (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7))))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) 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) x2)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) x2)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x2)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x2)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x2)
% Found ((eq_trans00000 ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x2))
% Found ((eq_trans00000 ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x2))
% Found (((fun (x5:(((eq (a->Prop)) Xq) b)) (x6:(((eq (a->Prop)) b) x2))=> (((eq_trans0000 x5) x6) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x2))
% Found (((fun (x5:(((eq (a->Prop)) Xq) b)) (x6:(((eq (a->Prop)) b) x2))=> ((((eq_trans000 x2) x5) x6) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x2))
% Found (((fun (x5:(((eq (a->Prop)) Xq) Xq)) (x6:(((eq (a->Prop)) Xq) x2))=> (((((eq_trans00 Xq) x2) x5) x6) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) as proof of ((P Xq)->(P x2))
% Found (((fun (x5:(((eq (a->Prop)) Xq) Xq)) (x6:(((eq (a->Prop)) Xq) x2))=> ((((((eq_trans0 Xq) Xq) x2) x5) x6) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) as proof of ((P Xq)->(P x2))
% Found (((fun (x5:(((eq (a->Prop)) Xq) Xq)) (x6:(((eq (a->Prop)) Xq) x2))=> (((((((eq_trans (a->Prop)) Xq) Xq) x2) x5) x6) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x5:(((eq (a->Prop)) Xq) Xq)) (x6:(((eq (a->Prop)) Xq) x2))=> (((((((eq_trans (a->Prop)) Xq) Xq) x2) x5) x6) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq))) as proof of ((P Xq)->(P x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x5:(((eq (a->Prop)) Xq) Xq)) (x6:(((eq (a->Prop)) Xq) x2))=> (((((((eq_trans (a->Prop)) Xq) Xq) x2) x5) x6) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x4))
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))):(((eq Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))
% Found (eq_ref0 ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) b)
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))):(((eq Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))
% Found (eq_ref0 ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x4))
% Found x3:((R Xx0) Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x3:((R Xx0) Xy))=> x3) as proof of (x2 Xy)
% Found (fun (x3:((R Xx0) Xy))=> x3) as proof of (((R Xx0) Xy)->(x2 Xy))
% Found x3:(x2 Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x3:(x2 Xy))=> x3) as proof of ((R Xx0) Xy)
% Found (fun (x3:(x2 Xy))=> x3) as proof of ((x2 Xy)->((R Xx0) Xy))
% Found ((conj20 (fun (x3:(x2 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x2 Xy))) (fun (x3:(x2 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x3:(x2 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x3:(x2 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x3:(x2 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion_dep0 (fun (x8:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x6:(Xq Xx))=> (((eta_expansion_dep a) (fun (x8:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion_dep a) (fun (x8:a)=> Prop)) Xq)) as proof of ((Xq Xx)->(((eq (a->Prop)) Xq) x0))
% Found (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion_dep a) (fun (x8:a)=> Prop)) Xq)) as proof of ((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->(((eq (a->Prop)) Xq) x0)))
% Found (and_rect20 (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion_dep a) (fun (x8:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((and_rect2 (((eq (a->Prop)) Xq) x0)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion_dep a) (fun (x8:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((fun (P0:Type) (x5:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x5) x00)) (((eq (a->Prop)) Xq) x0)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion_dep a) (fun (x8:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P0:Type) (x5:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x5) x00)) (((eq (a->Prop)) Xq) x0)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion_dep a) (fun (x8:a)=> Prop)) Xq)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq)) as proof of ((Xq Xx)->(((eq (a->Prop)) Xq) x2))
% Found (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq)) as proof of ((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->(((eq (a->Prop)) Xq) x2)))
% Found (and_rect20 (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((and_rect2 (((eq (a->Prop)) Xq) x2)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((fun (P0:Type) (x5:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x5) x00)) (((eq (a->Prop)) Xq) x2)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P0:Type) (x5:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x5) x00)) (((eq (a->Prop)) Xq) x2)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x3:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x3:(x0 Xy))=> x3) as proof of ((R Xx0) Xy)
% Found (fun (x3:(x0 Xy))=> x3) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found x3:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x3:((R Xx0) Xy))=> x3) as proof of (x0 Xy)
% Found (fun (x3:((R Xx0) Xy))=> x3) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found ((conj20 (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P0 Xq)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) 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) x4)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) x4)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x4)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x4)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x4)
% Found (((eq_trans00000 (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) b)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (((eq_trans00000 (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) b)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found ((((fun (x5:(((eq (a->Prop)) Xq) b)) (x6:(((eq (a->Prop)) b) x4))=> (((eq_trans0000 x5) x6) (fun (x8:(a->Prop))=> ((P Xq)->(P x8))))) (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) b)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found ((((fun (x5:(((eq (a->Prop)) Xq) b)) (x6:(((eq (a->Prop)) b) x4))=> ((((eq_trans000 x4) x5) x6) (fun (x8:(a->Prop))=> ((P Xq)->(P x8))))) (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) b)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found ((((fun (x5:(((eq (a->Prop)) Xq) (fun (x7:a)=> (Xq x7)))) (x6:(((eq (a->Prop)) (fun (x7:a)=> (Xq x7))) x4))=> (((((eq_trans00 (fun (x7:a)=> (Xq x7))) x4) x5) x6) (fun (x8:(a->Prop))=> ((P Xq)->(P x8))))) (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found ((((fun (x5:(((eq (a->Prop)) Xq) (fun (x7:a)=> (Xq x7)))) (x6:(((eq (a->Prop)) (fun (x7:a)=> (Xq x7))) x4))=> ((((((eq_trans0 Xq) (fun (x7:a)=> (Xq x7))) x4) x5) x6) (fun (x8:(a->Prop))=> ((P Xq)->(P x8))))) (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found ((((fun (x5:(((eq (a->Prop)) Xq) (fun (x7:a)=> (Xq x7)))) (x6:(((eq (a->Prop)) (fun (x7:a)=> (Xq x7))) x4))=> (((((((eq_trans (a->Prop)) Xq) (fun (x7:a)=> (Xq x7))) x4) x5) x6) (fun (x8:(a->Prop))=> ((P Xq)->(P x8))))) (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> ((((fun (x5:(((eq (a->Prop)) Xq) (fun (x7:a)=> (Xq x7)))) (x6:(((eq (a->Prop)) (fun (x7:a)=> (Xq x7))) x4))=> (((((((eq_trans (a->Prop)) Xq) (fun (x7:a)=> (Xq x7))) x4) x5) x6) (fun (x8:(a->Prop))=> ((P Xq)->(P x8))))) (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x4))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((fun (x5:(((eq (a->Prop)) Xq) (fun (x7:a)=> (Xq x7)))) (x6:(((eq (a->Prop)) (fun (x7:a)=> (Xq x7))) x4))=> (((((((eq_trans (a->Prop)) Xq) (fun (x7:a)=> (Xq x7))) x4) x5) x6) (fun (x8:(a->Prop))=> ((P Xq)->(P x8))))) (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x4:(Xq Xx))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x3:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x4:(Xq Xx))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) as proof of ((Xq Xx)->(((eq (a->Prop)) Xq) x0))
% Found (fun (x3:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x4:(Xq Xx))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) as proof of ((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->(((eq (a->Prop)) Xq) x0)))
% Found (and_rect10 (fun (x3:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x4:(Xq Xx))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((and_rect1 (((eq (a->Prop)) Xq) x0)) (fun (x3:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x4:(Xq Xx))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((fun (P0:Type) (x3:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x3) x00)) (((eq (a->Prop)) Xq) x0)) (fun (x3:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x4:(Xq Xx))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P0:Type) (x3:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x3) x00)) (((eq (a->Prop)) Xq) x0)) (fun (x3:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x4:(Xq Xx))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x3:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x3:(x0 Xy))=> x3) as proof of ((R Xx0) Xy)
% Found (fun (x3:(x0 Xy))=> x3) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found x3:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x3:((R Xx0) Xy))=> x3) as proof of (x0 Xy)
% Found (fun (x3:((R Xx0) Xy))=> x3) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found ((conj20 (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x4:(forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x3:(forall (Xx:a), ((R Xx) Xx))) (x4:(forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) as proof of ((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))->(((eq (a->Prop)) Xq) x0))
% Found (fun (x3:(forall (Xx:a), ((R Xx) Xx))) (x4:(forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) as proof of ((forall (Xx:a), ((R Xx) Xx))->((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))->(((eq (a->Prop)) Xq) x0)))
% Found (and_rect10 (fun (x3:(forall (Xx:a), ((R Xx) Xx))) (x4:(forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((and_rect1 (((eq (a->Prop)) Xq) x0)) (fun (x3:(forall (Xx:a), ((R Xx) Xx))) (x4:(forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((fun (P0:Type) (x3:((forall (Xx:a), ((R Xx) Xx))->((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))->P0)))=> (((((and_rect (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) P0) x3) x1)) (((eq (a->Prop)) Xq) x0)) (fun (x3:(forall (Xx:a), ((R Xx) Xx))) (x4:(forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P0:Type) (x3:((forall (Xx:a), ((R Xx) Xx))->((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))->P0)))=> (((((and_rect (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) P0) x3) x1)) (((eq (a->Prop)) Xq) x0)) (fun (x3:(forall (Xx:a), ((R Xx) Xx))) (x4:(forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found ((iff_sym0 (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq)) as proof of ((Xq Xx)->(((eq (a->Prop)) Xq) x0))
% Found (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq)) as proof of ((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->(((eq (a->Prop)) Xq) x0)))
% Found (and_rect20 (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((and_rect2 (((eq (a->Prop)) Xq) x0)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((fun (P0:Type) (x5:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x5) x00)) (((eq (a->Prop)) Xq) x0)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P0:Type) (x5:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x5) x00)) (((eq (a->Prop)) Xq) x0)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x6:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)) as proof of ((Xq Xx)->(((eq (a->Prop)) Xq) x0))
% Found (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)) as proof of ((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->(((eq (a->Prop)) Xq) x0)))
% Found (and_rect20 (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((and_rect2 (((eq (a->Prop)) Xq) x0)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((fun (P0:Type) (x5:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x5) x00)) (((eq (a->Prop)) Xq) x0)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P0:Type) (x5:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x5) x00)) (((eq (a->Prop)) Xq) x0)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq)) as proof of ((Xq Xx)->(((eq (a->Prop)) Xq) x2))
% Found (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq)) as proof of ((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->(((eq (a->Prop)) Xq) x2)))
% Found (and_rect20 (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((and_rect2 (((eq (a->Prop)) Xq) x2)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((fun (P0:Type) (x5:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x5) x00)) (((eq (a->Prop)) Xq) x2)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P0:Type) (x5:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x5) x00)) (((eq (a->Prop)) Xq) x2)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P0 Xq)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) 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) x0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) x0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x0)
% Found (((eq_trans00000 (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) b)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (((eq_trans00000 (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) b)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found ((((fun (x5:(((eq (a->Prop)) Xq) b)) (x6:(((eq (a->Prop)) b) x0))=> (((eq_trans0000 x5) x6) (fun (x8:(a->Prop))=> ((P Xq)->(P x8))))) (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) b)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found ((((fun (x5:(((eq (a->Prop)) Xq) b)) (x6:(((eq (a->Prop)) b) x0))=> ((((eq_trans000 x0) x5) x6) (fun (x8:(a->Prop))=> ((P Xq)->(P x8))))) (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) b)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found ((((fun (x5:(((eq (a->Prop)) Xq) (fun (x7:a)=> (Xq x7)))) (x6:(((eq (a->Prop)) (fun (x7:a)=> (Xq x7))) x0))=> (((((eq_trans00 (fun (x7:a)=> (Xq x7))) x0) x5) x6) (fun (x8:(a->Prop))=> ((P Xq)->(P x8))))) (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found ((((fun (x5:(((eq (a->Prop)) Xq) (fun (x7:a)=> (Xq x7)))) (x6:(((eq (a->Prop)) (fun (x7:a)=> (Xq x7))) x0))=> ((((((eq_trans0 Xq) (fun (x7:a)=> (Xq x7))) x0) x5) x6) (fun (x8:(a->Prop))=> ((P Xq)->(P x8))))) (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found ((((fun (x5:(((eq (a->Prop)) Xq) (fun (x7:a)=> (Xq x7)))) (x6:(((eq (a->Prop)) (fun (x7:a)=> (Xq x7))) x0))=> (((((((eq_trans (a->Prop)) Xq) (fun (x7:a)=> (Xq x7))) x0) x5) x6) (fun (x8:(a->Prop))=> ((P Xq)->(P x8))))) (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((fun (x5:(((eq (a->Prop)) Xq) (fun (x7:a)=> (Xq x7)))) (x6:(((eq (a->Prop)) (fun (x7:a)=> (Xq x7))) x0))=> (((((((eq_trans (a->Prop)) Xq) (fun (x7:a)=> (Xq x7))) x0) x5) x6) (fun (x8:(a->Prop))=> ((P Xq)->(P x8))))) (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((fun (x5:(((eq (a->Prop)) Xq) (fun (x7:a)=> (Xq x7)))) (x6:(((eq (a->Prop)) (fun (x7:a)=> (Xq x7))) x0))=> (((((((eq_trans (a->Prop)) Xq) (fun (x7:a)=> (Xq x7))) x0) x5) x6) (fun (x8:(a->Prop))=> ((P Xq)->(P x8))))) (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))):(((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found (eq_ref0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found x50:(P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P Xq)
% Found (fun (x50:(P Xq))=> x50) as proof of (P0 Xq)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) 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) x2)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) x2)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x2)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x2)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x2)
% Found (((eq_trans00000 (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) b)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (((eq_trans00000 (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) b)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found ((((fun (x5:(((eq (a->Prop)) Xq) b)) (x6:(((eq (a->Prop)) b) x2))=> (((eq_trans0000 x5) x6) (fun (x8:(a->Prop))=> ((P Xq)->(P x8))))) (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) b)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found ((((fun (x5:(((eq (a->Prop)) Xq) b)) (x6:(((eq (a->Prop)) b) x2))=> ((((eq_trans000 x2) x5) x6) (fun (x8:(a->Prop))=> ((P Xq)->(P x8))))) (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) b)) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found ((((fun (x5:(((eq (a->Prop)) Xq) (fun (x7:a)=> (Xq x7)))) (x6:(((eq (a->Prop)) (fun (x7:a)=> (Xq x7))) x2))=> (((((eq_trans00 (fun (x7:a)=> (Xq x7))) x2) x5) x6) (fun (x8:(a->Prop))=> ((P Xq)->(P x8))))) (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found ((((fun (x5:(((eq (a->Prop)) Xq) (fun (x7:a)=> (Xq x7)))) (x6:(((eq (a->Prop)) (fun (x7:a)=> (Xq x7))) x2))=> ((((((eq_trans0 Xq) (fun (x7:a)=> (Xq x7))) x2) x5) x6) (fun (x8:(a->Prop))=> ((P Xq)->(P x8))))) (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found ((((fun (x5:(((eq (a->Prop)) Xq) (fun (x7:a)=> (Xq x7)))) (x6:(((eq (a->Prop)) (fun (x7:a)=> (Xq x7))) x2))=> (((((((eq_trans (a->Prop)) Xq) (fun (x7:a)=> (Xq x7))) x2) x5) x6) (fun (x8:(a->Prop))=> ((P Xq)->(P x8))))) (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7)))) (fun (x50:(P Xq))=> x50)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((((fun (x5:(((eq (a->Prop)) Xq) (fun (x7:a)=> (Xq x7)))) (x6:(((eq (a->Prop)) (fun (x7:a)=> (Xq x7))) x2))=> (((((((eq_trans (a->Prop)) Xq) (fun (x7:a)=> (Xq x7))) x2) x5) x6) (fun (x8:(a->Prop))=> ((P Xq)->(P x8))))) (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7)))) (fun (x50:(P Xq))=> x50))) as proof of ((P Xq)->(P x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((fun (x5:(((eq (a->Prop)) Xq) (fun (x7:a)=> (Xq x7)))) (x6:(((eq (a->Prop)) (fun (x7:a)=> (Xq x7))) x2))=> (((((((eq_trans (a->Prop)) Xq) (fun (x7:a)=> (Xq x7))) x2) x5) x6) (fun (x8:(a->Prop))=> ((P Xq)->(P x8))))) (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) (fun (x7:a)=> (Xq x7)))) (fun (x50:(P Xq))=> x50))) as proof of (((eq (a->Prop)) Xq) x2)
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x2:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x2:((R Xx0) Xy))=> x2) as proof of (x0 Xy)
% Found (fun (x2:((R Xx0) Xy))=> x2) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found x2:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x2:(x0 Xy))=> x2) as proof of ((R Xx0) Xy)
% Found (fun (x2:(x0 Xy))=> x2) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found ((conj20 (fun (x2:(x0 Xy))=> x2)) (fun (x2:((R Xx0) Xy))=> x2)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x2:(x0 Xy))=> x2)) (fun (x2:((R Xx0) Xy))=> x2)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x2:(x0 Xy))=> x2)) (fun (x2:((R Xx0) Xy))=> x2)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x2:(x0 Xy))=> x2)) (fun (x2:((R Xx0) Xy))=> x2))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x2:(x0 Xy))=> x2)) (fun (x2:((R Xx0) Xy))=> x2))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x3:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x3:(x0 Xy))=> x3) as proof of ((R Xx0) Xy)
% Found (fun (x3:(x0 Xy))=> x3) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found x3:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x3:((R Xx0) Xy))=> x3) as proof of (x0 Xy)
% Found (fun (x3:((R Xx0) Xy))=> x3) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found ((conj20 (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x5:(P0 Xq)
% Instantiate: x0:=Xq:(a->Prop)
% Found (fun (x5:(P0 Xq))=> x5) as proof of (P0 x0)
% Found (fun (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of (((eq (a->Prop)) Xq) x0)
% Found iff_refl0:=(iff_refl (x4 Xy)):((iff (x4 Xy)) (x4 Xy))
% Found (iff_refl (x4 Xy)) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x4 Xy)) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x4 Xy))) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x4 Xx0)) (Xy:a)=> (iff_refl (x4 Xy))) as proof of (forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))
% Found x3:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x3:(x0 Xy))=> x3) as proof of ((R Xx0) Xy)
% Found (fun (x3:(x0 Xy))=> x3) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found x3:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x3:((R Xx0) Xy))=> x3) as proof of (x0 Xy)
% Found (fun (x3:((R Xx0) Xy))=> x3) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found ((conj20 (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x3:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x3:((R Xx0) Xy))=> x3) as proof of (x0 Xy)
% Found (fun (x3:((R Xx0) Xy))=> x3) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found x3:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x3:(x0 Xy))=> x3) as proof of ((R Xx0) Xy)
% Found (fun (x3:(x0 Xy))=> x3) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found ((conj20 (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x1:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x1:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x1:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x1:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x1:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found x5:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x0 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found x5:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found ((conj20 (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x5:((R Xx0) Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x2 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x2 Xy))
% Found x5:(x2 Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x5:(x2 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x2 Xy))=> x5) as proof of ((x2 Xy)->((R Xx0) Xy))
% Found ((conj20 (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found (((conj2 (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found (fun (Xy:a)=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found x3:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x3:(x0 Xy))=> x3) as proof of ((R Xx0) Xy)
% Found (fun (x3:(x0 Xy))=> x3) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found x3:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x3:((R Xx0) Xy))=> x3) as proof of (x0 Xy)
% Found (fun (x3:((R Xx0) Xy))=> x3) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found ((conj20 (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x3:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x3:(x0 Xy))=> x3) as proof of ((R Xx0) Xy)
% Found (fun (x3:(x0 Xy))=> x3) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found x3:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x3:((R Xx0) Xy))=> x3) as proof of (x0 Xy)
% Found (fun (x3:((R Xx0) Xy))=> x3) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found ((conj20 (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))) (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of ((forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz)))->((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))))
% Found (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))) (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of (((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))->((forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz)))->((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))))
% Found (and_rect00 (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))) (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found ((and_rect0 ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))) (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))) (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (((fun (P:Type) (x1:(((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->P)))=> (((((and_rect ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) P) x1) x)) ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))) (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))) (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (((fun (P:Type) (x1:(((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->P)))=> (((((and_rect ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) P) x1) x)) ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))) (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))) (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (Xy:a)=> (((fun (P:Type) (x1:(((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->P)))=> (((((and_rect ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) P) x1) x)) ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))) (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))) (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((fun (P:Type) (x1:(((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->P)))=> (((((and_rect ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) P) x1) x)) ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))) (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))) (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x2 Xy)):((iff (x2 Xy)) (x2 Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x2 Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (iff_refl (x2 Xy))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found x1:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x1:((R Xx0) Xy))=> x1) as proof of (x0 Xy)
% Found (fun (x1:((R Xx0) Xy))=> x1) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found x1:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x1:(x0 Xy))=> x1) as proof of ((R Xx0) Xy)
% Found (fun (x1:(x0 Xy))=> x1) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found ((conj20 (fun (x1:(x0 Xy))=> x1)) (fun (x1:((R Xx0) Xy))=> x1)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x1:(x0 Xy))=> x1)) (fun (x1:((R Xx0) Xy))=> x1)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x1:(x0 Xy))=> x1)) (fun (x1:((R Xx0) Xy))=> x1)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x1:(x0 Xy))=> x1)) (fun (x1:((R Xx0) Xy))=> x1))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x1:(x0 Xy))=> x1)) (fun (x1:((R Xx0) Xy))=> x1))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x4 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x4 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x4 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x4 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x4 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x4 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x4 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x4 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))
% Found x1:(P0 Xq)
% Instantiate: x0:=Xq:(a->Prop)
% Found (fun (x1:(P0 Xq))=> x1) as proof of (P0 x0)
% Found (fun (P0:((a->Prop)->Prop)) (x1:(P0 Xq))=> x1) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop)) (x1:(P0 Xq))=> x1) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x1:(P0 Xq))=> x1) as proof of (((eq (a->Prop)) Xq) x0)
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x4 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x4 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x4 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x4 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x4 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x4 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x4 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x4 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x3:(forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x2:(forall (Xx:a), ((R Xx) Xx))) (x3:(forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))=> (iff_refl (x0 Xy))) as proof of ((forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx)))->((iff (x0 Xy)) ((R Xx0) Xy)))
