%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV080^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n106.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 13:33:42 EDT 2014

% Result   : Timeout 300.00s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV080^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n106.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 07:57:06 CDT 2014
% % CPUTime  : 300.00 
% 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 0x105fb00>, <kernel.Type object at 0x105fef0>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (Xx:(a->Prop)), ((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_28)))))))))) of role conjecture named cEQP_1A_pme
% Conjecture to prove = (forall (Xx:(a->Prop)), ((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_28)))))))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (Xx:(a->Prop)), ((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_28))))))))))']
% Parameter a:Type.
% Trying to prove (forall (Xx:(a->Prop)), ((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_28))))))))))
% Found x0:(Xx Xx0)
% Instantiate: x:=(fun (x1:a)=> x1):(a->a)
% Found (fun (x0:(Xx Xx0))=> x0) as proof of (Xx (x Xx0))
% Found (fun (Xx0:a) (x0:(Xx Xx0))=> x0) as proof of ((Xx Xx0)->(Xx (x Xx0)))
% Found (fun (Xx0:a) (x0:(Xx Xx0))=> x0) as proof of (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))
% Found x0:(Xx Xx0)
% Instantiate: x:=(fun (x1:a)=> x1):(a->a)
% Found (fun (x0:(Xx Xx0))=> x0) as proof of (Xx (x Xx0))
% Found (fun (Xx0:a) (x0:(Xx Xx0))=> x0) as proof of ((Xx Xx0)->(Xx (x Xx0)))
% Found (fun (Xx0:a) (x0:(Xx Xx0))=> x0) as proof of (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))))))):(((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))))))) (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy))))))))))
% Found (eta_expansion_dep00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))))))) b)
% Found ((eta_expansion_dep0 (fun (x1:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))))))) b)
% Found (((eta_expansion_dep (a->a)) (fun (x1:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))))))) b)
% Found (((eta_expansion_dep (a->a)) (fun (x1:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))))))) b)
% Found (((eta_expansion_dep (a->a)) (fun (x1:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))))))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) ((eq a) Xy_28))))))))):(((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) ((eq a) Xy_28))))))))) (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28)))))))))
% Found (eta_expansion00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) ((eq a) Xy_28))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) ((eq a) Xy_28))))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) ((eq a) Xy_28))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) ((eq a) Xy_28))))))))) b)
% Found (((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) ((eq a) Xy_28))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) ((eq a) Xy_28))))))))) b)
% Found (((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) ((eq a) Xy_28))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) ((eq a) Xy_28))))))))) b)
% Found (((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) ((eq a) Xy_28))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) ((eq a) Xy_28))))))))) b)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))))))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))))))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))))))
% Found (fun (x:(a->a))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))))))
% Found (fun (x:(a->a))=> ((eq_ref Prop) (f x))) as proof of (forall (x:(a->a)), (((eq Prop) (f x)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy))))))))))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))))))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))))))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))))))
% Found (fun (x:(a->a))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))))))
% Found (fun (x:(a->a))=> ((eq_ref Prop) (f x))) as proof of (forall (x:(a->a)), (((eq Prop) (f x)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy))))))))))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28))))))))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28))))))))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28))))))))
% Found (fun (x:(a->a))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28))))))))
% Found (fun (x:(a->a))=> ((eq_ref Prop) (f x))) as proof of (forall (x:(a->a)), (((eq Prop) (f x)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28)))))))))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28))))))))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28))))))))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28))))))))
% Found (fun (x:(a->a))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28))))))))
% Found (fun (x:(a->a))=> ((eq_ref Prop) (f x))) as proof of (forall (x:(a->a)), (((eq Prop) (f x)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28)))))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq ((a->a)->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((a->a)->Prop)) a0) b)
% Found ((eq_ref ((a->a)->Prop)) a0) as proof of (((eq ((a->a)->Prop)) a0) b)
% Found ((eq_ref ((a->a)->Prop)) a0) as proof of (((eq ((a->a)->Prop)) a0) b)
% Found ((eq_ref ((a->a)->Prop)) a0) as proof of (((eq ((a->a)->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->a)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy))))))))))
% Found ((eq_ref ((a->a)->Prop)) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy))))))))))
% Found ((eq_ref ((a->a)->Prop)) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy))))))))))
% Found ((eq_ref ((a->a)->Prop)) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy))))))))))
% Found eq_ref00:=(eq_ref0 (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))))):(((eq Prop) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy))))))))
% Found (eq_ref0 (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))))) as proof of (((eq Prop) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))))) b)
% Found ((eq_ref Prop) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))))) as proof of (((eq Prop) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))))) b)
% Found ((eq_ref Prop) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))))) as proof of (((eq Prop) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))))) b)
% Found ((eq_ref Prop) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))))) as proof of (((eq Prop) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))))) b)
% Found eta_expansion000:=(eta_expansion00 a0):(((eq ((a->a)->Prop)) a0) (fun (x:(a->a))=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq ((a->a)->Prop)) a0) b)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq ((a->a)->Prop)) a0) b)
% Found (((eta_expansion (a->a)) Prop) a0) as proof of (((eq ((a->a)->Prop)) a0) b)
% Found (((eta_expansion (a->a)) Prop) a0) as proof of (((eq ((a->a)->Prop)) a0) b)
