%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEU864^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n103.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 13:33:18 EDT 2014

% Result   : Timeout 300.06s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEU864^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n103.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 11:37:36 CDT 2014
% % CPUTime  : 300.06 
% 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 0x8a06c8>, <kernel.Type object at 0x8a0ef0>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0xc78200>, <kernel.Constant object at 0x8a0830>) of role type named t
% Using role type
% Declaring t:a
% FOF formula (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt_0:a), ((((eq a) t) Xt_0)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt_0) Xz)))))))))->(X (fun (Xy:a)=> (((eq a) t) Xy))))) of role conjecture named cDOMLEMMA1_pme
% Conjecture to prove = (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt_0:a), ((((eq a) t) Xt_0)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt_0) Xz)))))))))->(X (fun (Xy:a)=> (((eq a) t) Xy))))):Prop
% We need to prove ['(forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt_0:a), ((((eq a) t) Xt_0)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt_0) Xz)))))))))->(X (fun (Xy:a)=> (((eq a) t) Xy)))))']
% Parameter a:Type.
% Parameter t:a.
% Trying to prove (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt_0:a), ((((eq a) t) Xt_0)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt_0) Xz)))))))))->(X (fun (Xy:a)=> (((eq a) t) Xy)))))
% Found eq_ref00:=(eq_ref0 (fun (Xy:a)=> (((eq a) t) Xy))):(((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) (fun (Xy:a)=> (((eq a) t) Xy)))
% Found (eq_ref0 (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((eq a) t)):(((eq (a->Prop)) ((eq a) t)) (fun (x:a)=> (((eq a) t) x)))
% Found (eta_expansion_dep00 ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (((eq a) t) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq a) t) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq a) t) x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((eq a) t) x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (((eq a) t) x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (((eq a) t) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq a) t) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq a) t) x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((eq a) t) x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (((eq a) t) x)))
% Found eq_ref00:=(eq_ref0 (fun (Xy:a)=> (((eq a) t) Xy))):(((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) (fun (Xy:a)=> (((eq a) t) Xy)))
% Found (eq_ref0 (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (((eq a) t) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq a) t) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq a) t) x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((eq a) t) x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (((eq a) t) x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (((eq a) t) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq a) t) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq a) t) x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((eq a) t) x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (((eq a) t) x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((eq a) t)):(((eq (a->Prop)) ((eq a) t)) (fun (x:a)=> (((eq a) t) x)))
% Found (eta_expansion_dep00 ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (((eq a) t) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((eq a) t) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((eq a) t) x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (((eq a) t) x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (((eq a) t) x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (((eq a) t) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((eq a) t) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((eq a) t) x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (((eq a) t) x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (((eq a) t) x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (((eq a) t) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((eq a) t) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((eq a) t) x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (((eq a) t) x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (((eq a) t) x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (((eq a) t) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((eq a) t) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((eq a) t) x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (((eq a) t) x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (((eq a) t) x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq a) t) Xy)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq a) t) Xy)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq a) t) Xy)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq a) t) 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:a)=> (((eq a) t) Xy)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq a) t) Xy)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq a) t) Xy)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq a) t) Xy)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1: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) ((eq a) t))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) ((eq a) t))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) ((eq a) t))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) ((eq a) t))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) ((eq a) t))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) ((eq a) t))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) ((eq a) t))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) ((eq a) t))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) ((eq a) t))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq a) t) Xy)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq a) t) Xy)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq a) t) Xy)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq a) t) Xy)))
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq a) t) Xy)))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq a) t) Xy)))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq a) t) Xy)))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq a) t) Xy)))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (((eq a) t) Xy)))
% 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_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) t))
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) ((eq a) t))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) ((eq a) t))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) ((eq a) t))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) ((eq a) t))
% 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) ((eq a) t))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) ((eq a) t))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) ((eq a) t))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) ((eq a) t))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) ((eq a) t))
% Found eq_ref00:=(eq_ref0 (fun (Xy:a)=> (((eq a) t) Xy))):(((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) (fun (Xy:a)=> (((eq a) t) Xy)))
% Found (eq_ref0 (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((eq a) t)):(((eq (a->Prop)) ((eq a) t)) (fun (x:a)=> (((eq a) t) x)))
% Found (eta_expansion_dep00 ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 f):(((eq (a->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) 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) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 f):(((eq (a->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 f):(((eq (a->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 f):(((eq (a->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eta_expansion000:=(eta_expansion00 (fun (Xy:a)=> (((eq a) t) Xy))):(((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) (fun (x:a)=> (((eq a) t) x)))
% Found (eta_expansion00 (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b0)
% Found ((eta_expansion0 Prop) (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b0)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b0)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b0)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b0)
% Found eq_ref00:=(eq_ref0 f):(((eq (a->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 f):(((eq (a->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((eq a) t)):(((eq (a->Prop)) ((eq a) t)) (fun (x:a)=> (((eq a) t) x)))
% Found (eta_expansion_dep00 ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b0)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b0)
% Found eq_ref00:=(eq_ref0 (fun (Xy:a)=> (((eq a) t) Xy))):(((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) (fun (Xy:a)=> (((eq a) t) Xy)))
% Found (eq_ref0 (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b0)