% Found (fun (x2:(forall (Xx:a), ((R Xx) Xx))) (x3:(forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))=> (iff_refl (x0 Xy))) as proof of ((forall (Xx:a), ((R Xx) Xx))->((forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx)))->((iff (x0 Xy)) ((R Xx0) Xy))))
% Found (and_rect10 (fun (x2:(forall (Xx:a), ((R Xx) Xx))) (x3:(forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((and_rect1 ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x2:(forall (Xx:a), ((R Xx) Xx))) (x3:(forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((fun (P0:Type) (x2:((forall (Xx:a), ((R Xx) Xx))->((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))->P0)))=> (((((and_rect (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) P0) x2) x1)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x2:(forall (Xx:a), ((R Xx) Xx))) (x3:(forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((fun (P0:Type) (x2:((forall (Xx:a), ((R Xx) Xx))->((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))->P0)))=> (((((and_rect (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) P0) x2) x1)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x2:(forall (Xx:a), ((R Xx) Xx))) (x3:(forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))=> (iff_refl (x0 Xy))))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((fun (P0:Type) (x2:((forall (Xx:a), ((R Xx) Xx))->((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))->P0)))=> (((((and_rect (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) P0) x2) x1)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x2:(forall (Xx:a), ((R Xx) Xx))) (x3:(forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))=> (iff_refl (x0 Xy))))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eq_ref000:=(eq_ref00 P0):((P0 Xq)->(P0 Xq))
% Found (eq_ref00 P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((eq_ref0 Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x1:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion_dep0 (fun (x3:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x1:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion_dep0 (fun (x3:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x1:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x1:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x1:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found x2:(P0 Xq)
% Instantiate: x0:=Xq:(a->Prop)
% Found (fun (x2:(P0 Xq))=> x2) as proof of (P0 x0)
% Found (fun (P0:((a->Prop)->Prop)) (x2:(P0 Xq))=> x2) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x1:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x2:(P0 Xq))=> x2) as proof of (((eq (a->Prop)) Xq) x0)
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))) as proof of ((forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00)))->((iff (x0 Xy)) ((R Xx0) Xy)))
% Found (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))) as proof of ((forall (Xx00:a), ((R Xx00) Xx00))->((forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00)))->((iff (x0 Xy)) ((R Xx0) Xy))))
% Found (and_rect10 (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((and_rect1 ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((fun (P0:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P0)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P0) x3) x1)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((fun (P0:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P0)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P0) x3) x1)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((fun (P0:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P0)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P0) x3) x1)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))) as proof of ((forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00)))->((iff (x0 Xy)) ((R Xx0) Xy)))
% Found (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))) as proof of ((forall (Xx00:a), ((R Xx00) Xx00))->((forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00)))->((iff (x0 Xy)) ((R Xx0) Xy))))
% Found (and_rect10 (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((and_rect1 ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((fun (P0:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P0)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P0) x3) x1)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((fun (P0:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P0)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P0) x3) x1)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((fun (P0:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P0)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P0) x3) x1)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x2 Xy)):((iff (x2 Xy)) (x2 Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x2 Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x2 Xy))) as proof of ((forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00)))->((iff (x2 Xy)) ((R Xx0) Xy)))
% Found (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x2 Xy))) as proof of ((forall (Xx00:a), ((R Xx00) Xx00))->((forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00)))->((iff (x2 Xy)) ((R Xx0) Xy))))
% Found (and_rect10 (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x2 Xy)))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found ((and_rect1 ((iff (x2 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x2 Xy)))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (((fun (P0:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P0)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P0) x3) x0)) ((iff (x2 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x2 Xy)))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((fun (P0:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P0)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P0) x3) x0)) ((iff (x2 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x2 Xy))))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (((fun (P0:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P0)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P0) x3) x0)) ((iff (x2 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x2 Xy))))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x4:(forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x3:(forall (Xx:a), ((R Xx) Xx))) (x4:(forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))=> (iff_refl (x0 Xy))) as proof of ((forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx)))->((iff (x0 Xy)) ((R Xx0) Xy)))
% Found (fun (x3:(forall (Xx:a), ((R Xx) Xx))) (x4:(forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))=> (iff_refl (x0 Xy))) as proof of ((forall (Xx:a), ((R Xx) Xx))->((forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx)))->((iff (x0 Xy)) ((R Xx0) Xy))))
% Found (and_rect10 (fun (x3:(forall (Xx:a), ((R Xx) Xx))) (x4:(forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((and_rect1 ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx:a), ((R Xx) Xx))) (x4:(forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((fun (P0:Type) (x3:((forall (Xx:a), ((R Xx) Xx))->((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))->P0)))=> (((((and_rect (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) P0) x3) x1)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx:a), ((R Xx) Xx))) (x4:(forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((fun (P0:Type) (x3:((forall (Xx:a), ((R Xx) Xx))->((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))->P0)))=> (((((and_rect (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) P0) x3) x1)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx:a), ((R Xx) Xx))) (x4:(forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))=> (iff_refl (x0 Xy))))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((fun (P0:Type) (x3:((forall (Xx:a), ((R Xx) Xx))->((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))->P0)))=> (((((and_rect (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) P0) x3) x1)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx:a), ((R Xx) Xx))) (x4:(forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))=> (iff_refl (x0 Xy))))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% 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) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))))
% Found (((eq_trans000 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))))
% Found ((((eq_trans00 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))))
% Found (((((eq_trans0 (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))))
% Found ((((((eq_trans Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))))
% Found ((((((eq_trans Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))))
% 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) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))))
% Found (((eq_trans000 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))))
% Found ((((eq_trans00 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))))
% Found (((((eq_trans0 (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))))
% Found ((((((eq_trans Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))))
% Found ((((((eq_trans Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))) as proof of ((forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00)))->((iff (x0 Xy)) ((R Xx0) Xy)))
% Found (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))) as proof of ((forall (Xx00:a), ((R Xx00) Xx00))->((forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00)))->((iff (x0 Xy)) ((R Xx0) Xy))))
% Found (and_rect10 (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((and_rect1 ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((fun (P0:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P0)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P0) x3) x1)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((fun (P0:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P0)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P0) x3) x1)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((fun (P0:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P0)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P0) x3) x1)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))) as proof of ((forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00)))->((iff (x0 Xy)) ((R Xx0) Xy)))
% Found (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))) as proof of ((forall (Xx00:a), ((R Xx00) Xx00))->((forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00)))->((iff (x0 Xy)) ((R Xx0) Xy))))
% Found (and_rect10 (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((and_rect1 ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((fun (P0:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P0)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P0) x3) x1)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((fun (P0:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P0)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P0) x3) x1)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((fun (P0:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P0)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P0) x3) x1)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x2 Xy)):((iff (x2 Xy)) (x2 Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x2 Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x2 Xy))) as proof of ((forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00)))->((iff (x2 Xy)) ((R Xx0) Xy)))
% Found (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x2 Xy))) as proof of ((forall (Xx00:a), ((R Xx00) Xx00))->((forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00)))->((iff (x2 Xy)) ((R Xx0) Xy))))
% Found (and_rect10 (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x2 Xy)))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found ((and_rect1 ((iff (x2 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x2 Xy)))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (((fun (P0:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P0)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P0) x3) x0)) ((iff (x2 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x2 Xy)))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((fun (P0:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P0)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P0) x3) x0)) ((iff (x2 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x2 Xy))))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (((fun (P0:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P0)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P0) x3) x0)) ((iff (x2 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x2 Xy))))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x4:(forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x3:(forall (Xx:a), ((R Xx) Xx))) (x4:(forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))=> (iff_refl (x0 Xy))) as proof of ((forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx)))->((iff (x0 Xy)) ((R Xx0) Xy)))
% Found (fun (x3:(forall (Xx:a), ((R Xx) Xx))) (x4:(forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))=> (iff_refl (x0 Xy))) as proof of ((forall (Xx:a), ((R Xx) Xx))->((forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx)))->((iff (x0 Xy)) ((R Xx0) Xy))))
% Found (and_rect10 (fun (x3:(forall (Xx:a), ((R Xx) Xx))) (x4:(forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((and_rect1 ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx:a), ((R Xx) Xx))) (x4:(forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((fun (P0:Type) (x3:((forall (Xx:a), ((R Xx) Xx))->((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))->P0)))=> (((((and_rect (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) P0) x3) x1)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx:a), ((R Xx) Xx))) (x4:(forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((fun (P0:Type) (x3:((forall (Xx:a), ((R Xx) Xx))->((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))->P0)))=> (((((and_rect (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) P0) x3) x1)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx:a), ((R Xx) Xx))) (x4:(forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))=> (iff_refl (x0 Xy))))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((fun (P0:Type) (x3:((forall (Xx:a), ((R Xx) Xx))->((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))->P0)))=> (((((and_rect (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) P0) x3) x1)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx:a), ((R Xx) Xx))) (x4:(forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))=> (iff_refl (x0 Xy))))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x2 Xy)):((iff (x2 Xy)) (x2 Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x2 Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (iff_refl (x2 Xy))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found x5:(x4 Xy)
% Instantiate: x4:=(R Xx0):(a->Prop)
% Found (fun (x5:(x4 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x4 Xy))=> x5) as proof of ((x4 Xy)->((R Xx0) Xy))
% Found x5:((R Xx0) Xy)
% Instantiate: x4:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x4 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x4 Xy))
% Found ((conj20 (fun (x5:(x4 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x4 Xy))) (fun (x5:(x4 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x4 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x4 Xy))) (fun (x5:(x4 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x4 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x4 Xy))) (fun (x5:(x4 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x4 Xx0)) (Xy:a)=> ((((conj ((x4 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x4 Xy))) (fun (x5:(x4 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))
% Found x5:(x4 Xy)
% Instantiate: x4:=(R Xx0):(a->Prop)
% Found (fun (x5:(x4 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x4 Xy))=> x5) as proof of ((x4 Xy)->((R Xx0) Xy))
% Found x5:((R Xx0) Xy)
% Instantiate: x4:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x4 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x4 Xy))
% Found ((conj20 (fun (x5:(x4 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x4 Xy))) (fun (x5:(x4 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x4 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x4 Xy))) (fun (x5:(x4 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x4 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x4 Xy))) (fun (x5:(x4 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x4 Xx0)) (Xy:a)=> ((((conj ((x4 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x4 Xy))) (fun (x5:(x4 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) b)
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% Found (eq_sym010 ((eq_ref Prop) (f x4))) as proof of (((eq Prop) b) (f x4))
% Found ((eq_sym01 b) ((eq_ref Prop) (f x4))) as proof of (((eq Prop) b) (f x4))
% Found (((eq_sym0 (f x4)) b) ((eq_ref Prop) (f x4))) as proof of (((eq Prop) b) (f x4))
% Found (((eq_sym0 (f x4)) b) ((eq_ref Prop) (f x4))) as proof of (((eq Prop) b) (f x4))
% Found ((eq_trans0000 ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))) (((eq_sym0 (f x4)) b) ((eq_ref Prop) (f x4)))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) (f x4))
% Found (((eq_trans000 (f x4)) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))) (((eq_sym0 (f x4)) b) ((eq_ref Prop) (f x4)))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) (f x4))
% Found ((((eq_trans00 ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) (f x4)) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))) (((eq_sym0 (f x4)) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((eq_ref Prop) (f x4)))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) (f x4))
% Found (((((eq_trans0 ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) (f x4)) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))) (((eq_sym0 (f x4)) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((eq_ref Prop) (f x4)))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) (f x4))
% Found ((((((eq_trans Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) (f x4)) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))) (((eq_sym0 (f x4)) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((eq_ref Prop) (f x4)))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) (f x4))
% Found ((((((eq_trans Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) (f x4)) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))) (((eq_sym0 (f x4)) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((eq_ref Prop) (f x4)))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) (f x4))
% Found (eq_sym000 ((((((eq_trans Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) (f x4)) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))) (((eq_sym0 (f x4)) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((eq_ref Prop) (f x4))))) as proof of (((eq Prop) (f x4)) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))
% Found ((eq_sym00 (f x4)) ((((((eq_trans Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) (f x4)) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))) (((eq_sym0 (f x4)) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((eq_ref Prop) (f x4))))) as proof of (((eq Prop) (f x4)) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))
% Found (((eq_sym0 ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) (f x4)) ((((((eq_trans Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) (f x4)) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))) (((eq_sym0 (f x4)) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((eq_ref Prop) (f x4))))) as proof of (((eq Prop) (f x4)) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))
% Found ((((eq_sym Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) (f x4)) ((((((eq_trans Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) (f x4)) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))) ((((eq_sym Prop) (f x4)) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((eq_ref Prop) (f x4))))) as proof of (((eq Prop) (f x4)) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq)) as proof of ((Xq Xx)->(((eq (a->Prop)) Xq) x0))
% Found (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq)) as proof of ((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->(((eq (a->Prop)) Xq) x0)))
% Found (and_rect20 (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((and_rect2 (((eq (a->Prop)) Xq) x0)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((fun (P0:Type) (x5:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x5) x00)) (((eq (a->Prop)) Xq) x0)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P0:Type) (x5:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x5) x00)) (((eq (a->Prop)) Xq) x0)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion_dep0 (fun (x8:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x6:(Xq Xx))=> (((eta_expansion_dep a) (fun (x8:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion_dep a) (fun (x8:a)=> Prop)) Xq)) as proof of ((Xq Xx)->(((eq (a->Prop)) Xq) x0))
% Found (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion_dep a) (fun (x8:a)=> Prop)) Xq)) as proof of ((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->(((eq (a->Prop)) Xq) x0)))
% Found (and_rect20 (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion_dep a) (fun (x8:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((and_rect2 (((eq (a->Prop)) Xq) x0)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion_dep a) (fun (x8:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((fun (P0:Type) (x5:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x5) x00)) (((eq (a->Prop)) Xq) x0)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion_dep a) (fun (x8:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P0:Type) (x5:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x5) x00)) (((eq (a->Prop)) Xq) x0)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion_dep a) (fun (x8:a)=> Prop)) Xq)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion_dep0 (fun (x8:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x6:(Xq Xx))=> (((eta_expansion_dep a) (fun (x8:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion_dep a) (fun (x8:a)=> Prop)) Xq)) as proof of ((Xq Xx)->(((eq (a->Prop)) Xq) x2))
% Found (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion_dep a) (fun (x8:a)=> Prop)) Xq)) as proof of ((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->(((eq (a->Prop)) Xq) x2)))
% Found (and_rect20 (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion_dep a) (fun (x8:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((and_rect2 (((eq (a->Prop)) Xq) x2)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion_dep a) (fun (x8:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((fun (P0:Type) (x5:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x5) x00)) (((eq (a->Prop)) Xq) x2)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion_dep a) (fun (x8:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P0:Type) (x5:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x5) x00)) (((eq (a->Prop)) Xq) x2)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion_dep a) (fun (x8:a)=> Prop)) Xq)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found x5:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x0 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found x5:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found ((conj20 (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x5:(x2 Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x5:(x2 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x2 Xy))=> x5) as proof of ((x2 Xy)->((R Xx0) Xy))
% Found x5:((R Xx0) Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x2 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x2 Xy))
% Found ((conj20 (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))):(((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found (eq_ref0 ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))):(((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found (eq_ref0 ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x10:(P0 (f x0))
% Found (fun (x10:(P0 (f x0)))=> x10) as proof of (P0 (f x0))
% Found (fun (x10:(P0 (f x0)))=> x10) as proof of (P1 (f x0))
% Found x10:(P0 (f x0))
% Found (fun (x10:(P0 (f x0)))=> x10) as proof of (P0 (f x0))
% Found (fun (x10:(P0 (f x0)))=> x10) as proof of (P1 (f x0))
% Found x5:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found x5:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x0 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found ((conj20 (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x5:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found x5:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x0 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found ((conj20 (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x5:(x2 Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x5:(x2 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x2 Xy))=> x5) as proof of ((x2 Xy)->((R Xx0) Xy))
% Found x5:((R Xx0) Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x2 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x2 Xy))
% Found ((conj20 (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found x20:(P0 Xq)
% Found (fun (x20:(P0 Xq))=> x20) as proof of (P0 Xq)
% Found (fun (x20:(P0 Xq))=> x20) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion_dep0000 (fun (x20:(P0 Xq))=> x20)) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion_dep000 (fun (x3:(a->Prop))=> ((P0 Xq)->(P0 x3)))) (fun (x20:(P0 Xq))=> x20)) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion_dep00 Xq) (fun (x3:(a->Prop))=> ((P0 Xq)->(P0 x3)))) (fun (x20:(P0 Xq))=> x20)) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion_dep0 (fun (x3:a)=> Prop)) Xq) (fun (x3:(a->Prop))=> ((P0 Xq)->(P0 x3)))) (fun (x20:(P0 Xq))=> x20)) as proof of ((P0 Xq)->(P0 x0))
% Found (((((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xq) (fun (x3:(a->Prop))=> ((P0 Xq)->(P0 x3)))) (fun (x20:(P0 Xq))=> x20)) as proof of ((P0 Xq)->(P0 x0))
% Found (((((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xq) (fun (x3:(a->Prop))=> ((P0 Xq)->(P0 x3)))) (fun (x20:(P0 Xq))=> x20)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xq) (fun (x3:(a->Prop))=> ((P0 Xq)->(P0 x3)))) (fun (x20:(P0 Xq))=> x20))) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x1:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xq) (fun (x3:(a->Prop))=> ((P0 Xq)->(P0 x3)))) (fun (x20:(P0 Xq))=> x20))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x20:(P0 Xq)
% Found (fun (x20:(P0 Xq))=> x20) as proof of (P0 Xq)
% Found (fun (x20:(P0 Xq))=> x20) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion0000 (fun (x20:(P0 Xq))=> x20)) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion000 (fun (x3:(a->Prop))=> ((P0 Xq)->(P0 x3)))) (fun (x20:(P0 Xq))=> x20)) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion00 Xq) (fun (x3:(a->Prop))=> ((P0 Xq)->(P0 x3)))) (fun (x20:(P0 Xq))=> x20)) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion0 Prop) Xq) (fun (x3:(a->Prop))=> ((P0 Xq)->(P0 x3)))) (fun (x20:(P0 Xq))=> x20)) as proof of ((P0 Xq)->(P0 x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x3:(a->Prop))=> ((P0 Xq)->(P0 x3)))) (fun (x20:(P0 Xq))=> x20)) as proof of ((P0 Xq)->(P0 x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x3:(a->Prop))=> ((P0 Xq)->(P0 x3)))) (fun (x20:(P0 Xq))=> x20)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x3:(a->Prop))=> ((P0 Xq)->(P0 x3)))) (fun (x20:(P0 Xq))=> x20))) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x1:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x3:(a->Prop))=> ((P0 Xq)->(P0 x3)))) (fun (x20:(P0 Xq))=> x20))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% 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_sym010 ((eq_ref Prop) (f x0))) as proof of (((eq Prop) b) (f x0))