% Found (((eta_expansion (a->a)) Prop) a0) as proof of (((eq ((a->a)->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->a)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) ((eq a) Xy_28)))))))))
% Found ((eq_ref ((a->a)->Prop)) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) ((eq a) Xy_28)))))))))
% Found ((eq_ref ((a->a)->Prop)) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) ((eq a) Xy_28)))))))))
% Found ((eq_ref ((a->a)->Prop)) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) ((eq a) Xy_28)))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->a)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) ((eq a) Xy_28)))))))))
% Found ((eq_ref ((a->a)->Prop)) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) ((eq a) Xy_28)))))))))
% Found ((eq_ref ((a->a)->Prop)) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) ((eq a) Xy_28)))))))))
% Found ((eq_ref ((a->a)->Prop)) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (Xs Xx0))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (Xs Xx0))))) ((eq a) Xy_28)))))))))
% Found eq_ref00:=(eq_ref0 (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28))))))):(((eq Prop) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28))))))) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28)))))))
% Found (eq_ref0 (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28))))))) as proof of (((eq Prop) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28))))))) b)
% Found ((eq_ref Prop) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28))))))) as proof of (((eq Prop) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28))))))) b)
% Found ((eq_ref Prop) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28))))))) as proof of (((eq Prop) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28))))))) b)
% Found ((eq_ref Prop) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28))))))) as proof of (((eq Prop) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28))))))) b)
% Found eq_ref00:=(eq_ref0 (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy))))):(((eq (a->Prop)) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy))))) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))
% Found (eq_ref0 (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy))))) b)
% Found eq_ref00:=(eq_ref0 (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy))))):(((eq (a->Prop)) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy))))) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))
% Found (eq_ref0 (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28)))):(((eq (a->Prop)) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28)))) (fun (x0:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) x0))))
% Found (eta_expansion_dep00 (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28)))) as proof of (((eq (a->Prop)) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28)))) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28)))) as proof of (((eq (a->Prop)) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28)))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28)))) as proof of (((eq (a->Prop)) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28)))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28)))) as proof of (((eq (a->Prop)) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28)))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28)))) as proof of (((eq (a->Prop)) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28)))) b)
% Found eq_ref00:=(eq_ref0 (f x00)):(((eq Prop) (f x00)) (f x00))
% Found (eq_ref0 (f x00)) as proof of (((eq Prop) (f x00)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) x00) Xy))))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) x00) Xy))))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) x00) Xy))))
% Found (fun (x00:a)=> ((eq_ref Prop) (f x00))) as proof of (((eq Prop) (f x00)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) x00) Xy))))
% Found (fun (x00:a)=> ((eq_ref Prop) (f x00))) as proof of (forall (x0:a), (((eq Prop) (f x0)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) x0) Xy)))))
% Found eq_ref00:=(eq_ref0 (f x00)):(((eq Prop) (f x00)) (f x00))
% Found (eq_ref0 (f x00)) as proof of (((eq Prop) (f x00)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) x00) Xy))))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) x00) Xy))))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) x00) Xy))))
% Found (fun (x00:a)=> ((eq_ref Prop) (f x00))) as proof of (((eq Prop) (f x00)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) x00) Xy))))
% Found (fun (x00:a)=> ((eq_ref Prop) (f x00))) as proof of (forall (x0:a), (((eq Prop) (f x0)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) x0) Xy)))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28)))):(((eq (a->Prop)) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28)))) (fun (x0:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) x0))))
% Found (eta_expansion00 (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28)))) as proof of (((eq (a->Prop)) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28)))) b)
% Found ((eta_expansion0 Prop) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28)))) as proof of (((eq (a->Prop)) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28)))) b)
% Found (((eta_expansion a) Prop) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28)))) as proof of (((eq (a->Prop)) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28)))) b)
% Found (((eta_expansion a) Prop) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28)))) as proof of (((eq (a->Prop)) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28)))) b)
% Found (((eta_expansion a) Prop) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28)))) as proof of (((eq (a->Prop)) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28)))) b)
% Found eq_ref00:=(eq_ref0 (f x00)):(((eq Prop) (f x00)) (f x00))
% Found (eq_ref0 (f x00)) as proof of (((eq Prop) (f x00)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) x00) Xy))))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) x00) Xy))))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) x00) Xy))))
% Found (fun (x00:a)=> ((eq_ref Prop) (f x00))) as proof of (((eq Prop) (f x00)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) x00) Xy))))
% Found (fun (x00:a)=> ((eq_ref Prop) (f x00))) as proof of (forall (x0:a), (((eq Prop) (f x0)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) x0) Xy)))))
% Found eq_ref00:=(eq_ref0 (f x00)):(((eq Prop) (f x00)) (f x00))
% Found (eq_ref0 (f x00)) as proof of (((eq Prop) (f x00)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) x00) Xy))))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) x00) Xy))))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) x00) Xy))))
% Found (fun (x00:a)=> ((eq_ref Prop) (f x00))) as proof of (((eq Prop) (f x00)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) x00) Xy))))
% Found (fun (x00:a)=> ((eq_ref Prop) (f x00))) as proof of (forall (x0:a), (((eq Prop) (f x0)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) x0) Xy)))))
% Found eq_ref00:=(eq_ref0 (f x00)):(((eq Prop) (f x00)) (f x00))