% 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) b0)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 f):(((eq (a->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 f):(((eq (a->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) 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) b0)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((eq a) t)):(((eq (a->Prop)) ((eq a) t)) (fun (x:a)=> (((eq a) t) x)))
% Found (eta_expansion_dep00 ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b0)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b0)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eta_expansion000:=(eta_expansion00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) 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) b0)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 f):(((eq (a->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eta_expansion000:=(eta_expansion00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eta_expansion000:=(eta_expansion00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eta_expansion000:=(eta_expansion00 (fun (Xy:a)=> (((eq a) t) Xy))):(((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) (fun (x:a)=> (((eq a) t) x)))
% Found (eta_expansion00 (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b0)
% Found ((eta_expansion0 Prop) (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b0)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b0)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b0)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b0)
% 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) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 ((eq a) t)):(((eq (a->Prop)) ((eq a) t)) ((eq a) t))
% Found (eq_ref0 ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b0)
% Found ((eq_ref (a->Prop)) ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b0)
% Found ((eq_ref (a->Prop)) ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b0)
% Found ((eq_ref (a->Prop)) ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b0)
% Found eq_ref00:=(eq_ref0 f):(((eq (a->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 f):(((eq (a->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) 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) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 f):(((eq (a->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 f):(((eq (a->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eta_expansion000:=(eta_expansion00 (fun (Xy:a)=> (((eq a) t) Xy))):(((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) (fun (x:a)=> (((eq a) t) x)))
% Found (eta_expansion00 (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b0)
% Found ((eta_expansion0 Prop) (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b0)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b0)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b0)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b0)
% Found eq_ref00:=(eq_ref0 f):(((eq (a->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 f):(((eq (a->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) 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) b0)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((eq a) t)):(((eq (a->Prop)) ((eq a) t)) (fun (x:a)=> (((eq a) t) x)))
% Found (eta_expansion_dep00 ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b0)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xy:a)=> (((eq a) t) Xy))):(((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) (fun (x:a)=> (((eq a) t) x)))
% Found (eta_expansion00 (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b0)
% Found ((eta_expansion0 Prop) (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b0)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b0)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b0)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b0)
% Found eq_ref00:=(eq_ref0 (fun (Xy:a)=> (((eq a) t) Xy))):(((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) (fun (Xy:a)=> (((eq a) t) Xy)))
% Found (eq_ref0 (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq a) t) Xy))) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 f):(((eq (a->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion000:=(eta_expansion00 ((eq a) t)):(((eq (a->Prop)) ((eq a) t)) (fun (x:a)=> (((eq a) t) x)))
% Found (eta_expansion00 ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b0)
% Found ((eta_expansion0 Prop) ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b0)
% Found (((eta_expansion a) Prop) ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b0)
% Found (((eta_expansion a) Prop) ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b0)
% Found (((eta_expansion a) Prop) ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b0)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((eq a) t)):(((eq (a->Prop)) ((eq a) t)) (fun (x:a)=> (((eq a) t) x)))
% Found (eta_expansion_dep00 ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b0)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) ((eq a) t)) as proof of (((eq (a->Prop)) ((eq a) t)) b0)
% 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) b0)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% 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) b0)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 f):(((eq (a->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eta_expansion000:=(eta_expansion00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eta_expansion000:=(eta_expansion00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eta_expansion000:=(eta_expansion00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 f):(((eq (a->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found ((eq_ref (a->Prop)) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eta_expansion000:=(eta_expansion00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (a->Prop)) f) (fun (x:a)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (a->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found (((eta_expansion a) Prop) f) as proof of (((eq (a->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% 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) b0)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% 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) b0)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% 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) b0)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) 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) f)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% 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) b0)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 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) b0)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% 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) b0)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% 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) b0)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% 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) b0)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% 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) f)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% 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) b0)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% 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) b0)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% 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) f)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) f)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) f)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) f)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% 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) b0)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% 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) b0)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% 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) f)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) f)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) f)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) f)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) 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) f)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) f)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) f)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) f)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) f)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) 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) f)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) 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) f)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% 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 eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (a0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eta_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) f)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found x3:(P2 (f x2))