% Found ((eq_sym01 b) ((eq_ref Prop) (f x0))) as proof of (((eq Prop) b) (f x0))
% Found (((eq_sym0 (f x0)) b) ((eq_ref Prop) (f x0))) as proof of (((eq Prop) b) (f x0))
% Found (((eq_sym0 (f x0)) b) ((eq_ref Prop) (f x0))) as proof of (((eq Prop) b) (f x0))
% Found ((eq_trans0000 ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (((eq_sym0 (f x0)) b) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0))
% Found (((eq_trans000 (f x0)) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (((eq_sym0 (f x0)) b) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0))
% Found ((((eq_trans00 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0)) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (((eq_sym0 (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0))
% Found (((((eq_trans0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0)) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (((eq_sym0 (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0))
% Found ((((((eq_trans Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0)) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (((eq_sym0 (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0))
% Found (fun (x2:(forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz))))=> ((((((eq_trans Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0)) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (((eq_sym0 (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0))
% Found (fun (x1:((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))) (x2:(forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz))))=> ((((((eq_trans Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0)) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (((eq_sym0 (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))))) as proof of ((forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz)))->(((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0)))
% Found (fun (x1:((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))) (x2:(forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz))))=> ((((((eq_trans Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0)) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (((eq_sym0 (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))))) as proof of (((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))->((forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz)))->(((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0))))
% Found (and_rect00 (fun (x1:((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))) (x2:(forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz))))=> ((((((eq_trans Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0)) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (((eq_sym0 (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0)))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0))
% Found ((and_rect0 (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0))) (fun (x1:((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))) (x2:(forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz))))=> ((((((eq_trans Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0)) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (((eq_sym0 (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0)))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0))
% Found (((fun (P0:Type) (x1:(((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->P0)))=> (((((and_rect ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) P0) x1) x)) (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0))) (fun (x1:((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))) (x2:(forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz))))=> ((((((eq_trans Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0)) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (((eq_sym0 (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0)))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0))
% Found (((fun (P0:Type) (x1:(((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->P0)))=> (((((and_rect ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) P0) x1) x)) (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0))) (fun (x1:((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))) (x2:(forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz))))=> ((((((eq_trans Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0)) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (((eq_sym0 (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0)))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0))
% Found (eq_sym000 (((fun (P0:Type) (x1:(((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->P0)))=> (((((and_rect ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) P0) x1) x)) (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0))) (fun (x1:((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))) (x2:(forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz))))=> ((((((eq_trans Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0)) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (((eq_sym0 (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))))))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_sym00 (f x0)) (((fun (P0:Type) (x1:(((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->P0)))=> (((((and_rect ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) P0) x1) x)) (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0))) (fun (x1:((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))) (x2:(forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz))))=> ((((((eq_trans Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0)) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (((eq_sym0 (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))))))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found (((eq_sym0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0)) (((fun (P0:Type) (x1:(((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))->((forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz)))->P0)))=> (((((and_rect ((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz)))) P0) x1) x)) (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0))) (fun (x1:((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))) (x2:(forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz))))=> ((((((eq_trans Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0)) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (((eq_sym0 (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))))))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((((eq_sym Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0)) (((fun (P0:Type) (x1:(((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))->((forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz)))->P0)))=> (((((and_rect ((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz)))) P0) x1) x)) (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0))) (fun (x1:((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))) (x2:(forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz))))=> ((((((eq_trans Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) (f x0)) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) ((((eq_sym Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((eq_ref Prop) (f x0))))))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found x4:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x4:((R Xx0) Xy))=> x4) as proof of (x0 Xy)
% Found (fun (x4:((R Xx0) Xy))=> x4) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found x4:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x4:(x0 Xy))=> x4) as proof of ((R Xx0) Xy)
% Found (fun (x4:(x0 Xy))=> x4) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found ((conj20 (fun (x4:(x0 Xy))=> x4)) (fun (x4:((R Xx0) Xy))=> x4)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x4:(x0 Xy))=> x4)) (fun (x4:((R Xx0) Xy))=> x4)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x4:(x0 Xy))=> x4)) (fun (x4:((R Xx0) Xy))=> x4)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x4:(x0 Xy))=> x4)) (fun (x4:((R Xx0) Xy))=> x4))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x4:(x0 Xy))=> x4)) (fun (x4:((R Xx0) Xy))=> x4))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x4:((R Xx0) Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x4:((R Xx0) Xy))=> x4) as proof of (x2 Xy)
% Found (fun (x4:((R Xx0) Xy))=> x4) as proof of (((R Xx0) Xy)->(x2 Xy))
% Found x4:(x2 Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x4:(x2 Xy))=> x4) as proof of ((R Xx0) Xy)
% Found (fun (x4:(x2 Xy))=> x4) as proof of ((x2 Xy)->((R Xx0) Xy))
% Found ((conj20 (fun (x4:(x2 Xy))=> x4)) (fun (x4:((R Xx0) Xy))=> x4)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x2 Xy))) (fun (x4:(x2 Xy))=> x4)) (fun (x4:((R Xx0) Xy))=> x4)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x4:(x2 Xy))=> x4)) (fun (x4:((R Xx0) Xy))=> x4)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x4:(x2 Xy))=> x4)) (fun (x4:((R Xx0) Xy))=> x4))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x4:(x2 Xy))=> x4)) (fun (x4:((R Xx0) Xy))=> x4))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found x4:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x4:(x0 Xy))=> x4) as proof of ((R Xx0) Xy)
% Found (fun (x4:(x0 Xy))=> x4) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found x4:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x4:((R Xx0) Xy))=> x4) as proof of (x0 Xy)
% Found (fun (x4:((R Xx0) Xy))=> x4) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found ((conj20 (fun (x4:(x0 Xy))=> x4)) (fun (x4:((R Xx0) Xy))=> x4)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x4:(x0 Xy))=> x4)) (fun (x4:((R Xx0) Xy))=> x4)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x4:(x0 Xy))=> x4)) (fun (x4:((R Xx0) Xy))=> x4)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x4:(x0 Xy))=> x4)) (fun (x4:((R Xx0) Xy))=> x4))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x4:(x0 Xy))=> x4)) (fun (x4:((R Xx0) Xy))=> x4))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x5:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x0 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found x5:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found ((conj20 (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x5:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found x5:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x0 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found ((conj20 (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x5:(x2 Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x5:(x2 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x2 Xy))=> x5) as proof of ((x2 Xy)->((R Xx0) Xy))
% Found x5:((R Xx0) Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x2 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x2 Xy))
% Found ((conj20 (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x3:(P0 Xq)
% Instantiate: x2:=Xq:(a->Prop)
% Found (fun (x3:(P0 Xq))=> x3) as proof of (P0 x2)
% Found (fun (P0:((a->Prop)->Prop)) (x3:(P0 Xq))=> x3) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop)) (x3:(P0 Xq))=> x3) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x3:(P0 Xq))=> x3) as proof of (((eq (a->Prop)) Xq) x2)
% Found x5:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x0 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found x5:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found ((conj20 (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x5:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x0 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found x5:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found ((conj20 (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x5:(x2 Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x5:(x2 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x2 Xy))=> x5) as proof of ((x2 Xy)->((R Xx0) Xy))
% Found x5:((R Xx0) Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x2 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x2 Xy))
% Found ((conj20 (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% 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)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x2))->(P ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))))
% Found (((eq_trans000 ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x2))->(P ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))))
% Found ((((eq_trans00 ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))) as proof of (forall (P:(Prop->Prop)), ((P (f x2))->(P ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))))
% Found (((((eq_trans0 (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))) as proof of (forall (P:(Prop->Prop)), ((P (f x2))->(P ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))))
% Found ((((((eq_trans Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))) as proof of (forall (P:(Prop->Prop)), ((P (f x2))->(P ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))))
% Found ((((((eq_trans Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))) as proof of (forall (P:(Prop->Prop)), ((P (f x2))->(P ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))))
% 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)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x2))->(P ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))))
% Found (((eq_trans000 ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x2))->(P ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))))
% Found ((((eq_trans00 ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))) as proof of (forall (P:(Prop->Prop)), ((P (f x2))->(P ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))))
% Found (((((eq_trans0 (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))) as proof of (forall (P:(Prop->Prop)), ((P (f x2))->(P ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))))
% Found ((((((eq_trans Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))) as proof of (forall (P:(Prop->Prop)), ((P (f x2))->(P ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))))
% Found ((((((eq_trans Prop) (f x2)) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))) as proof of (forall (P:(Prop->Prop)), ((P (f x2))->(P ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))))
% Found x5:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x0 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found x5:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found ((conj20 (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00)))->((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))))
% Found (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((forall (Xx00:a), ((R Xx00) Xx00))->((forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00)))->((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))))
% Found (and_rect10 (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found ((and_rect1 ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (((fun (P:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P) x3) x1)) ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (((fun (P:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P) x3) x1)) ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (Xy:a)=> (((fun (P:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P) x3) x1)) ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((fun (P:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P) x3) x1)) ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x5:((R Xx0) Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x2 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x2 Xy))
% Found x5:(x2 Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x5:(x2 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x2 Xy))=> x5) as proof of ((x2 Xy)->((R Xx0) Xy))
% Found ((conj20 (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found (((conj2 (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found (fun (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00)))->((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))))
% Found (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((forall (Xx00:a), ((R Xx00) Xx00))->((forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00)))->((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))))
% Found (and_rect10 (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)))) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found ((and_rect1 ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)))) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found (((fun (P:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P) x3) x0)) ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)))) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found (((fun (P:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P) x3) x0)) ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)))) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found (fun (Xy:a)=> (((fun (P:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P) x3) x0)) ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (((fun (P:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P) x3) x0)) ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found x4:(P0 Xq)
% Instantiate: x2:=Xq:(a->Prop)
% Found (fun (x4:(P0 Xq))=> x4) as proof of (P0 x2)
% Found (fun (P0:((a->Prop)->Prop)) (x4:(P0 Xq))=> x4) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x4:(P0 Xq))=> x4) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion_dep0 (fun (x5:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xq)->(P0 Xq))
% Found (eq_ref00 P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((eq_ref0 Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x2)
% Found x4:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x4:(x0 Xy))=> x4) as proof of ((R Xx0) Xy)
% Found (fun (x4:(x0 Xy))=> x4) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found x4:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x4:((R Xx0) Xy))=> x4) as proof of (x0 Xy)
% Found (fun (x4:((R Xx0) Xy))=> x4) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found ((conj20 (fun (x4:(x0 Xy))=> x4)) (fun (x4:((R Xx0) Xy))=> x4)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x4:(x0 Xy))=> x4)) (fun (x4:((R Xx0) Xy))=> x4)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x4:(x0 Xy))=> x4)) (fun (x4:((R Xx0) Xy))=> x4)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x4:(x0 Xy))=> x4)) (fun (x4:((R Xx0) Xy))=> x4))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x4:(x0 Xy))=> x4)) (fun (x4:((R Xx0) Xy))=> x4))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x3:((R Xx0) Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x3:((R Xx0) Xy))=> x3) as proof of (x2 Xy)
% Found (fun (x3:((R Xx0) Xy))=> x3) as proof of (((R Xx0) Xy)->(x2 Xy))
% Found x3:(x2 Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x3:(x2 Xy))=> x3) as proof of ((R Xx0) Xy)
% Found (fun (x3:(x2 Xy))=> x3) as proof of ((x2 Xy)->((R Xx0) Xy))
% Found ((conj20 (fun (x3:(x2 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x2 Xy))) (fun (x3:(x2 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x3:(x2 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x3:(x2 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x3:(x2 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found x5:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found x5:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x0 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found ((conj20 (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x5:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found x5:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x0 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found ((conj20 (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x5:((R Xx0) Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x2 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x2 Xy))
% Found x5:(x2 Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x5:(x2 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x2 Xy))=> x5) as proof of ((x2 Xy)->((R Xx0) Xy))
% Found ((conj20 (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found x5:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found x5:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x0 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found ((conj20 (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion_dep0 (fun (x5:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found x5:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x0 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found x5:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found ((conj20 (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x5:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found x5:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x0 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found ((conj20 (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x5:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x0 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found x5:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found ((conj20 (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x5:((R Xx0) Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x2 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x2 Xy))
% Found x5:(x2 Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x5:(x2 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x2 Xy))=> x5) as proof of ((x2 Xy)->((R Xx0) Xy))
% Found ((conj20 (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))) (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> (iff_refl (x0 Xy))) as proof of ((forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz)))->((iff (x0 Xy)) ((R Xx0) Xy)))
% Found (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))) (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> (iff_refl (x0 Xy))) as proof of (((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))->((forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz)))->((iff (x0 Xy)) ((R Xx0) Xy))))
% Found (and_rect00 (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))) (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((and_rect0 ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))) (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((fun (P0:Type) (x1:(((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->P0)))=> (((((and_rect ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) P0) x1) x)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))) (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((fun (P0:Type) (x1:(((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->P0)))=> (((((and_rect ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) P0) x1) x)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))) (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> (iff_refl (x0 Xy))))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((fun (P0:Type) (x1:(((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->P0)))=> (((((and_rect ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) P0) x1) x)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))) (x2:(forall (Xx:a) (Xy0:a) (Xz:a), (((and ((R Xx) Xy0)) ((R Xy0) Xz))->((R Xx) Xz))))=> (iff_refl (x0 Xy))))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x3:(P0 Xq)
% Instantiate: x0:=Xq:(a->Prop)
% Found (fun (x3:(P0 Xq))=> x3) as proof of (P0 x0)
% Found (fun (P0:((a->Prop)->Prop)) (x3:(P0 Xq))=> x3) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop)) (x3:(P0 Xq))=> x3) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x3:(P0 Xq))=> x3) as proof of (((eq (a->Prop)) Xq) x0)
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x2 Xy)):((iff (x2 Xy)) (x2 Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x2 Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (iff_refl (x2 Xy))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found x20:(P0 Xq)
% Found (fun (x20:(P0 Xq))=> x20) as proof of (P0 Xq)
% Found (fun (x20:(P0 Xq))=> x20) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion_dep0000 (fun (x20:(P0 Xq))=> x20)) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion_dep000 (fun (x3:(a->Prop))=> ((P0 Xq)->(P0 x3)))) (fun (x20:(P0 Xq))=> x20)) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion_dep00 Xq) (fun (x3:(a->Prop))=> ((P0 Xq)->(P0 x3)))) (fun (x20:(P0 Xq))=> x20)) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion_dep0 (fun (x3:a)=> Prop)) Xq) (fun (x3:(a->Prop))=> ((P0 Xq)->(P0 x3)))) (fun (x20:(P0 Xq))=> x20)) as proof of ((P0 Xq)->(P0 x0))
% Found (((((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xq) (fun (x3:(a->Prop))=> ((P0 Xq)->(P0 x3)))) (fun (x20:(P0 Xq))=> x20)) as proof of ((P0 Xq)->(P0 x0))
% Found (((((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xq) (fun (x3:(a->Prop))=> ((P0 Xq)->(P0 x3)))) (fun (x20:(P0 Xq))=> x20)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xq) (fun (x3:(a->Prop))=> ((P0 Xq)->(P0 x3)))) (fun (x20:(P0 Xq))=> x20))) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x1:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xq) (fun (x3:(a->Prop))=> ((P0 Xq)->(P0 x3)))) (fun (x20:(P0 Xq))=> x20))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x20:(P0 Xq)
% Found (fun (x20:(P0 Xq))=> x20) as proof of (P0 Xq)
% Found (fun (x20:(P0 Xq))=> x20) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion0000 (fun (x20:(P0 Xq))=> x20)) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion000 (fun (x3:(a->Prop))=> ((P0 Xq)->(P0 x3)))) (fun (x20:(P0 Xq))=> x20)) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion00 Xq) (fun (x3:(a->Prop))=> ((P0 Xq)->(P0 x3)))) (fun (x20:(P0 Xq))=> x20)) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion0 Prop) Xq) (fun (x3:(a->Prop))=> ((P0 Xq)->(P0 x3)))) (fun (x20:(P0 Xq))=> x20)) as proof of ((P0 Xq)->(P0 x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x3:(a->Prop))=> ((P0 Xq)->(P0 x3)))) (fun (x20:(P0 Xq))=> x20)) as proof of ((P0 Xq)->(P0 x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x3:(a->Prop))=> ((P0 Xq)->(P0 x3)))) (fun (x20:(P0 Xq))=> x20)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x3:(a->Prop))=> ((P0 Xq)->(P0 x3)))) (fun (x20:(P0 Xq))=> x20))) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x1:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x3:(a->Prop))=> ((P0 Xq)->(P0 x3)))) (fun (x20:(P0 Xq))=> x20))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x3:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x3:((R Xx0) Xy))=> x3) as proof of (x0 Xy)
% Found (fun (x3:((R Xx0) Xy))=> x3) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found x3:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x3:(x0 Xy))=> x3) as proof of ((R Xx0) Xy)
% Found (fun (x3:(x0 Xy))=> x3) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found ((conj20 (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq)) as proof of ((Xq Xx)->(((eq (a->Prop)) Xq) x0))
% Found (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq)) as proof of ((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->(((eq (a->Prop)) Xq) x0)))