% Found (eq_ref0 (f x00)) as proof of (((eq Prop) (f x00)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) x00)))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) x00)))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) x00)))
% Found (fun (x00:a)=> ((eq_ref Prop) (f x00))) as proof of (((eq Prop) (f x00)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) x00)))
% Found (fun (x00:a)=> ((eq_ref Prop) (f x00))) as proof of (forall (x0:a), (((eq Prop) (f x0)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) x0))))
% Found eq_ref00:=(eq_ref0 (f x00)):(((eq Prop) (f x00)) (f x00))
% Found (eq_ref0 (f x00)) as proof of (((eq Prop) (f x00)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) x00)))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) x00)))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) x00)))
% Found (fun (x00:a)=> ((eq_ref Prop) (f x00))) as proof of (((eq Prop) (f x00)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) x00)))
% Found (fun (x00:a)=> ((eq_ref Prop) (f x00))) as proof of (forall (x0:a), (((eq Prop) (f x0)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) x0))))
% Found eq_ref00:=(eq_ref0 (f x00)):(((eq Prop) (f x00)) (f x00))
% Found (eq_ref0 (f x00)) as proof of (((eq Prop) (f x00)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) x00)))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) x00)))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) x00)))
% Found (fun (x00:a)=> ((eq_ref Prop) (f x00))) as proof of (((eq Prop) (f x00)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) x00)))
% Found (fun (x00:a)=> ((eq_ref Prop) (f x00))) as proof of (forall (x0:a), (((eq Prop) (f x0)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) x0))))
% Found eq_ref00:=(eq_ref0 (f x00)):(((eq Prop) (f x00)) (f x00))
% Found (eq_ref0 (f x00)) as proof of (((eq Prop) (f x00)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) x00)))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) x00)))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) x00)))
% Found (fun (x00:a)=> ((eq_ref Prop) (f x00))) as proof of (((eq Prop) (f x00)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) x00)))
% Found (fun (x00:a)=> ((eq_ref Prop) (f x00))) as proof of (forall (x0:a), (((eq Prop) (f x0)) (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) x0))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))->(P (fun (x0:a)=> ((and (Xx x0)) (((eq a) Xy) (x x0))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eta_expansion_dep00 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found eq_ref00:=(eq_ref0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))):(((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (eq_ref0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq a) x00) Xy)))
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq a) x00) Xy)))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq a) x00) Xy)))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq a) x00) Xy)))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq a) x00) Xy)))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))->(P (fun (x0:a)=> ((and (Xx x0)) (((eq a) Xy) (x x0))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eta_expansion_dep00 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))):(((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (x0:a)=> ((and (Xx x0)) (((eq a) Xy) (x x0)))))
% Found (eta_expansion00 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq a) x00) Xy)))
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq a) x00) Xy)))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq a) x00) Xy)))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq a) x00) Xy)))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq a) x00) Xy)))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))->(P (fun (x0:a)=> ((and (Xx x0)) (((eq a) Xy) (x x0))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eta_expansion_dep00 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))->(P (fun (x0:a)=> ((and (Xx x0)) (((eq a) Xy) (x x0))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eta_expansion_dep00 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))->(P (fun (x0:a)=> ((and (Xx x0)) (((eq a) Xy) (x x0))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eta_expansion_dep00 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))->(P (fun (x0:a)=> ((and (Xx x0)) (((eq a) Xy) (x x0))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eta_expansion_dep00 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))->(P (fun (x0:a)=> ((and (Xx x0)) (((eq a) Xy) (x x0))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eta_expansion_dep00 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))):(((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (x0:a)=> ((and (Xx x0)) (((eq a) Xy) (x x0)))))
% Found (eta_expansion_dep00 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) 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) ((eq a) x00))
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) ((eq a) x00))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) ((eq a) x00))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) ((eq a) x00))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) ((eq a) x00))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))->(P (fun (x0:a)=> ((and (Xx x0)) (((eq a) Xy) (x x0))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eta_expansion_dep00 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))):(((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (x0:a)=> ((and (Xx x0)) (((eq a) Xy) (x x0)))))
% Found (eta_expansion_dep00 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) 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) ((eq a) x00))
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) ((eq a) x00))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) ((eq a) x00))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) ((eq a) x00))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) ((eq a) x00))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))->(P (fun (x0:a)=> ((and (Xx x0)) (((eq a) Xy) (x x0))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eta_expansion_dep00 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))->(P (fun (x0:a)=> ((and (Xx x0)) (((eq a) Xy) (x x0))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eta_expansion_dep00 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))->(P (fun (x0:a)=> ((and (Xx x0)) (((eq a) Xy) (x x0))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eta_expansion_dep00 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))->(P (fun (x0:a)=> ((and (Xx x0)) (((eq a) Xy) (x x0))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eta_expansion_dep00 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found eq_ref00:=(eq_ref0 (fun (Xy0:a)=> (((eq a) x00) Xy0))):(((eq (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found (eq_ref0 (fun (Xy0:a)=> (((eq a) x00) Xy0))) as proof of (((eq (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) as proof of (((eq (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) as proof of (((eq (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) as proof of (((eq (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy0:a)=> (((eq a) x00) Xy0))):(((eq (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) (fun (x:a)=> (((eq a) x00) x)))