% Instantiate: b:=f:(a->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (b x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (b x2)))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found x3:(P2 (f x2))
% Instantiate: b:=f:(a->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (b x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (b x2)))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found x3:(P2 (f x2))
% Instantiate: f0:=f:(a->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (f0 x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (f0 x2)))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:a), (((eq Prop) (f x)) (f0 x)))
% Found x3:(P2 (f x2))
% Instantiate: f0:=f:(a->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (f0 x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (f0 x2)))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:a), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found x3:(P2 (f x2))
% Instantiate: b:=f:(a->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (b x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (b x2)))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found x3:(P2 (f x2))
% Instantiate: b:=f:(a->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (b x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (b x2)))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found x3:(P2 (f x2))
% Instantiate: b:=f:(a->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (b x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (b x2)))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found x3:(P2 (f x2))
% Instantiate: b:=f:(a->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (b x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (b x2)))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found x3:(P2 (f x2))
% Instantiate: f0:=f:(a->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (f0 x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (f0 x2)))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:a), (((eq Prop) (f x)) (f0 x)))
% Found x3:(P2 (f x2))
% Instantiate: f0:=f:(a->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (f0 x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (f0 x2)))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:a), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found x3:(P2 (f x2))
% Instantiate: b:=f:(a->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (b x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (b x2)))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found x3:(P2 (f x2))
% Instantiate: b:=f:(a->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (b x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (b x2)))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found x3:(P2 (f x2))
% Instantiate: b:=f:(a->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (b x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (b x2)))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found x3:(P2 (f x2))
% Instantiate: b:=f:(a->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (b x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (b x2)))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (f0 x)))
% Found x3:(P2 (f x2))
% Instantiate: f0:=f:(a->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (f0 x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (f0 x2)))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:a), (((eq Prop) (f x)) (f0 x)))
% Found x3:(P2 (f x2))
% Instantiate: f0:=f:(a->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (f0 x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (f0 x2)))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:a), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found x3:(P2 (f x2))
% Instantiate: b:=f:(a->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (b x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (b x2)))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found x3:(P2 (f x2))
% Instantiate: b:=f:(a->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (b x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (b x2)))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (b x2))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found x3:(P2 (f x2))
% Instantiate: f0:=f:(a->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (f0 x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (f0 x2)))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:a), (((eq Prop) (f x)) (f0 x)))
% Found x3:(P2 (f x2))
% Instantiate: f0:=f:(a->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (f0 x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (f0 x2)))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:a), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found x3:(P2 (f x2))
% Instantiate: f0:=f:(a->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (f0 x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (f0 x2)))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:a), (((eq Prop) (f x)) (f0 x)))
% Found x3:(P2 (f x2))
% Instantiate: f0:=f:(a->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (f0 x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (f0 x2)))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:a), (((eq Prop) (f x)) (f0 x)))
% Found x1:(P2 (f x0))
% Instantiate: b:=f:(a->Prop)
% Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (b x0))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (b x0)))
% Found (fun (x0:a) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found x1:(P2 (f x0))
% Instantiate: b:=f:(a->Prop)
% Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (b x0))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (b x0)))
% Found (fun (x0:a) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found x1:(P2 (f x0))
% Instantiate: b:=f:(a->Prop)
% Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (b x0))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (b x0)))
% Found (fun (x0:a) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found x1:(P2 (f x0))
% Instantiate: b:=f:(a->Prop)
% Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (b x0))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (b x0)))
% Found (fun (x0:a) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found x3:(P2 (f x2))
% Instantiate: f0:=f:(a->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (f0 x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (f0 x2)))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:a), (((eq Prop) (f x)) (f0 x)))
% Found x3:(P2 (f x2))
% Instantiate: f0:=f:(a->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (f0 x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (f0 x2)))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (f0 x2))
% Found (fun (x2:a) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:a), (((eq Prop) (f x)) (f0 x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% 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) b0)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% 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) b0)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eta_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) f)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found x1:(P2 (f x0))
% Instantiate: f0:=f:(a->Prop)
% Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (f0 x0))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (f0 x0)))
% Found (fun (x0:a) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:a) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:a), (((eq Prop) (f x)) (f0 x)))
% Found x1:(P2 (f x0))
% Instantiate: f0:=f:(a->Prop)
% Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (f0 x0))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (f0 x0)))
% Found (fun (x0:a) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:a) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:a), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% 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) b0)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a0 x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (a0 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) b0)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% 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) b0)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% 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) f)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1: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) f)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) f)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) f)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) f)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% 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) b0)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eta_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 eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) f)
% Found ((eq_ref (a->Prop)) b) as proof
% EOF
%------------------------------------------------------------------------------