% Found (and_rect20 (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((and_rect2 (((eq (a->Prop)) Xq) x0)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((fun (P0:Type) (x5:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x5) x00)) (((eq (a->Prop)) Xq) x0)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P0:Type) (x5:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x5) x00)) (((eq (a->Prop)) Xq) x0)) (fun (x5:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x6:(Xq Xx))=> (((eta_expansion a) Prop) Xq)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x4:(P0 Xq)
% Instantiate: x2:=Xq:(a->Prop)
% Found (fun (x4:(P0 Xq))=> x4) as proof of (P0 x2)
% Found (fun (P0:((a->Prop)->Prop)) (x4:(P0 Xq))=> x4) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x4:(P0 Xq))=> x4) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xq)->(P0 Xq))
% Found (eq_ref00 P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((eq_ref0 Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion_dep0 (fun (x5:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion_dep0 (fun (x5:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion_dep0 (fun (x5:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xq)->(P0 Xq))
% Found (eq_ref00 P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((eq_ref0 Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x2)
% Found x4:(P0 Xq)
% Instantiate: x2:=Xq:(a->Prop)
% Found (fun (x4:(P0 Xq))=> x4) as proof of (P0 x2)
% Found (fun (P0:((a->Prop)->Prop)) (x4:(P0 Xq))=> x4) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x4:(P0 Xq))=> x4) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion_dep0 (fun (x5:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x2)
% Found x3:(P0 Xq)
% Instantiate: x0:=Xq:(a->Prop)
% Found (fun (x3:(P0 Xq))=> x3) as proof of (P0 x0)
% Found (fun (P0:((a->Prop)->Prop)) (x3:(P0 Xq))=> x3) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop)) (x3:(P0 Xq))=> x3) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x3:(P0 Xq))=> x3) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion:=(fun (A:Type) (B:Type)=> ((eta_expansion_dep A) (fun (x1:A)=> B))):(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x))))
% Instantiate: a0:=(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x)))):Prop
% Found eta_expansion as proof of a0
% Found x30:(P0 (f x2))
% Found (fun (x30:(P0 (f x2)))=> x30) as proof of (P0 (f x2))
% Found (fun (x30:(P0 (f x2)))=> x30) as proof of (P1 (f x2))
% Found x30:(P0 (f x2))
% Found (fun (x30:(P0 (f x2)))=> x30) as proof of (P0 (f x2))
% Found (fun (x30:(P0 (f x2)))=> x30) as proof of (P1 (f x2))
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x3:(Xq Xx))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x2:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x3:(Xq Xx))=> (((eta_expansion a) Prop) Xq)) as proof of ((Xq Xx)->(((eq (a->Prop)) Xq) x0))
% Found (fun (x2:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x3:(Xq Xx))=> (((eta_expansion a) Prop) Xq)) as proof of ((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->(((eq (a->Prop)) Xq) x0)))
% Found (and_rect00 (fun (x2:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x3:(Xq Xx))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((and_rect0 (((eq (a->Prop)) Xq) x0)) (fun (x2:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x3:(Xq Xx))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((fun (P0:Type) (x2:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x2) x1)) (((eq (a->Prop)) Xq) x0)) (fun (x2:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x3:(Xq Xx))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x1:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P0:Type) (x2:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x2) x1)) (((eq (a->Prop)) Xq) x0)) (fun (x2:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x3:(Xq Xx))=> (((eta_expansion a) Prop) Xq)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x3:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x3:(x0 Xy))=> x3) as proof of ((R Xx0) Xy)
% Found (fun (x3:(x0 Xy))=> x3) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found x3:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x3:((R Xx0) Xy))=> x3) as proof of (x0 Xy)
% Found (fun (x3:((R Xx0) Xy))=> x3) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found ((conj20 (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eq_ref00:=(eq_ref0 x0):(((eq (a->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found (eq_sym100 ((eq_ref (a->Prop)) x0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eq_sym10 Xq) ((eq_ref (a->Prop)) x0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eq_sym1 x0) Xq) ((eq_ref (a->Prop)) x0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((((eq_sym (a->Prop)) x0) Xq) ((eq_ref (a->Prop)) x0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x1:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((((eq_sym (a->Prop)) x0) Xq) ((eq_ref (a->Prop)) x0))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ex_intro0:=(ex_intro (a->Prop)):(forall (P:((a->Prop)->Prop)) (x:(a->Prop)), ((P x)->((ex (a->Prop)) P)))
% Instantiate: b:=(forall (P:((a->Prop)->Prop)) (x:(a->Prop)), ((P x)->((ex (a->Prop)) P))):Prop
% Found ex_intro0 as proof of b
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found x4:(P0 Xq)
% Instantiate: x0:=Xq:(a->Prop)
% Found (fun (x4:(P0 Xq))=> x4) as proof of (P0 x0)
% Found (fun (P0:((a->Prop)->Prop)) (x4:(P0 Xq))=> x4) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x4:(P0 Xq))=> x4) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion_dep0 (fun (x5:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xq)->(P0 Xq))
% Found (eq_ref00 P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((eq_ref0 Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_trans00000 (((eta_expansion a) Prop) Xq)) ((eq_ref (a->Prop)) b)) as proof of ((P Xq)->(P x0))
% Found ((eq_trans00000 (((eta_expansion a) Prop) Xq)) ((eq_ref (a->Prop)) b)) as proof of ((P Xq)->(P x0))
% Found (((fun (x1:(((eq (a->Prop)) Xq) b)) (x2:(((eq (a->Prop)) b) x0))=> (((eq_trans0000 x1) x2) P)) (((eta_expansion a) Prop) Xq)) ((eq_ref (a->Prop)) b)) as proof of ((P Xq)->(P x0))
% Found (((fun (x1:(((eq (a->Prop)) Xq) b)) (x2:(((eq (a->Prop)) b) x0))=> ((((eq_trans000 x0) x1) x2) P)) (((eta_expansion a) Prop) Xq)) ((eq_ref (a->Prop)) b)) as proof of ((P Xq)->(P x0))
% Found (((fun (x1:(((eq (a->Prop)) Xq) (fun (x3:a)=> (Xq x3)))) (x2:(((eq (a->Prop)) (fun (x3:a)=> (Xq x3))) x0))=> (((((eq_trans00 (fun (x3:a)=> (Xq x3))) x0) x1) x2) P)) (((eta_expansion a) Prop) Xq)) ((eq_ref (a->Prop)) (fun (x3:a)=> (Xq x3)))) as proof of ((P Xq)->(P x0))
% Found (((fun (x1:(((eq (a->Prop)) Xq) (fun (x3:a)=> (Xq x3)))) (x2:(((eq (a->Prop)) (fun (x3:a)=> (Xq x3))) x0))=> ((((((eq_trans0 Xq) (fun (x3:a)=> (Xq x3))) x0) x1) x2) P)) (((eta_expansion a) Prop) Xq)) ((eq_ref (a->Prop)) (fun (x3:a)=> (Xq x3)))) as proof of ((P Xq)->(P x0))
% Found (((fun (x1:(((eq (a->Prop)) Xq) (fun (x3:a)=> (Xq x3)))) (x2:(((eq (a->Prop)) (fun (x3:a)=> (Xq x3))) x0))=> (((((((eq_trans (a->Prop)) Xq) (fun (x3:a)=> (Xq x3))) x0) x1) x2) P)) (((eta_expansion a) Prop) Xq)) ((eq_ref (a->Prop)) (fun (x3:a)=> (Xq x3)))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x1:(((eq (a->Prop)) Xq) (fun (x3:a)=> (Xq x3)))) (x2:(((eq (a->Prop)) (fun (x3:a)=> (Xq x3))) x0))=> (((((((eq_trans (a->Prop)) Xq) (fun (x3:a)=> (Xq x3))) x0) x1) x2) P)) (((eta_expansion a) Prop) Xq)) ((eq_ref (a->Prop)) (fun (x3:a)=> (Xq x3))))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x1:(((eq (a->Prop)) Xq) (fun (x3:a)=> (Xq x3)))) (x2:(((eq (a->Prop)) (fun (x3:a)=> (Xq x3))) x0))=> (((((((eq_trans (a->Prop)) Xq) (fun (x3:a)=> (Xq x3))) x0) x1) x2) P)) (((eta_expansion a) Prop) Xq)) ((eq_ref (a->Prop)) (fun (x3:a)=> (Xq x3))))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x1:(((eq (a->Prop)) Xq) (fun (x3:a)=> (Xq x3)))) (x2:(((eq (a->Prop)) (fun (x3:a)=> (Xq x3))) x0))=> (((((((eq_trans (a->Prop)) Xq) (fun (x3:a)=> (Xq x3))) x0) x1) x2) P)) (((eta_expansion a) Prop) Xq)) ((eq_ref (a->Prop)) (fun (x3:a)=> (Xq x3))))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion_dep0 (fun (x5:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found x5:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found x5:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x0 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found ((conj20 (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00)))->((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))))
% Found (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((forall (Xx00:a), ((R Xx00) Xx00))->((forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00)))->((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))))
% Found (and_rect10 (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found ((and_rect1 ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (((fun (P:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P) x3) x1)) ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (((fun (P:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P) x3) x1)) ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (Xy:a)=> (((fun (P:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P) x3) x1)) ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((fun (P:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P) x3) x1)) ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found ex_intro0:=(ex_intro (a->Prop)):(forall (P:((a->Prop)->Prop)) (x:(a->Prop)), ((P x)->((ex (a->Prop)) P)))
% Instantiate: b:=(forall (P:((a->Prop)->Prop)) (x:(a->Prop)), ((P x)->((ex (a->Prop)) P))):Prop
% Found ex_intro0 as proof of b
% Found x40:(P0 Xq)
% Found (fun (x40:(P0 Xq))=> x40) as proof of (P0 Xq)
% Found (fun (x40:(P0 Xq))=> x40) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion_dep0000 (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion_dep000 (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion_dep00 Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion_dep0 (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40))) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x40:(P0 Xq)
% Found (fun (x40:(P0 Xq))=> x40) as proof of (P0 Xq)
% Found (fun (x40:(P0 Xq))=> x40) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion0000 (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion000 (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion00 Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion0 Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion a) Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion a) Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40))) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))):(((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found (eq_ref0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))):(((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found (eq_ref0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) b)
% Found x1:(P0 Xq)
% Instantiate: x0:=Xq:(a->Prop)
% Found (fun (x1:(P0 Xq))=> x1) as proof of (P0 x0)
% Found (fun (P0:((a->Prop)->Prop)) (x1:(P0 Xq))=> x1) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x1:(P0 Xq))=> x1) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found x1:(P0 Xq)
% Instantiate: x0:=Xq:(a->Prop)
% Found (fun (x1:(P0 Xq))=> x1) as proof of (P0 x0)
% Found (fun (P0:((a->Prop)->Prop)) (x1:(P0 Xq))=> x1) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x1:(P0 Xq))=> x1) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xq)->(P0 Xq))
% Found (eq_ref00 P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((eq_ref0 Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x4 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x4 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x4 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x4 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x4 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x4 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x4 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x4 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) b)
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x4))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x4))->(P ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))))
% Found (((eq_trans000 ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x4))->(P ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))))
% Found ((((eq_trans00 ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))) as proof of (forall (P:(Prop->Prop)), ((P (f x4))->(P ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))))
% Found (((((eq_trans0 (f x4)) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))) as proof of (forall (P:(Prop->Prop)), ((P (f x4))->(P ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))))
% Found ((((((eq_trans Prop) (f x4)) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))) as proof of (forall (P:(Prop->Prop)), ((P (f x4))->(P ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))))
% Found ((((((eq_trans Prop) (f x4)) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))) as proof of (forall (P:(Prop->Prop)), ((P (f x4))->(P ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))))
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) b)
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x4))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x4))->(P ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))))
% Found (((eq_trans000 ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x4))->(P ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))))
% Found ((((eq_trans00 ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))) as proof of (forall (P:(Prop->Prop)), ((P (f x4))->(P ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))))
% Found (((((eq_trans0 (f x4)) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))) as proof of (forall (P:(Prop->Prop)), ((P (f x4))->(P ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))))
% Found ((((((eq_trans Prop) (f x4)) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))) as proof of (forall (P:(Prop->Prop)), ((P (f x4))->(P ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))))
% Found ((((((eq_trans Prop) (f x4)) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))) as proof of (forall (P:(Prop->Prop)), ((P (f x4))->(P ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x10:(P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of (P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) x0)
% Found (((eq_trans00000 ((eq_ref (a->Prop)) Xq)) ((eq_ref (a->Prop)) b)) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (((eq_trans00000 ((eq_ref (a->Prop)) Xq)) ((eq_ref (a->Prop)) b)) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found ((((fun (x1:(((eq (a->Prop)) Xq) b)) (x2:(((eq (a->Prop)) b) x0))=> (((eq_trans0000 x1) x2) (fun (x4:(a->Prop))=> ((P Xq)->(P x4))))) ((eq_ref (a->Prop)) Xq)) ((eq_ref (a->Prop)) b)) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found ((((fun (x1:(((eq (a->Prop)) Xq) b)) (x2:(((eq (a->Prop)) b) x0))=> ((((eq_trans000 x0) x1) x2) (fun (x4:(a->Prop))=> ((P Xq)->(P x4))))) ((eq_ref (a->Prop)) Xq)) ((eq_ref (a->Prop)) b)) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found ((((fun (x1:(((eq (a->Prop)) Xq) Xq)) (x2:(((eq (a->Prop)) Xq) x0))=> (((((eq_trans00 Xq) x0) x1) x2) (fun (x4:(a->Prop))=> ((P Xq)->(P x4))))) ((eq_ref (a->Prop)) Xq)) ((eq_ref (a->Prop)) Xq)) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found ((((fun (x1:(((eq (a->Prop)) Xq) Xq)) (x2:(((eq (a->Prop)) Xq) x0))=> ((((((eq_trans0 Xq) Xq) x0) x1) x2) (fun (x4:(a->Prop))=> ((P Xq)->(P x4))))) ((eq_ref (a->Prop)) Xq)) ((eq_ref (a->Prop)) Xq)) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found ((((fun (x1:(((eq (a->Prop)) Xq) Xq)) (x2:(((eq (a->Prop)) Xq) x0))=> (((((((eq_trans (a->Prop)) Xq) Xq) x0) x1) x2) (fun (x4:(a->Prop))=> ((P Xq)->(P x4))))) ((eq_ref (a->Prop)) Xq)) ((eq_ref (a->Prop)) Xq)) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((fun (x1:(((eq (a->Prop)) Xq) Xq)) (x2:(((eq (a->Prop)) Xq) x0))=> (((((((eq_trans (a->Prop)) Xq) Xq) x0) x1) x2) (fun (x4:(a->Prop))=> ((P Xq)->(P x4))))) ((eq_ref (a->Prop)) Xq)) ((eq_ref (a->Prop)) Xq)) (fun (x10:(P Xq))=> x10))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((fun (x1:(((eq (a->Prop)) Xq) Xq)) (x2:(((eq (a->Prop)) Xq) x0))=> (((((((eq_trans (a->Prop)) Xq) Xq) x0) x1) x2) (fun (x4:(a->Prop))=> ((P Xq)->(P x4))))) ((eq_ref (a->Prop)) Xq)) ((eq_ref (a->Prop)) Xq)) (fun (x10:(P Xq))=> x10))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((fun (x1:(((eq (a->Prop)) Xq) Xq)) (x2:(((eq (a->Prop)) Xq) x0))=> (((((((eq_trans (a->Prop)) Xq) Xq) x0) x1) x2) (fun (x4:(a->Prop))=> ((P Xq)->(P x4))))) ((eq_ref (a->Prop)) Xq)) ((eq_ref (a->Prop)) Xq)) (fun (x10:(P Xq))=> x10))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x5:(P0 Xq)
% Instantiate: x4:=Xq:(a->Prop)
% Found (fun (x5:(P0 Xq))=> x5) as proof of (P0 x4)
% Found (fun (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of (((eq (a->Prop)) Xq) x4)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) b0)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) b0)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) b0)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) b0)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x4)
% Found x6:(P0 Xq)
% Instantiate: x4:=Xq:(a->Prop)
% Found (fun (x6:(P0 Xq))=> x6) as proof of (P0 x4)
% Found (fun (P0:((a->Prop)->Prop)) (x6:(P0 Xq))=> x6) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x6:(P0 Xq))=> x6) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xq)->(P0 Xq))
% Found (eq_ref00 P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((eq_ref0 Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found x40:(P0 Xq)
% Found (fun (x40:(P0 Xq))=> x40) as proof of (P0 Xq)
% Found (fun (x40:(P0 Xq))=> x40) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion_dep0000 (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion_dep000 (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion_dep00 Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion_dep0 (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40))) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))
% Found ex_intro0:=(ex_intro (a->Prop)):(forall (P:((a->Prop)->Prop)) (x:(a->Prop)), ((P x)->((ex (a->Prop)) P)))
% Instantiate: b:=(forall (P:((a->Prop)->Prop)) (x:(a->Prop)), ((P x)->((ex (a->Prop)) P))):Prop
% Found ex_intro0 as proof of b
% Found x40:(P0 Xq)
% Found (fun (x40:(P0 Xq))=> x40) as proof of (P0 Xq)
% Found (fun (x40:(P0 Xq))=> x40) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion0000 (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion000 (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion00 Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion0 Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion a) Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion a) Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40))) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 (Xq x2)):(((eq Prop) (Xq x2)) (Xq x2))
% Found (eq_ref0 (Xq x2)) as proof of (((eq Prop) (Xq x2)) (x0 x2))
% Found ((eq_ref Prop) (Xq x2)) as proof of (((eq Prop) (Xq x2)) (x0 x2))
% Found ((eq_ref Prop) (Xq x2)) as proof of (((eq Prop) (Xq x2)) (x0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (Xq x2))) as proof of (((eq Prop) (Xq x2)) (x0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (Xq x2))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found (functional_extensionality0000 (fun (x2:a)=> ((eq_ref Prop) (Xq x2)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((functional_extensionality000 x0) (fun (x2:a)=> ((eq_ref Prop) (Xq x2)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((functional_extensionality00 Xq) x0) (fun (x2:a)=> ((eq_ref Prop) (Xq x2)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((((functional_extensionality0 Prop) Xq) x0) (fun (x2:a)=> ((eq_ref Prop) (Xq x2)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((((functional_extensionality a) Prop) Xq) x0) (fun (x2:a)=> ((eq_ref Prop) (Xq x2)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x1:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((((functional_extensionality a) Prop) Xq) x0) (fun (x2:a)=> ((eq_ref Prop) (Xq x2))))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 (Xq x2)):(((eq Prop) (Xq x2)) (Xq x2))
% Found (eq_ref0 (Xq x2)) as proof of (((eq Prop) (Xq x2)) (x0 x2))
% Found ((eq_ref Prop) (Xq x2)) as proof of (((eq Prop) (Xq x2)) (x0 x2))
% Found ((eq_ref Prop) (Xq x2)) as proof of (((eq Prop) (Xq x2)) (x0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (Xq x2))) as proof of (((eq Prop) (Xq x2)) (x0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (Xq x2))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found (functional_extensionality_dep0000 (fun (x2:a)=> ((eq_ref Prop) (Xq x2)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((functional_extensionality_dep000 x0) (fun (x2:a)=> ((eq_ref Prop) (Xq x2)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((functional_extensionality_dep00 Xq) x0) (fun (x2:a)=> ((eq_ref Prop) (Xq x2)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((((functional_extensionality_dep0 (fun (x4:a)=> Prop)) Xq) x0) (fun (x2:a)=> ((eq_ref Prop) (Xq x2)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((((functional_extensionality_dep a) (fun (x4:a)=> Prop)) Xq) x0) (fun (x2:a)=> ((eq_ref Prop) (Xq x2)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x1:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((((functional_extensionality_dep a) (fun (x4:a)=> Prop)) Xq) x0) (fun (x2:a)=> ((eq_ref Prop) (Xq x2))))) as proof of (((eq (a->Prop)) Xq) x0)
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found ex_intro0:=(ex_intro (a->Prop)):(forall (P:((a->Prop)->Prop)) (x:(a->Prop)), ((P x)->((ex (a->Prop)) P)))
% Instantiate: b:=(forall (P:((a->Prop)->Prop)) (x:(a->Prop)), ((P x)->((ex (a->Prop)) P))):Prop
% Found ex_intro0 as proof of b
% Found x40:(P0 Xq)
% Found (fun (x40:(P0 Xq))=> x40) as proof of (P0 Xq)
% Found (fun (x40:(P0 Xq))=> x40) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion0000 (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion000 (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion00 Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion0 Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion a) Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion a) Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40))) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x40:(P0 Xq)
% Found (fun (x40:(P0 Xq))=> x40) as proof of (P0 Xq)
% Found (fun (x40:(P0 Xq))=> x40) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion_dep0000 (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion_dep000 (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion_dep00 Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion_dep0 (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40))) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found x5:(P0 Xq)
% Instantiate: x0:=Xq:(a->Prop)
% Found (fun (x5:(P0 Xq))=> x5) as proof of (P0 x0)
% Found (fun (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of (((eq (a->Prop)) Xq) x0)
% Found x5:(P0 Xq)
% Instantiate: x2:=Xq:(a->Prop)
% Found (fun (x5:(P0 Xq))=> x5) as proof of (P0 x2)
% Found (fun (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of (((eq (a->Prop)) Xq) x2)
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x1:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x5:((R Xx0) Xy)
% Instantiate: x4:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x4 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x4 Xy))
% Found x5:(x4 Xy)
% Instantiate: x4:=(R Xx0):(a->Prop)
% Found (fun (x5:(x4 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x4 Xy))=> x5) as proof of ((x4 Xy)->((R Xx0) Xy))
% Found ((conj20 (fun (x5:(x4 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x4 Xy))) (fun (x5:(x4 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x4 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x4 Xy))) (fun (x5:(x4 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x4 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x4 Xy))) (fun (x5:(x4 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x4 Xx0)) (Xy:a)=> ((((conj ((x4 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x4 Xy))) (fun (x5:(x4 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: a0:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of a0
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))) as proof of ((forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00)))->((iff (x0 Xy)) ((R Xx0) Xy)))
% Found (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))) as proof of ((forall (Xx00:a), ((R Xx00) Xx00))->((forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00)))->((iff (x0 Xy)) ((R Xx0) Xy))))
% Found (and_rect10 (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((and_rect1 ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((fun (P0:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P0)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P0) x3) x1)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((fun (P0:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P0)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P0) x3) x1)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((fun (P0:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P0)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P0) x3) x1)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x4:(forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x3:(forall (Xx:a), ((R Xx) Xx))) (x4:(forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))=> (iff_refl (x0 Xy))) as proof of ((forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx)))->((iff (x0 Xy)) ((R Xx0) Xy)))
% Found (fun (x3:(forall (Xx:a), ((R Xx) Xx))) (x4:(forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))=> (iff_refl (x0 Xy))) as proof of ((forall (Xx:a), ((R Xx) Xx))->((forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx)))->((iff (x0 Xy)) ((R Xx0) Xy))))
% Found (and_rect10 (fun (x3:(forall (Xx:a), ((R Xx) Xx))) (x4:(forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((and_rect1 ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx:a), ((R Xx) Xx))) (x4:(forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((fun (P0:Type) (x3:((forall (Xx:a), ((R Xx) Xx))->((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))->P0)))=> (((((and_rect (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) P0) x3) x1)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx:a), ((R Xx) Xx))) (x4:(forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((fun (P0:Type) (x3:((forall (Xx:a), ((R Xx) Xx))->((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))->P0)))=> (((((and_rect (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) P0) x3) x1)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx:a), ((R Xx) Xx))) (x4:(forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))=> (iff_refl (x0 Xy))))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((fun (P0:Type) (x3:((forall (Xx:a), ((R Xx) Xx))->((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))->P0)))=> (((((and_rect (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) P0) x3) x1)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx:a), ((R Xx) Xx))) (x4:(forall (Xx:a) (Xy0:a), (((R Xx) Xy0)->((R Xy0) Xx))))=> (iff_refl (x0 Xy))))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x2 Xy)):((iff (x2 Xy)) (x2 Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x2 Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x2 Xy))) as proof of ((forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00)))->((iff (x2 Xy)) ((R Xx0) Xy)))
% Found (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x2 Xy))) as proof of ((forall (Xx00:a), ((R Xx00) Xx00))->((forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00)))->((iff (x2 Xy)) ((R Xx0) Xy))))