% Found (eta_expansion_dep00 (fun (Xy0:a)=> (((eq a) x00) Xy0))) as proof of (((eq (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) as proof of (((eq (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) as proof of (((eq (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) as proof of (((eq (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) as proof of (((eq (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) b)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xy0:a)=> (((eq a) x00) Xy0)))->(P (fun (x:a)=> (((eq a) x00) x))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found ((eta_expansion_dep00 (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xy0:a)=> (((eq a) x00) Xy0)))->(P (fun (x:a)=> (((eq a) x00) x))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found ((eta_expansion_dep00 (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xy0:a)=> (((eq a) x00) Xy0)))->(P (fun (x:a)=> (((eq a) x00) x))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found ((eta_expansion_dep00 (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xy0:a)=> (((eq a) x00) Xy0)))->(P (fun (x:a)=> (((eq a) x00) x))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found ((eta_expansion_dep00 (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found eq_ref000:=(eq_ref00 P):((P ((and (Xx x01)) (((eq a) Xy) (x x01))))->(P ((and (Xx x01)) (((eq a) Xy) (x x01)))))
% Found (eq_ref00 P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref000:=(eq_ref00 P):((P ((and (Xx x01)) (((eq a) Xy) (x x01))))->(P ((and (Xx x01)) (((eq a) Xy) (x x01)))))
% Found (eq_ref00 P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref00:=(eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))):(((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found eq_ref00:=(eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))):(((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found eta_expansion000:=(eta_expansion00 ((eq a) x00)):(((eq (a->Prop)) ((eq a) x00)) (fun (x:a)=> (((eq a) x00) x)))
% Found (eta_expansion00 ((eq a) x00)) as proof of (((eq (a->Prop)) ((eq a) x00)) b)
% Found ((eta_expansion0 Prop) ((eq a) x00)) as proof of (((eq (a->Prop)) ((eq a) x00)) b)
% Found (((eta_expansion a) Prop) ((eq a) x00)) as proof of (((eq (a->Prop)) ((eq a) x00)) b)
% Found (((eta_expansion a) Prop) ((eq a) x00)) as proof of (((eq (a->Prop)) ((eq a) x00)) b)
% Found (((eta_expansion a) Prop) ((eq a) x00)) as proof of (((eq (a->Prop)) ((eq a) x00)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found eq_ref000:=(eq_ref00 P):((P ((and (Xx x01)) (((eq a) Xy) (x x01))))->(P ((and (Xx x01)) (((eq a) Xy) (x x01)))))
% Found (eq_ref00 P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref000:=(eq_ref00 P):((P ((and (Xx x01)) (((eq a) Xy) (x x01))))->(P ((and (Xx x01)) (((eq a) Xy) (x x01)))))
% Found (eq_ref00 P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref00:=(eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))):(((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found eq_ref00:=(eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))):(((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found eq_ref00:=(eq_ref0 ((eq a) x00)):(((eq (a->Prop)) ((eq a) x00)) ((eq a) x00))
% Found (eq_ref0 ((eq a) x00)) as proof of (((eq (a->Prop)) ((eq a) x00)) b)
% Found ((eq_ref (a->Prop)) ((eq a) x00)) as proof of (((eq (a->Prop)) ((eq a) x00)) b)
% Found ((eq_ref (a->Prop)) ((eq a) x00)) as proof of (((eq (a->Prop)) ((eq a) x00)) b)
% Found ((eq_ref (a->Prop)) ((eq a) x00)) as proof of (((eq (a->Prop)) ((eq a) x00)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P ((eq a) x00))->(P (fun (x:a)=> (((eq a) x00) x))))
% Found (eta_expansion_dep000 P) as proof of (P0 ((eq a) x00))
% Found ((eta_expansion_dep00 ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P ((eq a) x00))->(P (fun (x:a)=> (((eq a) x00) x))))
% Found (eta_expansion_dep000 P) as proof of (P0 ((eq a) x00))
% Found ((eta_expansion_dep00 ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P ((eq a) x00))->(P (fun (x:a)=> (((eq a) x00) x))))
% Found (eta_expansion_dep000 P) as proof of (P0 ((eq a) x00))
% Found ((eta_expansion_dep00 ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P ((eq a) x00))->(P (fun (x:a)=> (((eq a) x00) x))))
% Found (eta_expansion_dep000 P) as proof of (P0 ((eq a) x00))
% Found ((eta_expansion_dep00 ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found eq_ref000:=(eq_ref00 P):((P ((and (Xx x01)) (((eq a) Xy) (x x01))))->(P ((and (Xx x01)) (((eq a) Xy) (x x01)))))
% Found (eq_ref00 P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref000:=(eq_ref00 P):((P ((and (Xx x01)) (((eq a) Xy) (x x01))))->(P ((and (Xx x01)) (((eq a) Xy) (x x01)))))
% Found (eq_ref00 P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref00:=(eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))):(((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found eq_ref00:=(eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))):(((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found eq_ref000:=(eq_ref00 P):((P ((and (Xx x01)) (((eq a) Xy) (x x01))))->(P ((and (Xx x01)) (((eq a) Xy) (x x01)))))
% Found (eq_ref00 P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref000:=(eq_ref00 P):((P ((and (Xx x01)) (((eq a) Xy) (x x01))))->(P ((and (Xx x01)) (((eq a) Xy) (x x01)))))
% Found (eq_ref00 P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref00:=(eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))):(((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found eq_ref00:=(eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))):(((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found eq_ref00:=(eq_ref0 (((eq a) x00) x01)):(((eq Prop) (((eq a) x00) x01)) (((eq a) x00) x01))
% Found (eq_ref0 (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref00:=(eq_ref0 (((eq a) x00) x01)):(((eq Prop) (((eq a) x00) x01)) (((eq a) x00) x01))
% Found (eq_ref0 (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found eq_ref00:=(eq_ref0 (((eq a) x00) x01)):(((eq Prop) (((eq a) x00) x01)) (((eq a) x00) x01))
% Found (eq_ref0 (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref00:=(eq_ref0 (((eq a) x00) x01)):(((eq Prop) (((eq a) x00) x01)) (((eq a) x00) x01))
% Found (eq_ref0 (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref000:=(eq_ref00 P):((P (((eq a) x00) x01))->(P (((eq a) x00) x01)))
% Found (eq_ref00 P) as proof of (P0 (((eq a) x00) x01))
% Found ((eq_ref0 (((eq a) x00) x01)) P) as proof of (P0 (((eq a) x00) x01))
% Found (((eq_ref Prop) (((eq a) x00) x01)) P) as proof of (P0 (((eq a) x00) x01))
% Found (((eq_ref Prop) (((eq a) x00) x01)) P) as proof of (P0 (((eq a) x00) x01))
% Found eq_ref000:=(eq_ref00 P):((P (((eq a) x00) x01))->(P (((eq a) x00) x01)))
% Found (eq_ref00 P) as proof of (P0 (((eq a) x00) x01))
% Found ((eq_ref0 (((eq a) x00) x01)) P) as proof of (P0 (((eq a) x00) x01))