% Found (and_rect10 (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x2 Xy)))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found ((and_rect1 ((iff (x2 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x2 Xy)))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (((fun (P0:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P0)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P0) x3) x0)) ((iff (x2 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x2 Xy)))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((fun (P0:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P0)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P0) x3) x0)) ((iff (x2 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x2 Xy))))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (((fun (P0:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P0)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P0) x3) x0)) ((iff (x2 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x2 Xy))))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found x40:(P0 Xq)
% Found (fun (x40:(P0 Xq))=> x40) as proof of (P0 Xq)
% Found (fun (x40:(P0 Xq))=> x40) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion0000 (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion000 (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion00 Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion0 Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion a) Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion a) Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40))) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x40:(P0 Xq)
% Found (fun (x40:(P0 Xq))=> x40) as proof of (P0 Xq)
% Found (fun (x40:(P0 Xq))=> x40) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion_dep0000 (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion_dep000 (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion_dep00 Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion_dep0 (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40))) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x3:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))=> (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x2:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (x3:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))=> (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq)) as proof of ((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->(((eq (a->Prop)) Xq) x0))
% Found (fun (x2:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (x3:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))=> (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq)) as proof of (((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->(((eq (a->Prop)) Xq) x0)))
% Found (and_rect00 (fun (x2:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (x3:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))=> (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((and_rect0 (((eq (a->Prop)) Xq) x0)) (fun (x2:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (x3:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))=> (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((fun (P0:Type) (x2:(((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->P0)))=> (((((and_rect ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) P0) x2) x)) (((eq (a->Prop)) Xq) x0)) (fun (x2:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (x3:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))=> (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x1:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P0:Type) (x2:(((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->P0)))=> (((((and_rect ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) P0) x2) x)) (((eq (a->Prop)) Xq) x0)) (fun (x2:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (x3:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))=> (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x4)
% Found x6:(P0 Xq)
% Instantiate: x4:=Xq:(a->Prop)
% Found (fun (x6:(P0 Xq))=> x6) as proof of (P0 x4)
% Found (fun (P0:((a->Prop)->Prop)) (x6:(P0 Xq))=> x6) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x6:(P0 Xq))=> x6) as proof of (((eq (a->Prop)) Xq) x4)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xq)->(P0 Xq))
% Found (eq_ref00 P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((eq_ref0 Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x4)
% Found x40:(P0 Xq)
% Found (fun (x40:(P0 Xq))=> x40) as proof of (P0 Xq)
% Found (fun (x40:(P0 Xq))=> x40) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion_dep0000 (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion_dep000 (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion_dep00 Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion_dep0 (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x0))
% Found (((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x0))
% Found (((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40))) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x40:(P0 Xq)
% Found (fun (x40:(P0 Xq))=> x40) as proof of (P0 Xq)
% Found (fun (x40:(P0 Xq))=> x40) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion0000 (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion000 (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion00 Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion0 Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40))) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x4)
% Found x6:(P0 Xq)
% Instantiate: x4:=Xq:(a->Prop)
% Found (fun (x6:(P0 Xq))=> x6) as proof of (P0 x4)
% Found (fun (P0:((a->Prop)->Prop)) (x6:(P0 Xq))=> x6) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x6:(P0 Xq))=> x6) as proof of (((eq (a->Prop)) Xq) x4)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xq)->(P0 Xq))
% Found (eq_ref00 P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((eq_ref0 Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x4)
% Found x6:(P0 Xq)
% Instantiate: x4:=Xq:(a->Prop)
% Found (fun (x6:(P0 Xq))=> x6) as proof of (P0 x4)
% Found (fun (P0:((a->Prop)->Prop)) (x6:(P0 Xq))=> x6) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x6:(P0 Xq))=> x6) as proof of (((eq (a->Prop)) Xq) x4)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xq)->(P0 Xq))
% Found (eq_ref00 P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((eq_ref0 Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xq)->(P0 Xq))
% Found (eq_ref00 P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((eq_ref0 Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x4)
% Found x6:(P0 Xq)
% Instantiate: x4:=Xq:(a->Prop)
% Found (fun (x6:(P0 Xq))=> x6) as proof of (P0 x4)
% Found (fun (P0:((a->Prop)->Prop)) (x6:(P0 Xq))=> x6) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x6:(P0 Xq))=> x6) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x4)
% Found x6:(P0 Xq)
% Instantiate: x4:=Xq:(a->Prop)
% Found (fun (x6:(P0 Xq))=> x6) as proof of (P0 x4)
% Found (fun (P0:((a->Prop)->Prop)) (x6:(P0 Xq))=> x6) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x6:(P0 Xq))=> x6) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xq)->(P0 Xq))
% Found (eq_ref00 P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((eq_ref0 Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found x3:(P0 Xq)
% Instantiate: x0:=Xq:(a->Prop)
% Found (fun (x3:(P0 Xq))=> x3) as proof of (P0 x0)
% Found (fun (P0:((a->Prop)->Prop)) (x3:(P0 Xq))=> x3) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop)) (x3:(P0 Xq))=> x3) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x3:(P0 Xq))=> x3) as proof of (((eq (a->Prop)) Xq) x0)
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x4)
% Found x5:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x0 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found x5:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found ((conj20 (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x4)
% Found x5:(x2 Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x5:(x2 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x2 Xy))=> x5) as proof of ((x2 Xy)->((R Xx0) Xy))
% Found x5:((R Xx0) Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x2 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x2 Xy))
% Found ((conj20 (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: a0:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of a0
% Found x50:(P0 (f x4))
% Found (fun (x50:(P0 (f x4)))=> x50) as proof of (P0 (f x4))
% Found (fun (x50:(P0 (f x4)))=> x50) as proof of (P1 (f x4))
% Found x50:(P0 (f x4))
% Found (fun (x50:(P0 (f x4)))=> x50) as proof of (P0 (f x4))
% Found (fun (x50:(P0 (f x4)))=> x50) as proof of (P1 (f x4))
% Found x3:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x3:((R Xx0) Xy))=> x3) as proof of (x0 Xy)
% Found (fun (x3:((R Xx0) Xy))=> x3) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found x3:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x3:(x0 Xy))=> x3) as proof of ((R Xx0) Xy)
% Found (fun (x3:(x0 Xy))=> x3) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found ((conj20 (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x3:(x0 Xy))=> x3)) (fun (x3:((R Xx0) Xy))=> x3))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x10:(P0 Xq)
% Found (fun (x10:(P0 Xq))=> x10) as proof of (P0 Xq)
% Found (fun (x10:(P0 Xq))=> x10) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion_dep0000 (fun (x10:(P0 Xq))=> x10)) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion_dep000 (fun (x2:(a->Prop))=> ((P0 Xq)->(P0 x2)))) (fun (x10:(P0 Xq))=> x10)) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion_dep00 Xq) (fun (x2:(a->Prop))=> ((P0 Xq)->(P0 x2)))) (fun (x10:(P0 Xq))=> x10)) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P0 Xq)->(P0 x2)))) (fun (x10:(P0 Xq))=> x10)) as proof of ((P0 Xq)->(P0 x0))
% Found (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P0 Xq)->(P0 x2)))) (fun (x10:(P0 Xq))=> x10)) as proof of ((P0 Xq)->(P0 x0))
% Found (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P0 Xq)->(P0 x2)))) (fun (x10:(P0 Xq))=> x10)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P0 Xq)->(P0 x2)))) (fun (x10:(P0 Xq))=> x10))) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P0 Xq)->(P0 x2)))) (fun (x10:(P0 Xq))=> x10))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x10:(P0 Xq)
% Found (fun (x10:(P0 Xq))=> x10) as proof of (P0 Xq)
% Found (fun (x10:(P0 Xq))=> x10) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion0000 (fun (x10:(P0 Xq))=> x10)) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion000 (fun (x2:(a->Prop))=> ((P0 Xq)->(P0 x2)))) (fun (x10:(P0 Xq))=> x10)) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion00 Xq) (fun (x2:(a->Prop))=> ((P0 Xq)->(P0 x2)))) (fun (x10:(P0 Xq))=> x10)) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion0 Prop) Xq) (fun (x2:(a->Prop))=> ((P0 Xq)->(P0 x2)))) (fun (x10:(P0 Xq))=> x10)) as proof of ((P0 Xq)->(P0 x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P0 Xq)->(P0 x2)))) (fun (x10:(P0 Xq))=> x10)) as proof of ((P0 Xq)->(P0 x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P0 Xq)->(P0 x2)))) (fun (x10:(P0 Xq))=> x10)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P0 Xq)->(P0 x2)))) (fun (x10:(P0 Xq))=> x10))) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P0 Xq)->(P0 x2)))) (fun (x10:(P0 Xq))=> x10))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found x5:(P0 Xq)
% Instantiate: x0:=Xq:(a->Prop)
% Found (fun (x5:(P0 Xq))=> x5) as proof of (P0 x0)
% Found (fun (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b
% Found x5:(P0 Xq)
% Instantiate: x0:=Xq:(a->Prop)
% Found (fun (x5:(P0 Xq))=> x5) as proof of (P0 x0)
% Found (fun (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of (((eq (a->Prop)) Xq) x0)
% Found x5:(P0 Xq)
% Instantiate: x2:=Xq:(a->Prop)
% Found (fun (x5:(P0 Xq))=> x5) as proof of (P0 x2)
% Found (fun (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of (((eq (a->Prop)) Xq) x2)
% Found x1:(P0 Xq)
% Instantiate: x0:=Xq:(a->Prop)
% Found (fun (x1:(P0 Xq))=> x1) as proof of (P0 x0)
% Found (fun (P0:((a->Prop)->Prop)) (x1:(P0 Xq))=> x1) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x1:(P0 Xq))=> x1) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x5:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x4:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x5:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)) as proof of ((Xq Xx)->(((eq (a->Prop)) Xq) x2))
% Found (fun (x4:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x5:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)) as proof of ((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->(((eq (a->Prop)) Xq) x2)))
% Found (and_rect10 (fun (x4:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x5:(Xq Xx))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((and_rect1 (((eq (a->Prop)) Xq) x2)) (fun (x4:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x5:(Xq Xx))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((fun (P0:Type) (x4:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x4) x3)) (((eq (a->Prop)) Xq) x2)) (fun (x4:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x5:(Xq Xx))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P0:Type) (x4:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x4) x3)) (((eq (a->Prop)) Xq) x2)) (fun (x4:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x5:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) 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) x2)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) x2)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x2)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x2)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x2)
% Found ((eq_trans00000 (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x2))
% Found ((eq_trans00000 (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x2))
% Found (((fun (x3:(((eq (a->Prop)) Xq) b)) (x4:(((eq (a->Prop)) b) x2))=> (((eq_trans0000 x3) x4) P)) (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x2))
% Found (((fun (x3:(((eq (a->Prop)) Xq) b)) (x4:(((eq (a->Prop)) b) x2))=> ((((eq_trans000 x2) x3) x4) P)) (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x2))
% Found (((fun (x3:(((eq (a->Prop)) Xq) (fun (x5:a)=> (Xq x5)))) (x4:(((eq (a->Prop)) (fun (x5:a)=> (Xq x5))) x2))=> (((((eq_trans00 (fun (x5:a)=> (Xq x5))) x2) x3) x4) P)) (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) (fun (x5:a)=> (Xq x5)))) as proof of ((P Xq)->(P x2))
% Found (((fun (x3:(((eq (a->Prop)) Xq) (fun (x5:a)=> (Xq x5)))) (x4:(((eq (a->Prop)) (fun (x5:a)=> (Xq x5))) x2))=> ((((((eq_trans0 Xq) (fun (x5:a)=> (Xq x5))) x2) x3) x4) P)) (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) (fun (x5:a)=> (Xq x5)))) as proof of ((P Xq)->(P x2))
% Found (((fun (x3:(((eq (a->Prop)) Xq) (fun (x5:a)=> (Xq x5)))) (x4:(((eq (a->Prop)) (fun (x5:a)=> (Xq x5))) x2))=> (((((((eq_trans (a->Prop)) Xq) (fun (x5:a)=> (Xq x5))) x2) x3) x4) P)) (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) (fun (x5:a)=> (Xq x5)))) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x3:(((eq (a->Prop)) Xq) (fun (x5:a)=> (Xq x5)))) (x4:(((eq (a->Prop)) (fun (x5:a)=> (Xq x5))) x2))=> (((((((eq_trans (a->Prop)) Xq) (fun (x5:a)=> (Xq x5))) x2) x3) x4) P)) (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) (fun (x5:a)=> (Xq x5))))) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x3:(((eq (a->Prop)) Xq) (fun (x5:a)=> (Xq x5)))) (x4:(((eq (a->Prop)) (fun (x5:a)=> (Xq x5))) x2))=> (((((((eq_trans (a->Prop)) Xq) (fun (x5:a)=> (Xq x5))) x2) x3) x4) P)) (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) (fun (x5:a)=> (Xq x5))))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x3:(((eq (a->Prop)) Xq) (fun (x5:a)=> (Xq x5)))) (x4:(((eq (a->Prop)) (fun (x5:a)=> (Xq x5))) x2))=> (((((((eq_trans (a->Prop)) Xq) (fun (x5:a)=> (Xq x5))) x2) x3) x4) P)) (((eta_expansion a) Prop) Xq)) (((eta_expansion a) Prop) (fun (x5:a)=> (Xq x5))))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 x2):(((eq (a->Prop)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found (eq_sym100 ((eq_ref (a->Prop)) x2)) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eq_sym10 Xq) ((eq_ref (a->Prop)) x2)) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eq_sym1 x2) Xq) ((eq_ref (a->Prop)) x2)) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((((eq_sym (a->Prop)) x2) Xq) ((eq_ref (a->Prop)) x2)) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((((eq_sym (a->Prop)) x2) Xq) ((eq_ref (a->Prop)) x2))) as proof of (((eq (a->Prop)) Xq) x2)
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x40:(P0 Xq)
% Found (fun (x40:(P0 Xq))=> x40) as proof of (P0 Xq)
% Found (fun (x40:(P0 Xq))=> x40) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion_dep0000 (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion_dep000 (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion_dep00 Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion_dep0 (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40))) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x40:(P0 Xq)
% Found (fun (x40:(P0 Xq))=> x40) as proof of (P0 Xq)
% Found (fun (x40:(P0 Xq))=> x40) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion0000 (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion000 (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion00 Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion0 Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion a) Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion a) Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40))) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x6:(P0 Xq)
% Instantiate: x2:=Xq:(a->Prop)
% Found (fun (x6:(P0 Xq))=> x6) as proof of (P0 x2)
% Found (fun (P0:((a->Prop)->Prop)) (x6:(P0 Xq))=> x6) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x6:(P0 Xq))=> x6) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xq)->(P0 Xq))
% Found (eq_ref00 P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((eq_ref0 Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xq)->(P0 Xq))
% Found (eq_ref00 P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((eq_ref0 Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x2)
% Found x6:(P0 Xq)
% Instantiate: x2:=Xq:(a->Prop)
% Found (fun (x6:(P0 Xq))=> x6) as proof of (P0 x2)
% Found (fun (P0:((a->Prop)->Prop)) (x6:(P0 Xq))=> x6) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x6:(P0 Xq))=> x6) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xq)->(P0 Xq))
% Found (eq_ref00 P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((eq_ref0 Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found x6:(P0 Xq)
% Instantiate: x0:=Xq:(a->Prop)
% Found (fun (x6:(P0 Xq))=> x6) as proof of (P0 x0)
% Found (fun (P0:((a->Prop)->Prop)) (x6:(P0 Xq))=> x6) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x6:(P0 Xq))=> x6) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x2)
% Found x6:(P0 Xq)
% Instantiate: x2:=Xq:(a->Prop)
% Found (fun (x6:(P0 Xq))=> x6) as proof of (P0 x2)
% Found (fun (P0:((a->Prop)->Prop)) (x6:(P0 Xq))=> x6) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x6:(P0 Xq))=> x6) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xq)->(P0 Xq))
% Found (eq_ref00 P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((eq_ref0 Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x2)
% Found x40:(P0 Xq)
% Found (fun (x40:(P0 Xq))=> x40) as proof of (P0 Xq)
% Found (fun (x40:(P0 Xq))=> x40) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion_dep0000 (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion_dep000 (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion_dep00 Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion_dep0 (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40))) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x40:(P0 Xq)
% Found (fun (x40:(P0 Xq))=> x40) as proof of (P0 Xq)
% Found (fun (x40:(P0 Xq))=> x40) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion0000 (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion000 (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion00 Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion0 Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion a) Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion a) Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40))) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))):(((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) (fun (x:(a->Prop))=> ((and ((and (forall (Xx0:a), ((x Xx0)->(forall (Xy:a), ((iff (x Xy)) ((R Xx0) Xy)))))) (x Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x))))))
% Found (eta_expansion_dep00 (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) b0)
% Found ((eta_expansion_dep0 (fun (x3:(a->Prop))=> Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) b0)
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) b0)
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) b0)
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) b0)
% Found iff_refl0:=(iff_refl (x2 Xy)):((iff (x2 Xy)) (x2 Xy))
% Found (iff_refl (x2 Xy)) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found (iff_refl (x2 Xy)) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found (iff_refl (x2 Xy)) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found (fun (Xy:a)=> (iff_refl (x2 Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (iff_refl (x2 Xy))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion000:=(eta_expansion00 b):(((eq ((a->Prop)->Prop)) b) (fun (x:(a->Prop))=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq ((a->Prop)->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq ((a->Prop)->Prop)) b) b0)
% Found (((eta_expansion (a->Prop)) Prop) b) as proof of (((eq ((a->Prop)->Prop)) b) b0)
% Found (((eta_expansion (a->Prop)) Prop) b) as proof of (((eq ((a->Prop)->Prop)) b) b0)
% Found (((eta_expansion (a->Prop)) Prop) b) as proof of (((eq ((a->Prop)->Prop)) b) b0)
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x2)
% Found x3:(P0 Xq)
% Instantiate: x2:=Xq:(a->Prop)
% Found (fun (x3:(P0 Xq))=> x3) as proof of (P0 x2)
% Found (fun (P0:((a->Prop)->Prop)) (x3:(P0 Xq))=> x3) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x3:(P0 Xq))=> x3) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found iff_refl0:=(iff_refl (x2 Xy)):((iff (x2 Xy)) (x2 Xy))
% Found (iff_refl (x2 Xy)) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found (iff_refl (x2 Xy)) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found (iff_refl (x2 Xy)) as proof of ((and ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy)))
% Found (fun (Xy:a)=> (iff_refl (x2 Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (iff_refl (x2 Xy))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) 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) x0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) x0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_trans00000 (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x0))
% Found ((eq_trans00000 (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x0))
% Found (((fun (x3:(((eq (a->Prop)) Xq) b)) (x4:(((eq (a->Prop)) b) x0))=> (((eq_trans0000 x3) x4) P)) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x0))
% Found (((fun (x3:(((eq (a->Prop)) Xq) b)) (x4:(((eq (a->Prop)) b) x0))=> ((((eq_trans000 x0) x3) x4) P)) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x0))
% Found (((fun (x3:(((eq (a->Prop)) Xq) (fun (x5:a)=> (Xq x5)))) (x4:(((eq (a->Prop)) (fun (x5:a)=> (Xq x5))) x0))=> (((((eq_trans00 (fun (x5:a)=> (Xq x5))) x0) x3) x4) P)) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) (((eta_expansion a) Prop) (fun (x5:a)=> (Xq x5)))) as proof of ((P Xq)->(P x0))
% Found (((fun (x3:(((eq (a->Prop)) Xq) (fun (x5:a)=> (Xq x5)))) (x4:(((eq (a->Prop)) (fun (x5:a)=> (Xq x5))) x0))=> ((((((eq_trans0 Xq) (fun (x5:a)=> (Xq x5))) x0) x3) x4) P)) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) (((eta_expansion a) Prop) (fun (x5:a)=> (Xq x5)))) as proof of ((P Xq)->(P x0))
% Found (((fun (x3:(((eq (a->Prop)) Xq) (fun (x5:a)=> (Xq x5)))) (x4:(((eq (a->Prop)) (fun (x5:a)=> (Xq x5))) x0))=> (((((((eq_trans (a->Prop)) Xq) (fun (x5:a)=> (Xq x5))) x0) x3) x4) P)) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) (((eta_expansion a) Prop) (fun (x5:a)=> (Xq x5)))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x3:(((eq (a->Prop)) Xq) (fun (x5:a)=> (Xq x5)))) (x4:(((eq (a->Prop)) (fun (x5:a)=> (Xq x5))) x0))=> (((((((eq_trans (a->Prop)) Xq) (fun (x5:a)=> (Xq x5))) x0) x3) x4) P)) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) (((eta_expansion a) Prop) (fun (x5:a)=> (Xq x5))))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x3:(((eq (a->Prop)) Xq) (fun (x5:a)=> (Xq x5)))) (x4:(((eq (a->Prop)) (fun (x5:a)=> (Xq x5))) x0))=> (((((((eq_trans (a->Prop)) Xq) (fun (x5:a)=> (Xq x5))) x0) x3) x4) P)) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) (((eta_expansion a) Prop) (fun (x5:a)=> (Xq x5))))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x3:(((eq (a->Prop)) Xq) (fun (x5:a)=> (Xq x5)))) (x4:(((eq (a->Prop)) (fun (x5:a)=> (Xq x5))) x0))=> (((((((eq_trans (a->Prop)) Xq) (fun (x5:a)=> (Xq x5))) x0) x3) x4) P)) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) (((eta_expansion a) Prop) (fun (x5:a)=> (Xq x5))))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))):(((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found (eq_ref0 ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))):(((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found (eq_ref0 ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))):(((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found (eq_ref0 ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))):(((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found (eq_ref0 ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% Found x5:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x0 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found x5:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found ((conj20 (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x5:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x0 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found x5:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found ((conj20 (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x5:(x2 Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x5:(x2 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x2 Xy))=> x5) as proof of ((x2 Xy)->((R Xx0) Xy))
% Found x5:((R Xx0) Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x2 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x2 Xy))
% Found ((conj20 (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found x1:(P0 (f x0))
% Instantiate: b:=(f x0):Prop
% Found x1 as proof of (P1 b)
% Found x1:(P0 (f x0))
% Instantiate: b:=(f x0):Prop
% Found x1 as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))):(((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found (eq_ref0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) b)
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))):(((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found (eq_ref0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) b)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xq)->(P0 Xq))
% Found (eq_ref00 P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((eq_ref0 Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x2)
% Found x3:(P0 Xq)
% Instantiate: x2:=Xq:(a->Prop)
% Found (fun (x3:(P0 Xq))=> x3) as proof of (P0 x2)
% Found (fun (P0:((a->Prop)->Prop)) (x3:(P0 Xq))=> x3) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x3:(P0 Xq))=> x3) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x5:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x4:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x5:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)) as proof of ((Xq Xx)->(((eq (a->Prop)) Xq) x2))