% Found (((eq_ref Prop) (((eq a) x00) x01)) P) as proof of (P0 (((eq a) x00) x01))
% Found (((eq_ref Prop) (((eq a) x00) x01)) P) as proof of (P0 (((eq a) x00) x01))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref00:=(eq_ref0 (((eq a) x00) x01)):(((eq Prop) (((eq a) x00) x01)) (((eq a) x00) x01))
% Found (eq_ref0 (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref00:=(eq_ref0 (((eq a) x00) x01)):(((eq Prop) (((eq a) x00) x01)) (((eq a) x00) x01))
% Found (eq_ref0 (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found eq_ref00:=(eq_ref0 (((eq a) x00) x01)):(((eq Prop) (((eq a) x00) x01)) (((eq a) x00) x01))
% Found (eq_ref0 (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref00:=(eq_ref0 (((eq a) x00) x01)):(((eq Prop) (((eq a) x00) x01)) (((eq a) x00) x01))
% Found (eq_ref0 (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref000:=(eq_ref00 P):((P (((eq a) x00) x01))->(P (((eq a) x00) x01)))
% Found (eq_ref00 P) as proof of (P0 (((eq a) x00) x01))
% Found ((eq_ref0 (((eq a) x00) x01)) P) as proof of (P0 (((eq a) x00) x01))
% Found (((eq_ref Prop) (((eq a) x00) x01)) P) as proof of (P0 (((eq a) x00) x01))
% Found (((eq_ref Prop) (((eq a) x00) x01)) P) as proof of (P0 (((eq a) x00) x01))
% Found eq_ref000:=(eq_ref00 P):((P (((eq a) x00) x01))->(P (((eq a) x00) x01)))
% Found (eq_ref00 P) as proof of (P0 (((eq a) x00) x01))
% Found ((eq_ref0 (((eq a) x00) x01)) P) as proof of (P0 (((eq a) x00) x01))
% Found (((eq_ref Prop) (((eq a) x00) x01)) P) as proof of (P0 (((eq a) x00) x01))
% Found (((eq_ref Prop) (((eq a) x00) x01)) P) as proof of (P0 (((eq a) x00) x01))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref00:=(eq_ref0 (((eq a) x00) x01)):(((eq Prop) (((eq a) x00) x01)) (((eq a) x00) x01))
% Found (eq_ref0 (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found eq_ref00:=(eq_ref0 (((eq a) x00) x01)):(((eq Prop) (((eq a) x00) x01)) (((eq a) x00) x01))
% Found (eq_ref0 (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref00:=(eq_ref0 (((eq a) x00) x01)):(((eq Prop) (((eq a) x00) x01)) (((eq a) x00) x01))
% Found (eq_ref0 (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref00:=(eq_ref0 (((eq a) x00) x01)):(((eq Prop) (((eq a) x00) x01)) (((eq a) x00) x01))
% Found (eq_ref0 (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref000:=(eq_ref00 P):((P (((eq a) x00) x01))->(P (((eq a) x00) x01)))
% Found (eq_ref00 P) as proof of (P0 (((eq a) x00) x01))
% Found ((eq_ref0 (((eq a) x00) x01)) P) as proof of (P0 (((eq a) x00) x01))
% Found (((eq_ref Prop) (((eq a) x00) x01)) P) as proof of (P0 (((eq a) x00) x01))
% Found (((eq_ref Prop) (((eq a) x00) x01)) P) as proof of (P0 (((eq a) x00) x01))
% Found eq_ref000:=(eq_ref00 P):((P (((eq a) x00) x01))->(P (((eq a) x00) x01)))
% Found (eq_ref00 P) as proof of (P0 (((eq a) x00) x01))
% Found ((eq_ref0 (((eq a) x00) x01)) P) as proof of (P0 (((eq a) x00) x01))
% Found (((eq_ref Prop) (((eq a) x00) x01)) P) as proof of (P0 (((eq a) x00) x01))
% Found (((eq_ref Prop) (((eq a) x00) x01)) P) as proof of (P0 (((eq a) x00) x01))
% Found eq_ref00:=(eq_ref0 (((eq a) x00) x01)):(((eq Prop) (((eq a) x00) x01)) (((eq a) x00) x01))
% Found (eq_ref0 (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref00:=(eq_ref0 (((eq a) x00) x01)):(((eq Prop) (((eq a) x00) x01)) (((eq a) x00) x01))
% Found (eq_ref0 (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref00:=(eq_ref0 (((eq a) x00) x01)):(((eq Prop) (((eq a) x00) x01)) (((eq a) x00) x01))
% Found (eq_ref0 (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref00:=(eq_ref0 (((eq a) x00) x01)):(((eq Prop) (((eq a) x00) x01)) (((eq a) x00) x01))
% Found (eq_ref0 (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found ((eq_ref Prop) (((eq a) x00) x01)) as proof of (((eq Prop) (((eq a) x00) x01)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref000:=(eq_ref00 P):((P (((eq a) x00) x01))->(P (((eq a) x00) x01)))
% Found (eq_ref00 P) as proof of (P0 (((eq a) x00) x01))
% Found ((eq_ref0 (((eq a) x00) x01)) P) as proof of (P0 (((eq a) x00) x01))
% Found (((eq_ref Prop) (((eq a) x00) x01)) P) as proof of (P0 (((eq a) x00) x01))
% Found (((eq_ref Prop) (((eq a) x00) x01)) P) as proof of (P0 (((eq a) x00) x01))
% Found eq_ref000:=(eq_ref00 P):((P (((eq a) x00) x01))->(P (((eq a) x00) x01)))
% Found (eq_ref00 P) as proof of (P0 (((eq a) x00) x01))
% Found ((eq_ref0 (((eq a) x00) x01)) P) as proof of (P0 (((eq a) x00) x01))
% Found (((eq_ref Prop) (((eq a) x00) x01)) P) as proof of (P0 (((eq a) x00) x01))
% Found (((eq_ref Prop) (((eq a) x00) x01)) P) as proof of (P0 (((eq a) x00) x01))
% Found eq_sym0:=(eq_sym Prop):(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a)))
% Instantiate: b:=(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))):Prop
% Found eq_sym0 as proof of b
% Found x01:(Xx Xx0)
% Instantiate: x:=(fun (x0:a)=> x0):(a->a)
% Found (fun (x01:(Xx Xx0))=> x01) as proof of (Xx (x Xx0))
% Found (fun (Xx0:a) (x01:(Xx Xx0))=> x01) as proof of ((Xx Xx0)->(Xx (x Xx0)))
% Found (fun (Xx0:a) (x01:(Xx Xx0))=> x01) as proof of (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))
% Found ((conj00 (fun (Xx0:a) (x01:(Xx Xx0))=> x01)) eq_sym0) as proof of ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))) b)
% Found (((conj0 b) (fun (Xx0:a) (x01:(Xx Xx0))=> x01)) eq_sym0) as proof of ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))) b)
% Found ((((conj (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))) b) (fun (Xx0:a) (x01:(Xx Xx0))=> x01)) eq_sym0) as proof of ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))) b)
% Found ((((conj (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))) b) (fun (Xx0:a) (x01:(Xx Xx0))=> x01)) eq_sym0) as proof of ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))) b)
% Found ((((conj (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))) b) (fun (Xx0:a) (x01:(Xx Xx0))=> x01)) eq_sym0) as proof of (P b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found eq_ref00:=(eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))):(((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found eq_ref00:=(eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))):(((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found eq_sym0:=(eq_sym Prop):(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a)))
% Instantiate: b:=(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))):Prop
% Found eq_sym0 as proof of b
% Found x01:(Xx Xx0)
% Instantiate: x:=(fun (x0:a)=> x0):(a->a)
% Found (fun (x01:(Xx Xx0))=> x01) as proof of (Xx (x Xx0))
% Found (fun (Xx0:a) (x01:(Xx Xx0))=> x01) as proof of ((Xx Xx0)->(Xx (x Xx0)))
% Found (fun (Xx0:a) (x01:(Xx Xx0))=> x01) as proof of (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))
% Found ((conj00 (fun (Xx0:a) (x01:(Xx Xx0))=> x01)) eq_sym0) as proof of ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))) b)
% Found (((conj0 b) (fun (Xx0:a) (x01:(Xx Xx0))=> x01)) eq_sym0) as proof of ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))) b)
% Found ((((conj (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))) b) (fun (Xx0:a) (x01:(Xx Xx0))=> x01)) eq_sym0) as proof of ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))) b)