% Found (fun (x4:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x5:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)) as proof of ((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->(((eq (a->Prop)) Xq) x2)))
% Found (and_rect10 (fun (x4:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x5:(Xq Xx))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((and_rect1 (((eq (a->Prop)) Xq) x2)) (fun (x4:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x5:(Xq Xx))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((fun (P0:Type) (x4:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x4) x3)) (((eq (a->Prop)) Xq) x2)) (fun (x4:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x5:(Xq Xx))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P0:Type) (x4:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x4) x3)) (((eq (a->Prop)) Xq) x2)) (fun (x4:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x5:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: a0:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of a0
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 x2):(((eq (a->Prop)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found (eq_sym100 ((eq_ref (a->Prop)) x2)) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eq_sym10 Xq) ((eq_ref (a->Prop)) x2)) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eq_sym1 x2) Xq) ((eq_ref (a->Prop)) x2)) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((((eq_sym (a->Prop)) x2) Xq) ((eq_ref (a->Prop)) x2)) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((((eq_sym (a->Prop)) x2) Xq) ((eq_ref (a->Prop)) x2))) as proof of (((eq (a->Prop)) Xq) x2)
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x5:(Xq Xx))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x4:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x5:(Xq Xx))=> (((eta_expansion a) Prop) Xq)) as proof of ((Xq Xx)->(((eq (a->Prop)) Xq) x2))
% Found (fun (x4:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x5:(Xq Xx))=> (((eta_expansion a) Prop) Xq)) as proof of ((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->(((eq (a->Prop)) Xq) x2)))
% Found (and_rect10 (fun (x4:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x5:(Xq Xx))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((and_rect1 (((eq (a->Prop)) Xq) x2)) (fun (x4:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x5:(Xq Xx))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((fun (P0:Type) (x4:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x4) x3)) (((eq (a->Prop)) Xq) x2)) (fun (x4:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x5:(Xq Xx))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P0:Type) (x4:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x4) x3)) (((eq (a->Prop)) Xq) x2)) (fun (x4:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x5:(Xq Xx))=> (((eta_expansion a) Prop) Xq)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x40:(P0 Xq)
% Found (fun (x40:(P0 Xq))=> x40) as proof of (P0 Xq)
% Found (fun (x40:(P0 Xq))=> x40) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion_dep0000 (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion_dep000 (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion_dep00 Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion_dep0 (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x0))
% Found (((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x0))
% Found (((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40))) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x40:(P0 Xq)
% Found (fun (x40:(P0 Xq))=> x40) as proof of (P0 Xq)
% Found (fun (x40:(P0 Xq))=> x40) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion0000 (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion000 (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion00 Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion0 Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40))) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x5:(a->Prop))=> ((P0 Xq)->(P0 x5)))) (fun (x40:(P0 Xq))=> x40))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 x2):(((eq (a->Prop)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found ((eq_ref (a->Prop)) x2) as proof of (((eq (a->Prop)) x2) Xq)
% Found (eq_sym100 ((eq_ref (a->Prop)) x2)) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eq_sym10 Xq) ((eq_ref (a->Prop)) x2)) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eq_sym1 x2) Xq) ((eq_ref (a->Prop)) x2)) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((((eq_sym (a->Prop)) x2) Xq) ((eq_ref (a->Prop)) x2)) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((((eq_sym (a->Prop)) x2) Xq) ((eq_ref (a->Prop)) x2))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x30:(P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) 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) x2)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) x2)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x2)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x2)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x2)
% Found (((eq_trans00000 ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (((eq_trans00000 ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found ((((fun (x3:(((eq (a->Prop)) Xq) b)) (x4:(((eq (a->Prop)) b) x2))=> (((eq_trans0000 x3) x4) (fun (x6:(a->Prop))=> ((P Xq)->(P x6))))) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found ((((fun (x3:(((eq (a->Prop)) Xq) b)) (x4:(((eq (a->Prop)) b) x2))=> ((((eq_trans000 x2) x3) x4) (fun (x6:(a->Prop))=> ((P Xq)->(P x6))))) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found ((((fun (x3:(((eq (a->Prop)) Xq) Xq)) (x4:(((eq (a->Prop)) Xq) x2))=> (((((eq_trans00 Xq) x2) x3) x4) (fun (x6:(a->Prop))=> ((P Xq)->(P x6))))) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found ((((fun (x3:(((eq (a->Prop)) Xq) Xq)) (x4:(((eq (a->Prop)) Xq) x2))=> ((((((eq_trans0 Xq) Xq) x2) x3) x4) (fun (x6:(a->Prop))=> ((P Xq)->(P x6))))) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found ((((fun (x3:(((eq (a->Prop)) Xq) Xq)) (x4:(((eq (a->Prop)) Xq) x2))=> (((((((eq_trans (a->Prop)) Xq) Xq) x2) x3) x4) (fun (x6:(a->Prop))=> ((P Xq)->(P x6))))) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((((fun (x3:(((eq (a->Prop)) Xq) Xq)) (x4:(((eq (a->Prop)) Xq) x2))=> (((((((eq_trans (a->Prop)) Xq) Xq) x2) x3) x4) (fun (x6:(a->Prop))=> ((P Xq)->(P x6))))) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) (fun (x30:(P Xq))=> x30))) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> ((((fun (x3:(((eq (a->Prop)) Xq) Xq)) (x4:(((eq (a->Prop)) Xq) x2))=> (((((((eq_trans (a->Prop)) Xq) Xq) x2) x3) x4) (fun (x6:(a->Prop))=> ((P Xq)->(P x6))))) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) (fun (x30:(P Xq))=> x30))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((fun (x3:(((eq (a->Prop)) Xq) Xq)) (x4:(((eq (a->Prop)) Xq) x2))=> (((((((eq_trans (a->Prop)) Xq) Xq) x2) x3) x4) (fun (x6:(a->Prop))=> ((P Xq)->(P x6))))) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) (fun (x30:(P Xq))=> x30))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))):(((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))
% Found (eq_ref0 (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) b0)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) b0)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) b0)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) b0)
% Found x3:(P0 Xq)
% Instantiate: x0:=Xq:(a->Prop)
% Found (fun (x3:(P0 Xq))=> x3) as proof of (P0 x0)
% Found (fun (P0:((a->Prop)->Prop)) (x3:(P0 Xq))=> x3) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x3:(P0 Xq))=> x3) as proof of (((eq (a->Prop)) Xq) x0)
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))
% Found x10:(P0 Xq)
% Found (fun (x10:(P0 Xq))=> x10) as proof of (P0 Xq)
% Found (fun (x10:(P0 Xq))=> x10) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion0000 (fun (x10:(P0 Xq))=> x10)) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion000 (fun (x2:(a->Prop))=> ((P0 Xq)->(P0 x2)))) (fun (x10:(P0 Xq))=> x10)) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion00 Xq) (fun (x2:(a->Prop))=> ((P0 Xq)->(P0 x2)))) (fun (x10:(P0 Xq))=> x10)) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion0 Prop) Xq) (fun (x2:(a->Prop))=> ((P0 Xq)->(P0 x2)))) (fun (x10:(P0 Xq))=> x10)) as proof of ((P0 Xq)->(P0 x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P0 Xq)->(P0 x2)))) (fun (x10:(P0 Xq))=> x10)) as proof of ((P0 Xq)->(P0 x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P0 Xq)->(P0 x2)))) (fun (x10:(P0 Xq))=> x10)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P0 Xq)->(P0 x2)))) (fun (x10:(P0 Xq))=> x10))) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P0 Xq)->(P0 x2)))) (fun (x10:(P0 Xq))=> x10))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x10:(P0 Xq)
% Found (fun (x10:(P0 Xq))=> x10) as proof of (P0 Xq)
% Found (fun (x10:(P0 Xq))=> x10) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion_dep0000 (fun (x10:(P0 Xq))=> x10)) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion_dep000 (fun (x2:(a->Prop))=> ((P0 Xq)->(P0 x2)))) (fun (x10:(P0 Xq))=> x10)) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion_dep00 Xq) (fun (x2:(a->Prop))=> ((P0 Xq)->(P0 x2)))) (fun (x10:(P0 Xq))=> x10)) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P0 Xq)->(P0 x2)))) (fun (x10:(P0 Xq))=> x10)) as proof of ((P0 Xq)->(P0 x0))
% Found (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P0 Xq)->(P0 x2)))) (fun (x10:(P0 Xq))=> x10)) as proof of ((P0 Xq)->(P0 x0))
% Found (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P0 Xq)->(P0 x2)))) (fun (x10:(P0 Xq))=> x10)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P0 Xq)->(P0 x2)))) (fun (x10:(P0 Xq))=> x10))) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P0 Xq)->(P0 x2)))) (fun (x10:(P0 Xq))=> x10))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xq)->(P0 Xq))
% Found (eq_ref00 P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((eq_ref0 Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found x3:(P0 Xq)
% Instantiate: x0:=Xq:(a->Prop)
% Found (fun (x3:(P0 Xq))=> x3) as proof of (P0 x0)
% Found (fun (P0:((a->Prop)->Prop)) (x3:(P0 Xq))=> x3) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x3:(P0 Xq))=> x3) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) b0)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) b0)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) b0)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found x60:(P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of (P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion0000 (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion000 (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion00 Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion0 Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of (((eq (a->Prop)) Xq) x4)
% Found x60:(P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of (P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion_dep0000 (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion_dep000 (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion_dep00 Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))) as proof of ((forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00)))->((iff (x0 Xy)) ((R Xx0) Xy)))
% Found (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))) as proof of ((forall (Xx00:a), ((R Xx00) Xx00))->((forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00)))->((iff (x0 Xy)) ((R Xx0) Xy))))
% Found (and_rect10 (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((and_rect1 ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((fun (P0:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P0)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P0) x3) x1)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((fun (P0:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P0)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P0) x3) x1)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((fun (P0:Type) (x3:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P0)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P0) x3) x1)) ((iff (x0 Xy)) ((R Xx0) Xy))) (fun (x3:(forall (Xx00:a), ((R Xx00) Xx00))) (x4:(forall (Xx00:a) (Xy0:a), (((R Xx00) Xy0)->((R Xy0) Xx00))))=> (iff_refl (x0 Xy))))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x5:(Xq Xx))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x4:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x5:(Xq Xx))=> (((eta_expansion a) Prop) Xq)) as proof of ((Xq Xx)->(((eq (a->Prop)) Xq) x0))
% Found (fun (x4:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x5:(Xq Xx))=> (((eta_expansion a) Prop) Xq)) as proof of ((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->(((eq (a->Prop)) Xq) x0)))
% Found (and_rect10 (fun (x4:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x5:(Xq Xx))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((and_rect1 (((eq (a->Prop)) Xq) x0)) (fun (x4:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x5:(Xq Xx))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((fun (P0:Type) (x4:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x4) x3)) (((eq (a->Prop)) Xq) x0)) (fun (x4:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x5:(Xq Xx))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P0:Type) (x4:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x4) x3)) (((eq (a->Prop)) Xq) x0)) (fun (x4:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x5:(Xq Xx))=> (((eta_expansion a) Prop) Xq)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x30:(P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P Xq)
% Found (fun (x30:(P Xq))=> x30) as proof of (P0 Xq)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) 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) x0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) x0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x0)
% Found (((eq_trans00000 ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (((eq_trans00000 ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found ((((fun (x3:(((eq (a->Prop)) Xq) b)) (x4:(((eq (a->Prop)) b) x0))=> (((eq_trans0000 x3) x4) (fun (x6:(a->Prop))=> ((P Xq)->(P x6))))) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found ((((fun (x3:(((eq (a->Prop)) Xq) b)) (x4:(((eq (a->Prop)) b) x0))=> ((((eq_trans000 x0) x3) x4) (fun (x6:(a->Prop))=> ((P Xq)->(P x6))))) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found ((((fun (x3:(((eq (a->Prop)) Xq) Xq)) (x4:(((eq (a->Prop)) Xq) x0))=> (((((eq_trans00 Xq) x0) x3) x4) (fun (x6:(a->Prop))=> ((P Xq)->(P x6))))) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found ((((fun (x3:(((eq (a->Prop)) Xq) Xq)) (x4:(((eq (a->Prop)) Xq) x0))=> ((((((eq_trans0 Xq) Xq) x0) x3) x4) (fun (x6:(a->Prop))=> ((P Xq)->(P x6))))) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found ((((fun (x3:(((eq (a->Prop)) Xq) Xq)) (x4:(((eq (a->Prop)) Xq) x0))=> (((((((eq_trans (a->Prop)) Xq) Xq) x0) x3) x4) (fun (x6:(a->Prop))=> ((P Xq)->(P x6))))) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) (fun (x30:(P Xq))=> x30)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((fun (x3:(((eq (a->Prop)) Xq) Xq)) (x4:(((eq (a->Prop)) Xq) x0))=> (((((((eq_trans (a->Prop)) Xq) Xq) x0) x3) x4) (fun (x6:(a->Prop))=> ((P Xq)->(P x6))))) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) (fun (x30:(P Xq))=> x30))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> ((((fun (x3:(((eq (a->Prop)) Xq) Xq)) (x4:(((eq (a->Prop)) Xq) x0))=> (((((((eq_trans (a->Prop)) Xq) Xq) x0) x3) x4) (fun (x6:(a->Prop))=> ((P Xq)->(P x6))))) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) (fun (x30:(P Xq))=> x30))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((fun (x3:(((eq (a->Prop)) Xq) Xq)) (x4:(((eq (a->Prop)) Xq) x0))=> (((((((eq_trans (a->Prop)) Xq) Xq) x0) x3) x4) (fun (x6:(a->Prop))=> ((P Xq)->(P x6))))) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) (fun (x30:(P Xq))=> x30))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 x0):(((eq (a->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found (eq_sym100 ((eq_ref (a->Prop)) x0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eq_sym10 Xq) ((eq_ref (a->Prop)) x0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eq_sym1 x0) Xq) ((eq_ref (a->Prop)) x0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((((eq_sym (a->Prop)) x0) Xq) ((eq_ref (a->Prop)) x0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((((eq_sym (a->Prop)) x0) Xq) ((eq_ref (a->Prop)) x0))) as proof of (((eq (a->Prop)) Xq) x0)
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eq_ref00:=(eq_ref0 (Xq x4)):(((eq Prop) (Xq x4)) (Xq x4))
% Found (eq_ref0 (Xq x4)) as proof of (((eq Prop) (Xq x4)) (x2 x4))
% Found ((eq_ref Prop) (Xq x4)) as proof of (((eq Prop) (Xq x4)) (x2 x4))
% Found ((eq_ref Prop) (Xq x4)) as proof of (((eq Prop) (Xq x4)) (x2 x4))
% Found (fun (x4:a)=> ((eq_ref Prop) (Xq x4))) as proof of (((eq Prop) (Xq x4)) (x2 x4))
% Found (fun (x4:a)=> ((eq_ref Prop) (Xq x4))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x2 x)))
% Found (functional_extensionality0000 (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((functional_extensionality000 x2) (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((functional_extensionality00 Xq) x2) (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((((functional_extensionality0 Prop) Xq) x2) (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((((functional_extensionality a) Prop) Xq) x2) (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((((functional_extensionality a) Prop) Xq) x2) (fun (x4:a)=> ((eq_ref Prop) (Xq x4))))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 (Xq x4)):(((eq Prop) (Xq x4)) (Xq x4))
% Found (eq_ref0 (Xq x4)) as proof of (((eq Prop) (Xq x4)) (x2 x4))
% Found ((eq_ref Prop) (Xq x4)) as proof of (((eq Prop) (Xq x4)) (x2 x4))
% Found ((eq_ref Prop) (Xq x4)) as proof of (((eq Prop) (Xq x4)) (x2 x4))
% Found (fun (x4:a)=> ((eq_ref Prop) (Xq x4))) as proof of (((eq Prop) (Xq x4)) (x2 x4))
% Found (fun (x4:a)=> ((eq_ref Prop) (Xq x4))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x2 x)))
% Found (functional_extensionality_dep0000 (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((functional_extensionality_dep000 x2) (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((functional_extensionality_dep00 Xq) x2) (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((((functional_extensionality_dep0 (fun (x6:a)=> Prop)) Xq) x2) (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((((functional_extensionality_dep a) (fun (x6:a)=> Prop)) Xq) x2) (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((((functional_extensionality_dep a) (fun (x6:a)=> Prop)) Xq) x2) (fun (x4:a)=> ((eq_ref Prop) (Xq x4))))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found or_comm_i:=(fun (A:Prop) (B:Prop) (H:((or A) B))=> ((((((or_ind A) B) ((or B) A)) ((or_intror B) A)) ((or_introl B) A)) H)):(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A)))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A))):Prop
% Found or_comm_i as proof of a0
% Found iff_refl0:=(iff_refl (x2 Xy)):((iff (x2 Xy)) (x2 Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x2 Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x3:(x2 Xx0)) (Xy:a)=> (iff_refl (x2 Xy))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x2:(Xq Xx))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x1:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x2:(Xq Xx))=> (((eta_expansion a) Prop) Xq)) as proof of ((Xq Xx)->(((eq (a->Prop)) Xq) x0))
% Found (fun (x1:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x2:(Xq Xx))=> (((eta_expansion a) Prop) Xq)) as proof of ((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->(((eq (a->Prop)) Xq) x0)))
% Found (and_rect00 (fun (x1:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x2:(Xq Xx))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((and_rect0 (((eq (a->Prop)) Xq) x0)) (fun (x1:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x2:(Xq Xx))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((fun (P0:Type) (x1:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x1) x00)) (((eq (a->Prop)) Xq) x0)) (fun (x1:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x2:(Xq Xx))=> (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P0:Type) (x1:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x1) x00)) (((eq (a->Prop)) Xq) x0)) (fun (x1:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x2:(Xq Xx))=> (((eta_expansion a) Prop) Xq)))) as proof of (((eq (a->Prop)) Xq) x0)
% 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) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of ((P0 (f x0))->(P0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of ((P0 (f x0))->(P0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))
% Found (((fun (x1:(((eq Prop) (f x0)) b)) (x2:(((eq Prop) b) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))=> (((eq_trans0000 x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of ((P0 (f x0))->(P0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))
% Found (((fun (x1:(((eq Prop) (f x0)) b)) (x2:(((eq Prop) b) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))=> ((((eq_trans000 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of ((P0 (f x0))->(P0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))
% Found (((fun (x1:(((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (x2:(((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))=> (((((eq_trans00 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) as proof of ((P0 (f x0))->(P0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))
% Found (((fun (x1:(((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (x2:(((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))=> ((((((eq_trans0 (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) as proof of ((P0 (f x0))->(P0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))
% Found (((fun (x1:(((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (x2:(((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))=> (((((((eq_trans Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) as proof of ((P0 (f x0))->(P0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))
% Found (fun (P0:(Prop->Prop))=> (((fun (x1:(((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (x2:(((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))=> (((((((eq_trans Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))) as proof of ((P0 (f x0))->(P0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))
% 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) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of ((P0 (f x0))->(P0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of ((P0 (f x0))->(P0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))
% Found (((fun (x1:(((eq Prop) (f x0)) b)) (x2:(((eq Prop) b) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))=> (((eq_trans0000 x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of ((P0 (f x0))->(P0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))
% Found (((fun (x1:(((eq Prop) (f x0)) b)) (x2:(((eq Prop) b) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))=> ((((eq_trans000 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of ((P0 (f x0))->(P0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))
% Found (((fun (x1:(((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (x2:(((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))=> (((((eq_trans00 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) as proof of ((P0 (f x0))->(P0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))
% Found (((fun (x1:(((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (x2:(((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))=> ((((((eq_trans0 (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) as proof of ((P0 (f x0))->(P0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))
% Found (((fun (x1:(((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (x2:(((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))=> (((((((eq_trans Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) as proof of ((P0 (f x0))->(P0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))
% Found (fun (P0:(Prop->Prop))=> (((fun (x1:(((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (x2:(((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))=> (((((((eq_trans Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))) as proof of ((P0 (f x0))->(P0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))
% Found x3:(P0 Xq)
% Instantiate: x0:=Xq:(a->Prop)
% Found (fun (x3:(P0 Xq))=> x3) as proof of (P0 x0)
% Found (fun (P0:((a->Prop)->Prop)) (x3:(P0 Xq))=> x3) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x3:(P0 Xq))=> x3) as proof of (((eq (a->Prop)) Xq) x0)
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x2:(Xq Xx))=> (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x1:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x2:(Xq Xx))=> (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) as proof of ((Xq Xx)->(((eq (a->Prop)) Xq) x0))
% Found (fun (x1:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x2:(Xq Xx))=> (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)) as proof of ((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->(((eq (a->Prop)) Xq) x0)))
% Found (and_rect00 (fun (x1:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x2:(Xq Xx))=> (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((and_rect0 (((eq (a->Prop)) Xq) x0)) (fun (x1:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x2:(Xq Xx))=> (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((fun (P0:Type) (x1:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x1) x00)) (((eq (a->Prop)) Xq) x0)) (fun (x1:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x2:(Xq Xx))=> (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P0:Type) (x1:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x1) x00)) (((eq (a->Prop)) Xq) x0)) (fun (x1:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x2:(Xq Xx))=> (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x3:(P0 Xq)
% Instantiate: x0:=Xq:(a->Prop)
% Found (fun (x3:(P0 Xq))=> x3) as proof of (P0 x0)
% Found (fun (P0:((a->Prop)->Prop)) (x3:(P0 Xq))=> x3) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x3:(P0 Xq))=> x3) as proof of (((eq (a->Prop)) Xq) x0)
% Found x60:(P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of (P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion_dep0000 (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion_dep000 (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion_dep00 Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of (((eq (a->Prop)) Xq) x4)
% Found x60:(P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of (P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion0000 (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion000 (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion00 Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion0 Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found x3:(P0 Xq)
% Instantiate: x0:=Xq:(a->Prop)
% Found (fun (x3:(P0 Xq))=> x3) as proof of (P0 x0)
% Found (fun (P0:((a->Prop)->Prop)) (x3:(P0 Xq))=> x3) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x3:(P0 Xq))=> x3) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xq)->(P0 Xq))
% Found (eq_ref00 P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((eq_ref0 Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found x60:(P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of (P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion0000 (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion000 (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion00 Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion0 Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of (((eq (a->Prop)) Xq) x4)
% Found x60:(P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of (P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion_dep0000 (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion_dep000 (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion_dep00 Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of (((eq (a->Prop)) Xq) x4)
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eq_ref00:=(eq_ref0 x0):(((eq (a->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found ((eq_ref (a->Prop)) x0) as proof of (((eq (a->Prop)) x0) Xq)
% Found (eq_sym100 ((eq_ref (a->Prop)) x0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eq_sym10 Xq) ((eq_ref (a->Prop)) x0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eq_sym1 x0) Xq) ((eq_ref (a->Prop)) x0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((((eq_sym (a->Prop)) x0) Xq) ((eq_ref (a->Prop)) x0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((((eq_sym (a->Prop)) x0) Xq) ((eq_ref (a->Prop)) x0))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x60:(P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of (P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion0000 (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion000 (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion00 Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion0 Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of (((eq (a->Prop)) Xq) x4)
% Found x60:(P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of (P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion_dep0000 (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion_dep000 (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion_dep00 Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of (((eq (a->Prop)) Xq) x4)
% Found x60:(P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of (P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion_dep0000 (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion_dep000 (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion_dep00 Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of (((eq (a->Prop)) Xq) x4)
% Found x60:(P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of (P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion0000 (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion000 (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion00 Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion0 Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x0)
% Found x60:(P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of (P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion_dep0000 (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion_dep000 (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion_dep00 Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of (((eq (a->Prop)) Xq) x4)
% Found x60:(P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of (P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion0000 (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion000 (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion00 Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion0 Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of (((eq (a->Prop)) Xq) x4)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x5:(forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x4:(forall (Xx:a), ((R Xx) Xx))) (x5:(forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))=> ((eq_ref (a->Prop)) Xq)) as proof of ((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))->(((eq (a->Prop)) Xq) x2))
% Found (fun (x4:(forall (Xx:a), ((R Xx) Xx))) (x5:(forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))=> ((eq_ref (a->Prop)) Xq)) as proof of ((forall (Xx:a), ((R Xx) Xx))->((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))->(((eq (a->Prop)) Xq) x2)))
% Found (and_rect10 (fun (x4:(forall (Xx:a), ((R Xx) Xx))) (x5:(forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((and_rect1 (((eq (a->Prop)) Xq) x2)) (fun (x4:(forall (Xx:a), ((R Xx) Xx))) (x5:(forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((fun (P0:Type) (x4:((forall (Xx:a), ((R Xx) Xx))->((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))->P0)))=> (((((and_rect (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) P0) x4) x0)) (((eq (a->Prop)) Xq) x2)) (fun (x4:(forall (Xx:a), ((R Xx) Xx))) (x5:(forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P0:Type) (x4:((forall (Xx:a), ((R Xx) Xx))->((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))->P0)))=> (((((and_rect (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) P0) x4) x0)) (((eq (a->Prop)) Xq) x2)) (fun (x4:(forall (Xx:a), ((R Xx) Xx))) (x5:(forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))=> ((eq_ref (a->Prop)) Xq)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x5:(forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))=> (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x4:(forall (Xx:a), ((R Xx) Xx))) (x5:(forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))=> (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq)) as proof of ((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))->(((eq (a->Prop)) Xq) x2))
% Found (fun (x4:(forall (Xx:a), ((R Xx) Xx))) (x5:(forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))=> (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq)) as proof of ((forall (Xx:a), ((R Xx) Xx))->((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))->(((eq (a->Prop)) Xq) x2)))
% Found (and_rect10 (fun (x4:(forall (Xx:a), ((R Xx) Xx))) (x5:(forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))=> (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((and_rect1 (((eq (a->Prop)) Xq) x2)) (fun (x4:(forall (Xx:a), ((R Xx) Xx))) (x5:(forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))=> (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((fun (P0:Type) (x4:((forall (Xx:a), ((R Xx) Xx))->((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))->P0)))=> (((((and_rect (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) P0) x4) x0)) (((eq (a->Prop)) Xq) x2)) (fun (x4:(forall (Xx:a), ((R Xx) Xx))) (x5:(forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))=> (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P0:Type) (x4:((forall (Xx:a), ((R Xx) Xx))->((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))->P0)))=> (((((and_rect (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) P0) x4) x0)) (((eq (a->Prop)) Xq) x2)) (fun (x4:(forall (Xx:a), ((R Xx) Xx))) (x5:(forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))=> (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x5:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x4:(forall (Xx0:a), ((R Xx0) Xx0))) (x5:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> ((eq_ref (a->Prop)) Xq)) as proof of ((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->(((eq (a->Prop)) Xq) x2))
% Found (fun (x4:(forall (Xx0:a), ((R Xx0) Xx0))) (x5:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> ((eq_ref (a->Prop)) Xq)) as proof of ((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->(((eq (a->Prop)) Xq) x2)))
% Found (and_rect10 (fun (x4:(forall (Xx0:a), ((R Xx0) Xx0))) (x5:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((and_rect1 (((eq (a->Prop)) Xq) x2)) (fun (x4:(forall (Xx0:a), ((R Xx0) Xx0))) (x5:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((fun (P0:Type) (x4:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P0)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P0) x4) x0)) (((eq (a->Prop)) Xq) x2)) (fun (x4:(forall (Xx0:a), ((R Xx0) Xx0))) (x5:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P0:Type) (x4:((forall (Xx0:a), ((R Xx0) Xx0))->((forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))->P0)))=> (((((and_rect (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))) P0) x4) x0)) (((eq (a->Prop)) Xq) x2)) (fun (x4:(forall (Xx0:a), ((R Xx0) Xx0))) (x5:(forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))=> ((eq_ref (a->Prop)) Xq)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x0 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x0 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x5:(forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x4:(forall (Xx:a), ((R Xx) Xx))) (x5:(forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))=> ((eq_ref (a->Prop)) Xq)) as proof of ((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))->(((eq (a->Prop)) Xq) x0))
% Found (fun (x4:(forall (Xx:a), ((R Xx) Xx))) (x5:(forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))=> ((eq_ref (a->Prop)) Xq)) as proof of ((forall (Xx:a), ((R Xx) Xx))->((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))->(((eq (a->Prop)) Xq) x0)))
% Found (and_rect10 (fun (x4:(forall (Xx:a), ((R Xx) Xx))) (x5:(forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((and_rect1 (((eq (a->Prop)) Xq) x0)) (fun (x4:(forall (Xx:a), ((R Xx) Xx))) (x5:(forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((fun (P0:Type) (x4:((forall (Xx:a), ((R Xx) Xx))->((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))->P0)))=> (((((and_rect (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) P0) x4) x1)) (((eq (a->Prop)) Xq) x0)) (fun (x4:(forall (Xx:a), ((R Xx) Xx))) (x5:(forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P0:Type) (x4:((forall (Xx:a), ((R Xx) Xx))->((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))->P0)))=> (((((and_rect (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) P0) x4) x1)) (((eq (a->Prop)) Xq) x0)) (fun (x4:(forall (Xx:a), ((R Xx) Xx))) (x5:(forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))=> ((eq_ref (a->Prop)) Xq)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found iff_refl0:=(iff_refl ((R Xx0) Xy)):((iff ((R Xx0) Xy)) ((R Xx0) Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_refl ((R Xx0) Xy)) as proof of ((iff ((R Xx0) Xy)) (x2 Xy))
% Found (iff_sym00 (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found ((iff_sym0 (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (((iff_sym ((R Xx0) Xy)) (x2 Xy)) (iff_refl ((R Xx0) Xy)))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (iff_refl (x0 Xy)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found or_comm_i:=(fun (A:Prop) (B:Prop) (H:((or A) B))=> ((((((or_ind A) B) ((or B) A)) ((or_intror B) A)) ((or_introl B) A)) H)):(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A)))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A))):Prop
% Found or_comm_i as proof of a0
% Found x3:(P0 Xq)
% Instantiate: x2:=Xq:(a->Prop)
% Found (fun (x3:(P0 Xq))=> x3) as proof of (P0 x2)
% Found (fun (P0:((a->Prop)->Prop)) (x3:(P0 Xq))=> x3) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x3:(P0 Xq))=> x3) as proof of (((eq (a->Prop)) Xq) x2)
% Found or_comm_i:=(fun (A:Prop) (B:Prop) (H:((or A) B))=> ((((((or_ind A) B) ((or B) A)) ((or_intror B) A)) ((or_introl B) A)) H)):(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A)))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A))):Prop
% Found or_comm_i as proof of a0
% Found eq_ref00:=(eq_ref0 (Xq x4)):(((eq Prop) (Xq x4)) (Xq x4))
% Found (eq_ref0 (Xq x4)) as proof of (((eq Prop) (Xq x4)) (x2 x4))
% Found ((eq_ref Prop) (Xq x4)) as proof of (((eq Prop) (Xq x4)) (x2 x4))
% Found ((eq_ref Prop) (Xq x4)) as proof of (((eq Prop) (Xq x4)) (x2 x4))
% Found (fun (x4:a)=> ((eq_ref Prop) (Xq x4))) as proof of (((eq Prop) (Xq x4)) (x2 x4))
% Found (fun (x4:a)=> ((eq_ref Prop) (Xq x4))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x2 x)))
% Found (functional_extensionality_dep0000 (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((functional_extensionality_dep000 x2) (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((functional_extensionality_dep00 Xq) x2) (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((((functional_extensionality_dep0 (fun (x6:a)=> Prop)) Xq) x2) (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((((functional_extensionality_dep a) (fun (x6:a)=> Prop)) Xq) x2) (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((((functional_extensionality_dep a) (fun (x6:a)=> Prop)) Xq) x2) (fun (x4:a)=> ((eq_ref Prop) (Xq x4))))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 (Xq x4)):(((eq Prop) (Xq x4)) (Xq x4))
% Found (eq_ref0 (Xq x4)) as proof of (((eq Prop) (Xq x4)) (x2 x4))
% Found ((eq_ref Prop) (Xq x4)) as proof of (((eq Prop) (Xq x4)) (x2 x4))
% Found ((eq_ref Prop) (Xq x4)) as proof of (((eq Prop) (Xq x4)) (x2 x4))
% Found (fun (x4:a)=> ((eq_ref Prop) (Xq x4))) as proof of (((eq Prop) (Xq x4)) (x2 x4))
% Found (fun (x4:a)=> ((eq_ref Prop) (Xq x4))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x2 x)))
% Found (functional_extensionality0000 (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((functional_extensionality000 x2) (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((functional_extensionality00 Xq) x2) (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((((functional_extensionality0 Prop) Xq) x2) (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((((functional_extensionality a) Prop) Xq) x2) (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((((functional_extensionality a) Prop) Xq) x2) (fun (x4:a)=> ((eq_ref Prop) (Xq x4))))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x1:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x1:(x0 Xy))=> x1) as proof of ((R Xx0) Xy)
% Found (fun (x1:(x0 Xy))=> x1) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found x1:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x1:((R Xx0) Xy))=> x1) as proof of (x0 Xy)
% Found (fun (x1:((R Xx0) Xy))=> x1) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found ((conj20 (fun (x1:(x0 Xy))=> x1)) (fun (x1:((R Xx0) Xy))=> x1)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x1:(x0 Xy))=> x1)) (fun (x1:((R Xx0) Xy))=> x1)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x1:(x0 Xy))=> x1)) (fun (x1:((R Xx0) Xy))=> x1)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x1:(x0 Xy))=> x1)) (fun (x1:((R Xx0) Xy))=> x1)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x1:(x0 Xy))=> x1)) (fun (x1:((R Xx0) Xy))=> x1))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x1:(x0 Xy))=> x1)) (fun (x1:((R Xx0) Xy))=> x1))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x5:(P0 Xq)
% Instantiate: x0:=Xq:(a->Prop)
% Found (fun (x5:(P0 Xq))=> x5) as proof of (P0 x0)
% Found (fun (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of (((eq (a->Prop)) Xq) x0)
% Found x5:(P0 Xq)
% Instantiate: x0:=Xq:(a->Prop)
% Found (fun (x5:(P0 Xq))=> x5) as proof of (P0 x0)
% Found (fun (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of (((eq (a->Prop)) Xq) x0)
% Found x5:(P0 Xq)
% Instantiate: x2:=Xq:(a->Prop)
% Found (fun (x5:(P0 Xq))=> x5) as proof of (P0 x2)
% Found (fun (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion000:=(eta_expansion00 (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))):(((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) (fun (x:(a->Prop))=> ((and ((and (forall (Xx0:a), ((x Xx0)->(forall (Xy:a), ((iff (x Xy)) ((R Xx0) Xy)))))) (x Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x))))))
% Found (eta_expansion00 (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) b0)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) b0)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) b0)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (forall (Xx0:a), ((Xp Xx0)->(forall (Xy:a), ((iff (Xp Xy)) ((R Xx0) Xy)))))) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) b0)
% Found eq_ref00:=(eq_ref0 (Xq x4)):(((eq Prop) (Xq x4)) (Xq x4))
% Found (eq_ref0 (Xq x4)) as proof of (((eq Prop) (Xq x4)) (x2 x4))
% Found ((eq_ref Prop) (Xq x4)) as proof of (((eq Prop) (Xq x4)) (x2 x4))
% Found ((eq_ref Prop) (Xq x4)) as proof of (((eq Prop) (Xq x4)) (x2 x4))
% Found (fun (x4:a)=> ((eq_ref Prop) (Xq x4))) as proof of (((eq Prop) (Xq x4)) (x2 x4))
% Found (fun (x4:a)=> ((eq_ref Prop) (Xq x4))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x2 x)))
% Found (functional_extensionality_dep0000 (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((functional_extensionality_dep000 x2) (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((functional_extensionality_dep00 Xq) x2) (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((((functional_extensionality_dep0 (fun (x6:a)=> Prop)) Xq) x2) (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((((functional_extensionality_dep a) (fun (x6:a)=> Prop)) Xq) x2) (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((((functional_extensionality_dep a) (fun (x6:a)=> Prop)) Xq) x2) (fun (x4:a)=> ((eq_ref Prop) (Xq x4))))) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 (Xq x4)):(((eq Prop) (Xq x4)) (Xq x4))
% Found (eq_ref0 (Xq x4)) as proof of (((eq Prop) (Xq x4)) (x2 x4))
% Found ((eq_ref Prop) (Xq x4)) as proof of (((eq Prop) (Xq x4)) (x2 x4))
% Found ((eq_ref Prop) (Xq x4)) as proof of (((eq Prop) (Xq x4)) (x2 x4))
% Found (fun (x4:a)=> ((eq_ref Prop) (Xq x4))) as proof of (((eq Prop) (Xq x4)) (x2 x4))
% Found (fun (x4:a)=> ((eq_ref Prop) (Xq x4))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x2 x)))
% Found (functional_extensionality0000 (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((functional_extensionality000 x2) (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((functional_extensionality00 Xq) x2) (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((((functional_extensionality0 Prop) Xq) x2) (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((((functional_extensionality a) Prop) Xq) x2) (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((((functional_extensionality a) Prop) Xq) x2) (fun (x4:a)=> ((eq_ref Prop) (Xq x4))))) as proof of (((eq (a->Prop)) Xq) x2)
% Found iff_refl0:=(iff_refl (x2 Xy)):((iff (x2 Xy)) (x2 Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x2 Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x3:(x2 Xx0)) (Xy:a)=> (iff_refl (x2 Xy))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found x60:(P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of (P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion0000 (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion000 (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion00 Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion0 Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of (((eq (a->Prop)) Xq) x4)
% Found x60:(P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of (P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion_dep0000 (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion_dep000 (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion_dep00 Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of (((eq (a->Prop)) Xq) x4)
% Found x1:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x1:(x0 Xy))=> x1) as proof of ((R Xx0) Xy)
% Found (fun (x1:(x0 Xy))=> x1) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found x1:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x1:((R Xx0) Xy))=> x1) as proof of (x0 Xy)
% Found (fun (x1:((R Xx0) Xy))=> x1) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found ((conj20 (fun (x1:(x0 Xy))=> x1)) (fun (x1:((R Xx0) Xy))=> x1)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x1:(x0 Xy))=> x1)) (fun (x1:((R Xx0) Xy))=> x1)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x1:(x0 Xy))=> x1)) (fun (x1:((R Xx0) Xy))=> x1)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x1:(x0 Xy))=> x1)) (fun (x1:((R Xx0) Xy))=> x1)) as proof of ((and ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy)))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x1:(x0 Xy))=> x1)) (fun (x1:((R Xx0) Xy))=> x1))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x1:(x0 Xy))=> x1)) (fun (x1:((R Xx0) Xy))=> x1))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x2 Xy)):((iff (x2 Xy)) (x2 Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x2 Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x3:(x2 Xx0)) (Xy:a)=> (iff_refl (x2 Xy))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found x60:(P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of (P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion_dep0000 (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion_dep000 (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion_dep00 Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x60:(P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of (P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion0000 (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion000 (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion00 Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion0 Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x60:(P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of (P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion0000 (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion000 (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion00 Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion0 Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x60:(P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of (P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion_dep0000 (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion_dep000 (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion_dep00 Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x30:(P0 Xq)
% Found (fun (x30:(P0 Xq))=> x30) as proof of (P0 Xq)
% Found (fun (x30:(P0 Xq))=> x30) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion_dep0000 (fun (x30:(P0 Xq))=> x30)) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion_dep000 (fun (x4:(a->Prop))=> ((P0 Xq)->(P0 x4)))) (fun (x30:(P0 Xq))=> x30)) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion_dep00 Xq) (fun (x4:(a->Prop))=> ((P0 Xq)->(P0 x4)))) (fun (x30:(P0 Xq))=> x30)) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P0 Xq)->(P0 x4)))) (fun (x30:(P0 Xq))=> x30)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P0 Xq)->(P0 x4)))) (fun (x30:(P0 Xq))=> x30)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P0 Xq)->(P0 x4)))) (fun (x30:(P0 Xq))=> x30)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P0 Xq)->(P0 x4)))) (fun (x30:(P0 Xq))=> x30))) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P0 Xq)->(P0 x4)))) (fun (x30:(P0 Xq))=> x30))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x60:(P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of (P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion0000 (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion000 (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion00 Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion0 Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x30:(P0 Xq)
% Found (fun (x30:(P0 Xq))=> x30) as proof of (P0 Xq)
% Found (fun (x30:(P0 Xq))=> x30) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion0000 (fun (x30:(P0 Xq))=> x30)) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion000 (fun (x4:(a->Prop))=> ((P0 Xq)->(P0 x4)))) (fun (x30:(P0 Xq))=> x30)) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion00 Xq) (fun (x4:(a->Prop))=> ((P0 Xq)->(P0 x4)))) (fun (x30:(P0 Xq))=> x30)) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion0 Prop) Xq) (fun (x4:(a->Prop))=> ((P0 Xq)->(P0 x4)))) (fun (x30:(P0 Xq))=> x30)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P0 Xq)->(P0 x4)))) (fun (x30:(P0 Xq))=> x30)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P0 Xq)->(P0 x4)))) (fun (x30:(P0 Xq))=> x30)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P0 Xq)->(P0 x4)))) (fun (x30:(P0 Xq))=> x30))) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P0 Xq)->(P0 x4)))) (fun (x30:(P0 Xq))=> x30))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x60:(P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of (P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion_dep0000 (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion_dep000 (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion_dep00 Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x0))
% Found (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x0))
% Found (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) b0)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) b0)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) b0)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) b0)
% Found x60:(P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of (P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion0000 (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion000 (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion00 Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion0 Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x60:(P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of (P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion_dep0000 (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x2))
% Found ((eta_expansion_dep000 (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x2))
% Found (((eta_expansion_dep00 Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x2))
% Found ((((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x2))
% Found (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of ((P0 Xq)->(P0 x2))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of (((eq (a->Prop)) Xq) x2)
% Found iff_refl0:=(iff_refl (x2 Xy)):((iff (x2 Xy)) (x2 Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x2 Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (iff_refl (x2 Xy))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(a->Prop))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(a->Prop))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(a->Prop))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(a->Prop))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) 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) x4)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) x4)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x4)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x4)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x4)
% Found ((eq_trans00000 ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x4))
% Found ((eq_trans00000 ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x4))
% Found (((fun (x5:(((eq (a->Prop)) Xq) b)) (x6:(((eq (a->Prop)) b) x4))=> (((eq_trans0000 x5) x6) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x4))
% Found (((fun (x5:(((eq (a->Prop)) Xq) b)) (x6:(((eq (a->Prop)) b) x4))=> ((((eq_trans000 x4) x5) x6) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x4))
% Found (((fun (x5:(((eq (a->Prop)) Xq) Xq)) (x6:(((eq (a->Prop)) Xq) x4))=> (((((eq_trans00 Xq) x4) x5) x6) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) as proof of ((P Xq)->(P x4))
% Found (((fun (x5:(((eq (a->Prop)) Xq) Xq)) (x6:(((eq (a->Prop)) Xq) x4))=> ((((((eq_trans0 Xq) Xq) x4) x5) x6) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) as proof of ((P Xq)->(P x4))
% Found (((fun (x5:(((eq (a->Prop)) Xq) Xq)) (x6:(((eq (a->Prop)) Xq) x4))=> (((((((eq_trans (a->Prop)) Xq) Xq) x4) x5) x6) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x5:(((eq (a->Prop)) Xq) Xq)) (x6:(((eq (a->Prop)) Xq) x4))=> (((((((eq_trans (a->Prop)) Xq) Xq) x4) x5) x6) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq))) as proof of ((P Xq)->(P x4))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x5:(((eq (a->Prop)) Xq) Xq)) (x6:(((eq (a->Prop)) Xq) x4))=> (((((((eq_trans (a->Prop)) Xq) Xq) x4) x5) x6) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x4)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x5:(((eq (a->Prop)) Xq) Xq)) (x6:(((eq (a->Prop)) Xq) x4))=> (((((((eq_trans (a->Prop)) Xq) Xq) x4) x5) x6) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x4)