% Found ((((conj (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))) b) (fun (Xx0:a) (x01:(Xx Xx0))=> x01)) eq_sym0) as proof of ((and (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))) b)
% Found ((((conj (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))) b) (fun (Xx0:a) (x01:(Xx Xx0))=> x01)) eq_sym0) as proof of (P b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found eq_ref00:=(eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))):(((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found eq_ref00:=(eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))):(((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found x01:(Xx Xx0)
% Instantiate: x:=(fun (x0:a)=> x0):(a->a)
% Found (fun (x01:(Xx Xx0))=> x01) as proof of (Xx (x Xx0))
% Found (fun (Xx0:a) (x01:(Xx Xx0))=> x01) as proof of ((Xx Xx0)->(Xx (x Xx0)))
% Found (fun (Xx0:a) (x01:(Xx Xx0))=> x01) as proof of (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found eq_ref00:=(eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))):(((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found eq_ref00:=(eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))):(((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found eq_ref00:=(eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))):(((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found eq_ref00:=(eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))):(((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))->(P (fun (x0:a)=> ((and (Xx x0)) (((eq a) Xy) (x x0))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eta_expansion_dep00 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))):(((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (x0:a)=> ((and (Xx x0)) (((eq a) Xy) (x x0)))))
% Found (eta_expansion_dep00 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq a) x00) Xy)))
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq a) x00) Xy)))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq a) x00) Xy)))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq a) x00) Xy)))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq a) x00) Xy)))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))->(P (fun (x0:a)=> ((and (Xx x0)) (((eq a) Xy) (x x0))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eta_expansion_dep00 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found x01:(Xx Xx0)
% Instantiate: x:=(fun (x0:a)=> x0):(a->a)
% Found (fun (x01:(Xx Xx0))=> x01) as proof of (Xx (x Xx0))
% Found (fun (Xx0:a) (x01:(Xx Xx0))=> x01) as proof of ((Xx Xx0)->(Xx (x Xx0)))
% Found (fun (Xx0:a) (x01:(Xx Xx0))=> x01) as proof of (forall (Xx0:a), ((Xx Xx0)->(Xx (x Xx0))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))):(((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (x0:a)=> ((and (Xx x0)) (((eq a) Xy) (x x0)))))
% Found (eta_expansion_dep00 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq a) x00) Xy)))
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq a) x00) Xy)))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq a) x00) Xy)))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq a) x00) Xy)))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq a) x00) Xy)))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))->(P (fun (x0:a)=> ((and (Xx x0)) (((eq a) Xy) (x x0))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eta_expansion_dep00 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))->(P (fun (x0:a)=> ((and (Xx x0)) (((eq a) Xy) (x x0))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eta_expansion_dep00 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))->(P (fun (x0:a)=> ((and (Xx x0)) (((eq a) Xy) (x x0))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eta_expansion_dep00 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))->(P (fun (x0:a)=> ((and (Xx x0)) (((eq a) Xy) (x x0))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eta_expansion_dep00 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))->(P (fun (x0:a)=> ((and (Xx x0)) (((eq a) Xy) (x x0))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eta_expansion_dep00 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found eq_ref00:=(eq_ref0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))):(((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (eq_ref0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) 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) ((eq a) x00))
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) ((eq a) x00))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) ((eq a) x00))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) ((eq a) x00))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) ((eq a) x00))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))->(P (fun (x0:a)=> ((and (Xx x0)) (((eq a) Xy) (x x0))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eta_expansion_dep00 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found eq_ref00:=(eq_ref0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))):(((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (eq_ref0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) 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) ((eq a) x00))
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) ((eq a) x00))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) ((eq a) x00))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) ((eq a) x00))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) ((eq a) x00))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))->(P (fun (x0:a)=> ((and (Xx x0)) (((eq a) Xy) (x x0))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eta_expansion_dep00 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))->(P (fun (x0:a)=> ((and (Xx x0)) (((eq a) Xy) (x x0))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eta_expansion_dep00 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))->(P (fun (x0:a)=> ((and (Xx x0)) (((eq a) Xy) (x x0))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eta_expansion_dep00 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))->(P (fun (x0:a)=> ((and (Xx x0)) (((eq a) Xy) (x x0))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eta_expansion_dep00 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found eq_ref00:=(eq_ref0 (fun (Xy0:a)=> (((eq a) x00) Xy0))):(((eq (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found (eq_ref0 (fun (Xy0:a)=> (((eq a) x00) Xy0))) as proof of (((eq (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) as proof of (((eq (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) as proof of (((eq (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) as proof of (((eq (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found eq_ref00:=(eq_ref0 (fun (Xy0:a)=> (((eq a) x00) Xy0))):(((eq (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found (eq_ref0 (fun (Xy0:a)=> (((eq a) x00) Xy0))) as proof of (((eq (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) as proof of (((eq (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) as proof of (((eq (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) as proof of (((eq (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found eq_ref00:=(eq_ref0 (fun (Xy0:a)=> (((eq a) x00) Xy0))):(((eq (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found (eq_ref0 (fun (Xy0:a)=> (((eq a) x00) Xy0))) as proof of (((eq (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) as proof of (((eq (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) as proof of (((eq (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) as proof of (((eq (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xy0:a)=> (((eq a) x00) Xy0))):(((eq (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) (fun (x:a)=> (((eq a) x00) x)))
% Found (eta_expansion00 (fun (Xy0:a)=> (((eq a) x00) Xy0))) as proof of (((eq (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) b)
% Found ((eta_expansion0 Prop) (fun (Xy0:a)=> (((eq a) x00) Xy0))) as proof of (((eq (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) b)
% Found (((eta_expansion a) Prop) (fun (Xy0:a)=> (((eq a) x00) Xy0))) as proof of (((eq (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) b)
% Found (((eta_expansion a) Prop) (fun (Xy0:a)=> (((eq a) x00) Xy0))) as proof of (((eq (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) b)
% Found (((eta_expansion a) Prop) (fun (Xy0:a)=> (((eq a) x00) Xy0))) as proof of (((eq (a->Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% 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 (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xy0:a)=> (((eq a) x00) Xy0)))->(P (fun (x:a)=> (((eq a) x00) x))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found ((eta_expansion_dep00 (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xy0:a)=> (((eq a) x00) Xy0)))->(P (fun (x:a)=> (((eq a) x00) x))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found ((eta_expansion_dep00 (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xy0:a)=> (((eq a) x00) Xy0)))->(P (fun (x:a)=> (((eq a) x00) x))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found ((eta_expansion_dep00 (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xy0:a)=> (((eq a) x00) Xy0)))->(P (fun (x:a)=> (((eq a) x00) x))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found ((eta_expansion_dep00 (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xy0:a)=> (((eq a) x00) Xy0)))->(P (fun (x:a)=> (((eq a) x00) x))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found ((eta_expansion_dep00 (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xy0:a)=> (((eq a) x00) Xy0)))->(P (fun (x:a)=> (((eq a) x00) x))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found ((eta_expansion_dep00 (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xy0:a)=> (((eq a) x00) Xy0)))->(P (fun (x:a)=> (((eq a) x00) x))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found ((eta_expansion_dep00 (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xy0:a)=> (((eq a) x00) Xy0)))->(P (fun (x:a)=> (((eq a) x00) x))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found ((eta_expansion_dep00 (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xy0:a)=> (((eq a) x00) Xy0))) P) as proof of (P0 (fun (Xy0:a)=> (((eq a) x00) Xy0)))
% Found eq_ref000:=(eq_ref00 P):((P ((and (Xx x01)) (((eq a) Xy) (x x01))))->(P ((and (Xx x01)) (((eq a) Xy) (x x01)))))
% Found (eq_ref00 P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref000:=(eq_ref00 P):((P ((and (Xx x01)) (((eq a) Xy) (x x01))))->(P ((and (Xx x01)) (((eq a) Xy) (x x01)))))
% Found (eq_ref00 P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref00:=(eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))):(((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found eq_ref00:=(eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))):(((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found eq_ref000:=(eq_ref00 P):((P ((and (Xx x01)) (((eq a) Xy) (x x01))))->(P ((and (Xx x01)) (((eq a) Xy) (x x01)))))
% Found (eq_ref00 P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref000:=(eq_ref00 P):((P ((and (Xx x01)) (((eq a) Xy) (x x01))))->(P ((and (Xx x01)) (((eq a) Xy) (x x01)))))
% Found (eq_ref00 P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref00:=(eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))):(((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found eq_ref00:=(eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))):(((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((eq a) x00)):(((eq (a->Prop)) ((eq a) x00)) (fun (x:a)=> (((eq a) x00) x)))
% Found (eta_expansion_dep00 ((eq a) x00)) as proof of (((eq (a->Prop)) ((eq a) x00)) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) ((eq a) x00)) as proof of (((eq (a->Prop)) ((eq a) x00)) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) ((eq a) x00)) as proof of (((eq (a->Prop)) ((eq a) x00)) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) ((eq a) x00)) as proof of (((eq (a->Prop)) ((eq a) x00)) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) ((eq a) x00)) as proof of (((eq (a->Prop)) ((eq a) x00)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((eq a) x00)):(((eq (a->Prop)) ((eq a) x00)) (fun (x:a)=> (((eq a) x00) x)))
% Found (eta_expansion_dep00 ((eq a) x00)) as proof of (((eq (a->Prop)) ((eq a) x00)) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) ((eq a) x00)) as proof of (((eq (a->Prop)) ((eq a) x00)) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) ((eq a) x00)) as proof of (((eq (a->Prop)) ((eq a) x00)) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) ((eq a) x00)) as proof of (((eq (a->Prop)) ((eq a) x00)) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) ((eq a) x00)) as proof of (((eq (a->Prop)) ((eq a) x00)) b)
% 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 (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:a), ((Xx Xy)->((ex a) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) ((eq a) Xy_28)))))))
% Found eq_ref000:=(eq_ref00 P):((P ((and (Xx x01)) (((eq a) Xy) (x x01))))->(P ((and (Xx x01)) (((eq a) Xy) (x x01)))))
% Found (eq_ref00 P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref000:=(eq_ref00 P):((P ((and (Xx x01)) (((eq a) Xy) (x x01))))->(P ((and (Xx x01)) (((eq a) Xy) (x x01)))))
% Found (eq_ref00 P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref00:=(eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))):(((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found eq_ref00:=(eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))):(((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found eq_ref000:=(eq_ref00 P):((P ((and (Xx x01)) (((eq a) Xy) (x x01))))->(P ((and (Xx x01)) (((eq a) Xy) (x x01)))))
% Found (eq_ref00 P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref000:=(eq_ref00 P):((P ((and (Xx x01)) (((eq a) Xy) (x x01))))->(P ((and (Xx x01)) (((eq a) Xy) (x x01)))))
% Found (eq_ref00 P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref00:=(eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))):(((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found eq_ref00:=(eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))):(((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found eta_expansion000:=(eta_expansion00 ((eq a) x00)):(((eq (a->Prop)) ((eq a) x00)) (fun (x:a)=> (((eq a) x00) x)))