% Found x10:(P0 (f x0))
% Found (fun (x10:(P0 (f x0)))=> x10) as proof of (P0 (f x0))
% Found (fun (x10:(P0 (f x0)))=> x10) as proof of (P1 (f x0))
% 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) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) (fun (x10:(P0 (f x0)))=> x10)) as proof of ((P0 (f x0))->(P0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) (fun (x10:(P0 (f x0)))=> x10)) as proof of ((P0 (f x0))->(P0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))
% Found ((((fun (x1:(((eq Prop) (f x0)) b)) (x2:(((eq Prop) b) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))=> (((eq_trans0000 x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) (fun (x10:(P0 (f x0)))=> x10)) as proof of ((P0 (f x0))->(P0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))
% Found ((((fun (x1:(((eq Prop) (f x0)) b)) (x2:(((eq Prop) b) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))=> ((((eq_trans000 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) (fun (x10:(P0 (f x0)))=> x10)) as proof of ((P0 (f x0))->(P0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))
% Found ((((fun (x1:(((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (x2:(((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))=> (((((eq_trans00 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (fun (x10:(P0 (f x0)))=> x10)) as proof of ((P0 (f x0))->(P0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))
% Found ((((fun (x1:(((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (x2:(((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))=> ((((((eq_trans0 (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (fun (x10:(P0 (f x0)))=> x10)) as proof of ((P0 (f x0))->(P0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))
% Found ((((fun (x1:(((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (x2:(((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))=> (((((((eq_trans Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (fun (x10:(P0 (f x0)))=> x10)) as proof of ((P0 (f x0))->(P0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))
% Found (fun (P0:(Prop->Prop))=> ((((fun (x1:(((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (x2:(((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))=> (((((((eq_trans Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (fun (x10:(P0 (f x0)))=> x10))) as proof of ((P0 (f x0))->(P0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))
% Found x10:(P0 (f x0))
% Found (fun (x10:(P0 (f x0)))=> x10) as proof of (P0 (f x0))
% Found (fun (x10:(P0 (f x0)))=> x10) as proof of (P1 (f x0))
% 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) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) (fun (x10:(P0 (f x0)))=> x10)) as proof of ((P0 (f x0))->(P0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) (fun (x10:(P0 (f x0)))=> x10)) as proof of ((P0 (f x0))->(P0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))
% Found ((((fun (x1:(((eq Prop) (f x0)) b)) (x2:(((eq Prop) b) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))=> (((eq_trans0000 x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) (fun (x10:(P0 (f x0)))=> x10)) as proof of ((P0 (f x0))->(P0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))
% Found ((((fun (x1:(((eq Prop) (f x0)) b)) (x2:(((eq Prop) b) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))=> ((((eq_trans000 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) (fun (x10:(P0 (f x0)))=> x10)) as proof of ((P0 (f x0))->(P0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))
% Found ((((fun (x1:(((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (x2:(((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))=> (((((eq_trans00 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (fun (x10:(P0 (f x0)))=> x10)) as proof of ((P0 (f x0))->(P0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))
% Found ((((fun (x1:(((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (x2:(((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))=> ((((((eq_trans0 (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (fun (x10:(P0 (f x0)))=> x10)) as proof of ((P0 (f x0))->(P0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))
% Found ((((fun (x1:(((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (x2:(((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))=> (((((((eq_trans Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (fun (x10:(P0 (f x0)))=> x10)) as proof of ((P0 (f x0))->(P0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))
% Found (fun (P0:(Prop->Prop))=> ((((fun (x1:(((eq Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (x2:(((eq Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))=> (((((((eq_trans Prop) (f x0)) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0)))))) (fun (x10:(P0 (f x0)))=> x10))) as proof of ((P0 (f x0))->(P0 ((and ((and (forall (Xx0:a), ((x0 Xx0)->(forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))))) (x0 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))
% Found conj10:=(conj1 (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))):(((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))))
% Found (conj1 (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) as proof of (((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->a0))
% Found ((conj ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) as proof of (((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->a0))
% Found ((conj ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) as proof of (((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->a0))
% Found ((conj ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) as proof of (((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->a0))
% Found (and_rect00 ((conj ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) as proof of a0
% Found ((and_rect0 a0) ((conj ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) as proof of a0
% Found (((fun (P0:Type) (x1:(((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->P0)))=> (((((and_rect ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) P0) x1) x)) a0) ((conj ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) as proof of a0
% Found (((fun (P0:Type) (x1:(((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->P0)))=> (((((and_rect ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) P0) x1) x)) a0) ((conj ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) as proof of a0
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found x3:(P0 (f x2))
% Instantiate: b:=(f x2):Prop
% Found x3 as proof of (P1 b)
% Found x3:(P0 (f x2))
% Instantiate: b:=(f x2):Prop
% Found x3 as proof of (P1 b)
% Found x3:(P0 Xq)
% Instantiate: x0:=Xq:(a->Prop)
% Found (fun (x3:(P0 Xq))=> x3) as proof of (P0 x0)
% Found (fun (P0:((a->Prop)->Prop)) (x3:(P0 Xq))=> x3) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x3:(P0 Xq))=> x3) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))):(((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found (eq_ref0 ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))):(((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2)))))
% Found (eq_ref0 ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x2 Xx0)->(forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))))) (x2 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x2))))) b)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x7:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x6:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x7:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)) as proof of ((Xq Xx)->(((eq (a->Prop)) Xq) x4))
% Found (fun (x6:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x7:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)) as proof of ((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->(((eq (a->Prop)) Xq) x4)))
% Found (and_rect20 (fun (x6:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x7:(Xq Xx))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((and_rect2 (((eq (a->Prop)) Xq) x4)) (fun (x6:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x7:(Xq Xx))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((fun (P0:Type) (x6:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x6) x5)) (((eq (a->Prop)) Xq) x4)) (fun (x6:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x7:(Xq Xx))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((fun (P0:Type) (x6:((forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))->((Xq Xx)->P0)))=> (((((and_rect (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)) P0) x6) x5)) (((eq (a->Prop)) Xq) x4)) (fun (x6:(forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (x7:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)))) as proof of (((eq (a->Prop)) Xq) x4)
% Found iff_refl0:=(iff_refl (x2 Xy)):((iff (x2 Xy)) (x2 Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x2 Xy)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x2 Xy))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> (iff_refl (x2 Xy))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found eq_ref00:=(eq_ref0 x4):(((eq (a->Prop)) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq (a->Prop)) x4) Xq)
% Found ((eq_ref (a->Prop)) x4) as proof of (((eq (a->Prop)) x4) Xq)
% Found ((eq_ref (a->Prop)) x4) as proof of (((eq (a->Prop)) x4) Xq)
% Found ((eq_ref (a->Prop)) x4) as proof of (((eq (a->Prop)) x4) Xq)
% Found (eq_sym100 ((eq_ref (a->Prop)) x4)) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((eq_sym10 Xq) ((eq_ref (a->Prop)) x4)) as proof of (((eq (a->Prop)) Xq) x4)
% Found (((eq_sym1 x4) Xq) ((eq_ref (a->Prop)) x4)) as proof of (((eq (a->Prop)) Xq) x4)
% Found ((((eq_sym (a->Prop)) x4) Xq) ((eq_ref (a->Prop)) x4)) as proof of (((eq (a->Prop)) Xq) x4)
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((((eq_sym (a->Prop)) x4) Xq) ((eq_ref (a->Prop)) x4))) as proof of (((eq (a->Prop)) Xq) x4)
% Found eq_ref00:=(eq_ref0 (Xq x4)):(((eq Prop) (Xq x4)) (Xq x4))
% Found (eq_ref0 (Xq x4)) as proof of (((eq Prop) (Xq x4)) (x0 x4))
% Found ((eq_ref Prop) (Xq x4)) as proof of (((eq Prop) (Xq x4)) (x0 x4))
% Found ((eq_ref Prop) (Xq x4)) as proof of (((eq Prop) (Xq x4)) (x0 x4))
% Found (fun (x4:a)=> ((eq_ref Prop) (Xq x4))) as proof of (((eq Prop) (Xq x4)) (x0 x4))
% Found (fun (x4:a)=> ((eq_ref Prop) (Xq x4))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found (functional_extensionality0000 (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((functional_extensionality000 x0) (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((functional_extensionality00 Xq) x0) (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((((functional_extensionality0 Prop) Xq) x0) (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((((functional_extensionality a) Prop) Xq) x0) (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((((functional_extensionality a) Prop) Xq) x0) (fun (x4:a)=> ((eq_ref Prop) (Xq x4))))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 (Xq x4)):(((eq Prop) (Xq x4)) (Xq x4))
% Found (eq_ref0 (Xq x4)) as proof of (((eq Prop) (Xq x4)) (x0 x4))
% Found ((eq_ref Prop) (Xq x4)) as proof of (((eq Prop) (Xq x4)) (x0 x4))
% Found ((eq_ref Prop) (Xq x4)) as proof of (((eq Prop) (Xq x4)) (x0 x4))
% Found (fun (x4:a)=> ((eq_ref Prop) (Xq x4))) as proof of (((eq Prop) (Xq x4)) (x0 x4))
% Found (fun (x4:a)=> ((eq_ref Prop) (Xq x4))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x0 x)))
% Found (functional_extensionality_dep0000 (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((functional_extensionality_dep000 x0) (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((functional_extensionality_dep00 Xq) x0) (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((((functional_extensionality_dep0 (fun (x6:a)=> Prop)) Xq) x0) (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((((functional_extensionality_dep a) (fun (x6:a)=> Prop)) Xq) x0) (fun (x4:a)=> ((eq_ref Prop) (Xq x4)))) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x3:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((((functional_extensionality_dep a) (fun (x6:a)=> Prop)) Xq) x0) (fun (x4:a)=> ((eq_ref Prop) (Xq x4))))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x5:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found x5:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x0 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found ((conj20 (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x5:((R Xx0) Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x0 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x0 Xy))
% Found x5:(x0 Xy)
% Instantiate: x0:=(R Xx0):(a->Prop)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x0 Xy))=> x5) as proof of ((x0 Xy)->((R Xx0) Xy))
% Found ((conj20 (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> ((((conj ((x0 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x0 Xy))) (fun (x5:(x0 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found x5:(x2 Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x5:(x2 Xy))=> x5) as proof of ((R Xx0) Xy)
% Found (fun (x5:(x2 Xy))=> x5) as proof of ((x2 Xy)->((R Xx0) Xy))
% Found x5:((R Xx0) Xy)
% Instantiate: x2:=(R Xx0):(a->Prop)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (x2 Xy)
% Found (fun (x5:((R Xx0) Xy))=> x5) as proof of (((R Xx0) Xy)->(x2 Xy))
% Found ((conj20 (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (((conj2 (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5)) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of ((iff (x2 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x2 Xx0)) (Xy:a)=> ((((conj ((x2 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x2 Xy))) (fun (x5:(x2 Xy))=> x5)) (fun (x5:((R Xx0) Xy))=> x5))) as proof of (forall (Xy:a), ((iff (x2 Xy)) ((R Xx0) Xy)))
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found x5:(P0 Xq)
% Instantiate: x4:=Xq:(a->Prop)
% Found (fun (x5:(P0 Xq))=> x5) as proof of (P0 x4)
% Found (fun (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of (((eq (a->Prop)) Xq) x4)
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x3:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found or_comm_i:=(fun (A:Prop) (B:Prop) (H:((or A) B))=> ((((((or_ind A) B) ((or B) A)) ((or_intror B) A)) ((or_introl B) A)) H)):(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A)))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A))):Prop
% Found or_comm_i as proof of a0
% Found iff_refl0:=(iff_refl (x4 Xy)):((iff (x4 Xy)) (x4 Xy))
% Found (iff_refl (x4 Xy)) as proof of ((and ((x4 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x4 Xy)))
% Found (iff_refl (x4 Xy)) as proof of ((and ((x4 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x4 Xy)))
% Found (iff_refl (x4 Xy)) as proof of ((and ((x4 Xy)->((R Xx0) Xy))) (((R Xx0) Xy)->(x4 Xy)))
% Found (fun (Xy:a)=> (iff_refl (x4 Xy))) as proof of ((iff (x4 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x4 Xx0)) (Xy:a)=> (iff_refl (x4 Xy))) as proof of (forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))
% Found or_comm_i:=(fun (A:Prop) (B:Prop) (H:((or A) B))=> ((((((or_ind A) B) ((or B) A)) ((or_intror B) A)) ((or_introl B) A)) H)):(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A)))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A))):Prop
% Found or_comm_i as proof of a0
% Found x30:(P0 Xq)
% Found (fun (x30:(P0 Xq))=> x30) as proof of (P0 Xq)
% Found (fun (x30:(P0 Xq))=> x30) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion_dep0000 (fun (x30:(P0 Xq))=> x30)) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion_dep000 (fun (x4:(a->Prop))=> ((P0 Xq)->(P0 x4)))) (fun (x30:(P0 Xq))=> x30)) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion_dep00 Xq) (fun (x4:(a->Prop))=> ((P0 Xq)->(P0 x4)))) (fun (x30:(P0 Xq))=> x30)) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P0 Xq)->(P0 x4)))) (fun (x30:(P0 Xq))=> x30)) as proof of ((P0 Xq)->(P0 x0))
% Found (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P0 Xq)->(P0 x4)))) (fun (x30:(P0 Xq))=> x30)) as proof of ((P0 Xq)->(P0 x0))
% Found (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P0 Xq)->(P0 x4)))) (fun (x30:(P0 Xq))=> x30)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P0 Xq)->(P0 x4)))) (fun (x30:(P0 Xq))=> x30))) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xq) (fun (x4:(a->Prop))=> ((P0 Xq)->(P0 x4)))) (fun (x30:(P0 Xq))=> x30))) as proof of (((eq (a->Prop)) Xq) x0)
% Found x30:(P0 Xq)
% Found (fun (x30:(P0 Xq))=> x30) as proof of (P0 Xq)
% Found (fun (x30:(P0 Xq))=> x30) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion0000 (fun (x30:(P0 Xq))=> x30)) as proof of ((P0 Xq)->(P0 x0))
% Found ((eta_expansion000 (fun (x4:(a->Prop))=> ((P0 Xq)->(P0 x4)))) (fun (x30:(P0 Xq))=> x30)) as proof of ((P0 Xq)->(P0 x0))
% Found (((eta_expansion00 Xq) (fun (x4:(a->Prop))=> ((P0 Xq)->(P0 x4)))) (fun (x30:(P0 Xq))=> x30)) as proof of ((P0 Xq)->(P0 x0))
% Found ((((eta_expansion0 Prop) Xq) (fun (x4:(a->Prop))=> ((P0 Xq)->(P0 x4)))) (fun (x30:(P0 Xq))=> x30)) as proof of ((P0 Xq)->(P0 x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P0 Xq)->(P0 x4)))) (fun (x30:(P0 Xq))=> x30)) as proof of ((P0 Xq)->(P0 x0))
% Found (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P0 Xq)->(P0 x4)))) (fun (x30:(P0 Xq))=> x30)) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P0 Xq)->(P0 x4)))) (fun (x30:(P0 Xq))=> x30))) as proof of ((P0 Xq)->(P0 x0))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x4:(a->Prop))=> ((P0 Xq)->(P0 x4)))) (fun (x30:(P0 Xq))=> x30))) as proof of (((eq (a->Prop)) Xq) x0)
% Found or_comm_i:=(fun (A:Prop) (B:Prop) (H:((or A) B))=> ((((((or_ind A) B) ((or B) A)) ((or_intror B) A)) ((or_introl B) A)) H)):(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A)))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A))):Prop
% Found or_comm_i as proof of a0
% Found iff_refl0:=(iff_refl (x0 Xy)):((iff (x0 Xy)) (x0 Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (iff_refl (x0 Xy)) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (Xy:a)=> (iff_refl (x0 Xy))) as proof of ((iff (x0 Xy)) ((R Xx0) Xy))
% Found (fun (x00:(x0 Xx0)) (Xy:a)=> (iff_refl (x0 Xy))) as proof of (forall (Xy:a), ((iff (x0 Xy)) ((R Xx0) Xy)))
% Found x60:(P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of (P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion0000 (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion000 (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion00 Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion0 Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of (((eq (a->Prop)) Xq) x4)
% Found x60:(P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of (P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion_dep0000 (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion_dep000 (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion_dep00 Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of (((eq (a->Prop)) Xq) x4)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) 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) x0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) x0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x0)
% Found ((eq_trans00000 ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x0))
% Found ((eq_trans00000 ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x0))
% Found (((fun (x5:(((eq (a->Prop)) Xq) b)) (x6:(((eq (a->Prop)) b) x0))=> (((eq_trans0000 x5) x6) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x0))
% Found (((fun (x5:(((eq (a->Prop)) Xq) b)) (x6:(((eq (a->Prop)) b) x0))=> ((((eq_trans000 x0) x5) x6) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x0))
% Found (((fun (x5:(((eq (a->Prop)) Xq) Xq)) (x6:(((eq (a->Prop)) Xq) x0))=> (((((eq_trans00 Xq) x0) x5) x6) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) as proof of ((P Xq)->(P x0))
% Found (((fun (x5:(((eq (a->Prop)) Xq) Xq)) (x6:(((eq (a->Prop)) Xq) x0))=> ((((((eq_trans0 Xq) Xq) x0) x5) x6) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) as proof of ((P Xq)->(P x0))
% Found (((fun (x5:(((eq (a->Prop)) Xq) Xq)) (x6:(((eq (a->Prop)) Xq) x0))=> (((((((eq_trans (a->Prop)) Xq) Xq) x0) x5) x6) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x5:(((eq (a->Prop)) Xq) Xq)) (x6:(((eq (a->Prop)) Xq) x0))=> (((((((eq_trans (a->Prop)) Xq) Xq) x0) x5) x6) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq))) as proof of ((P Xq)->(P x0))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x5:(((eq (a->Prop)) Xq) Xq)) (x6:(((eq (a->Prop)) Xq) x0))=> (((((((eq_trans (a->Prop)) Xq) Xq) x0) x5) x6) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x0)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x5:(((eq (a->Prop)) Xq) Xq)) (x6:(((eq (a->Prop)) Xq) x0))=> (((((((eq_trans (a->Prop)) Xq) Xq) x0) x5) x6) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x4))
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))):(((eq Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))
% Found (eq_ref0 ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) b)
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))):(((eq Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))
% Found (eq_ref0 ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x4))
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) b)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) 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) x2)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) x2)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x2)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x2)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) x2)
% Found ((eq_trans00000 ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x2))
% Found ((eq_trans00000 ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x2))
% Found (((fun (x5:(((eq (a->Prop)) Xq) b)) (x6:(((eq (a->Prop)) b) x2))=> (((eq_trans0000 x5) x6) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x2))
% Found (((fun (x5:(((eq (a->Prop)) Xq) b)) (x6:(((eq (a->Prop)) b) x2))=> ((((eq_trans000 x2) x5) x6) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) b)) as proof of ((P Xq)->(P x2))
% Found (((fun (x5:(((eq (a->Prop)) Xq) Xq)) (x6:(((eq (a->Prop)) Xq) x2))=> (((((eq_trans00 Xq) x2) x5) x6) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) as proof of ((P Xq)->(P x2))
% Found (((fun (x5:(((eq (a->Prop)) Xq) Xq)) (x6:(((eq (a->Prop)) Xq) x2))=> ((((((eq_trans0 Xq) Xq) x2) x5) x6) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) as proof of ((P Xq)->(P x2))
% Found (((fun (x5:(((eq (a->Prop)) Xq) Xq)) (x6:(((eq (a->Prop)) Xq) x2))=> (((((((eq_trans (a->Prop)) Xq) Xq) x2) x5) x6) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq)) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x5:(((eq (a->Prop)) Xq) Xq)) (x6:(((eq (a->Prop)) Xq) x2))=> (((((((eq_trans (a->Prop)) Xq) Xq) x2) x5) x6) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq))) as proof of ((P Xq)->(P x2))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x5:(((eq (a->Prop)) Xq) Xq)) (x6:(((eq (a->Prop)) Xq) x2))=> (((((((eq_trans (a->Prop)) Xq) Xq) x2) x5) x6) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq))) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x2)))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P:((a->Prop)->Prop))=> (((fun (x5:(((eq (a->Prop)) Xq) Xq)) (x6:(((eq (a->Prop)) Xq) x2))=> (((((((eq_trans (a->Prop)) Xq) Xq) x2) x5) x6) P)) ((eq_ref (a->Prop)) Xq)) (((eta_expansion a) Prop) Xq))) as proof of (((eq (a->Prop)) Xq) x2)
% Found x60:(P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of (P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion_dep0000 (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion_dep000 (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion_dep00 Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of (((eq (a->Prop)) Xq) x4)
% Found x60:(P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of (P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion0000 (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion000 (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion00 Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion0 Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x4))
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))):(((eq Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4)))))
% Found (eq_ref0 ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) as proof of (((eq Prop) ((and ((and (forall (Xx0:a), ((x4 Xx0)->(forall (Xy:a), ((iff (x4 Xy)) ((R Xx0) Xy)))))) (x4 Xx))) (forall (Xq:(a->Prop)), (((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))->(((eq (a->Prop)) Xq) x4))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x0)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x0)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x2)
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x2)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xq)->(P0 Xq))
% Found (eq_ref00 P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((eq_ref0 Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eq_ref (a->Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion_dep00 Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P0)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x4)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 Xq)->(P0 (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion00 Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion0 Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion a) Prop) Xq) P0) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P0)) as proof of (((eq (a->Prop)) Xq) x4)
% Found x5:(P0 Xq)
% Instantiate: x4:=Xq:(a->Prop)
% Found (fun (x5:(P0 Xq))=> x5) as proof of (P0 x4)
% Found (fun (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x00:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop)) (x5:(P0 Xq))=> x5) as proof of (((eq (a->Prop)) Xq) x4)
% Found x60:(P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of (P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion_dep0000 (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion_dep000 (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion_dep00 Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of (((eq (a->Prop)) Xq) x4)
% Found x60:(P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of (P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion_dep0000 (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion_dep000 (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion_dep00 Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of (((eq (a->Prop)) Xq) x4)
% Found x60:(P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of (P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion0000 (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion000 (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion00 Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion0 Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of ((P0 Xq)->(P0 x4))
% Found (fun (x5:((and (forall (Xx0:a), ((Xq Xx0)->(forall (Xy:a), ((iff (Xq Xy)) ((R Xx0) Xy)))))) (Xq Xx))) (P0:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60))) as proof of (((eq (a->Prop)) Xq) x4)
% Found x60:(P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of (P0 Xq)
% Found (fun (x60:(P0 Xq))=> x60) as proof of ((P0 Xq)->(P0 Xq))
% Found (eta_expansion0000 (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((eta_expansion000 (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((eta_expansion00 Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found ((((eta_expansion0 Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 Xq)->(P0 x4))
% Found (((((eta_expansion a) Prop) Xq) (fun (x7:(a->Prop))=> ((P0 Xq)->(P0 x7)))) (fun (x60:(P0 Xq))=> x60)) as proof of ((P0 X
% EOF
%------------------------------------------------------------------------------