% Found (eta_expansion00 ((eq a) x00)) as proof of (((eq (a->Prop)) ((eq a) x00)) b)
% Found ((eta_expansion0 Prop) ((eq a) x00)) as proof of (((eq (a->Prop)) ((eq a) x00)) b)
% Found (((eta_expansion a) Prop) ((eq a) x00)) as proof of (((eq (a->Prop)) ((eq a) x00)) b)
% Found (((eta_expansion a) Prop) ((eq a) x00)) as proof of (((eq (a->Prop)) ((eq a) x00)) b)
% Found (((eta_expansion a) Prop) ((eq a) x00)) as proof of (((eq (a->Prop)) ((eq a) x00)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0)))))
% Found eq_ref00:=(eq_ref0 ((eq a) x00)):(((eq (a->Prop)) ((eq a) x00)) ((eq a) x00))
% Found (eq_ref0 ((eq a) x00)) as proof of (((eq (a->Prop)) ((eq a) x00)) b)
% Found ((eq_ref (a->Prop)) ((eq a) x00)) as proof of (((eq (a->Prop)) ((eq a) x00)) b)
% Found ((eq_ref (a->Prop)) ((eq a) x00)) as proof of (((eq (a->Prop)) ((eq a) x00)) b)
% Found ((eq_ref (a->Prop)) ((eq a) x00)) as proof of (((eq (a->Prop)) ((eq a) x00)) b)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P ((eq a) x00))->(P (fun (x:a)=> (((eq a) x00) x))))
% Found (eta_expansion_dep000 P) as proof of (P0 ((eq a) x00))
% Found ((eta_expansion_dep00 ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P ((eq a) x00))->(P (fun (x:a)=> (((eq a) x00) x))))
% Found (eta_expansion_dep000 P) as proof of (P0 ((eq a) x00))
% Found ((eta_expansion_dep00 ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P ((eq a) x00))->(P (fun (x:a)=> (((eq a) x00) x))))
% Found (eta_expansion_dep000 P) as proof of (P0 ((eq a) x00))
% Found ((eta_expansion_dep00 ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P ((eq a) x00))->(P (fun (x:a)=> (((eq a) x00) x))))
% Found (eta_expansion_dep000 P) as proof of (P0 ((eq a) x00))
% Found ((eta_expansion_dep00 ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P ((eq a) x00))->(P (fun (x:a)=> (((eq a) x00) x))))
% Found (eta_expansion_dep000 P) as proof of (P0 ((eq a) x00))
% Found ((eta_expansion_dep00 ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P ((eq a) x00))->(P (fun (x:a)=> (((eq a) x00) x))))
% Found (eta_expansion_dep000 P) as proof of (P0 ((eq a) x00))
% Found ((eta_expansion_dep00 ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P ((eq a) x00))->(P (fun (x:a)=> (((eq a) x00) x))))
% Found (eta_expansion_dep000 P) as proof of (P0 ((eq a) x00))
% Found ((eta_expansion_dep00 ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P ((eq a) x00))->(P (fun (x:a)=> (((eq a) x00) x))))
% Found (eta_expansion_dep000 P) as proof of (P0 ((eq a) x00))
% Found ((eta_expansion_dep00 ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) ((eq a) x00)) P) as proof of (P0 ((eq a) x00))
% Found eq_ref00:=(eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))):(((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found eq_ref00:=(eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))):(((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found eq_ref000:=(eq_ref00 P):((P ((and (Xx x01)) (((eq a) Xy) (x x01))))->(P ((and (Xx x01)) (((eq a) Xy) (x x01)))))
% Found (eq_ref00 P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref000:=(eq_ref00 P):((P ((and (Xx x01)) (((eq a) Xy) (x x01))))->(P ((and (Xx x01)) (((eq a) Xy) (x x01)))))
% Found (eq_ref00 P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref000:=(eq_ref00 P):((P ((and (Xx x01)) (((eq a) Xy) (x x01))))->(P ((and (Xx x01)) (((eq a) Xy) (x x01)))))
% Found (eq_ref00 P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref000:=(eq_ref00 P):((P ((and (Xx x01)) (((eq a) Xy) (x x01))))->(P ((and (Xx x01)) (((eq a) Xy) (x x01)))))
% Found (eq_ref00 P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref00:=(eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))):(((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found eq_ref00:=(eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))):(((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found eq_ref000:=(eq_ref00 P):((P ((and (Xx x01)) (((eq a) Xy) (x x01))))->(P ((and (Xx x01)) (((eq a) Xy) (x x01)))))
% Found (eq_ref00 P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref00:=(eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))):(((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found eq_ref00:=(eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))):(((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found eq_ref000:=(eq_ref00 P):((P ((and (Xx x01)) (((eq a) Xy) (x x01))))->(P ((and (Xx x01)) (((eq a) Xy) (x x01)))))
% Found (eq_ref00 P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref00:=(eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))):(((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found eq_ref00:=(eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))):(((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found ((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) as proof of (((eq Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) x00) x01))
% Found eq_ref000:=(eq_ref00 P):((P ((and (Xx x01)) (((eq a) Xy) (x x01))))->(P ((and (Xx x01)) (((eq a) Xy) (x x01)))))
% Found (eq_ref00 P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eq_ref000:=(eq_ref00 P):((P ((and (Xx x01)) (((eq a) Xy) (x x01))))->(P ((and (Xx x01)) (((eq a) Xy) (x x01)))))
% Found (eq_ref00 P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found ((eq_ref0 ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found (((eq_ref Prop) ((and (Xx x01)) (((eq a) Xy) (x x01)))) P) as proof of (P0 ((and (Xx x01)) (((eq a) Xy) (x x01))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq ((a->a)->Prop)) b) (fun (x:(a->a))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq ((a->a)->Prop)) b) b0)
% Found ((eta_expansion_dep0 (fun (x1:(a->a))=> Prop)) b) as proof of (((eq ((a->a)->Prop)) b) b0)
% Found (((eta_expansion_dep (a->a)) (fun (x1:(a->a))=> Prop)) b) as proof of (((eq ((a->a)->Prop)) b) b0)
% Found (((eta_expansion_dep (a->a)) (fun (x1:(a->a))=> Prop)) b) as proof of (((eq ((a->a)->Prop)) b) b0)
% Found (((eta_expansion_dep (a->a)) (fun (x1:(a->a))=> Prop)) b) as proof of (((eq ((a->a)->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->a)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->a)->Prop)) b) b0)
% Found ((eq_ref ((a->a)->Prop)) b) as proof of (((eq ((a->a)->Prop)) b) b0)
% Found ((eq_ref ((a->a)->Prop)) b) as proof of (((eq ((a->a)->Prop)) b) b0)
% Found ((eq_ref ((a->a)->Prop)) b) as proof of (((eq ((a->a)->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(a->a))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(a->a))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(a->a)), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(a->a))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(a->a))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(a->a)), (((eq Prop) (f0 x)) (f x)))
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_28) Xy)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy_28:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy) (x X
% EOF
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