TSTP Solution File: SEU897^5 by cocATP---0.2.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEU897^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n100.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 13:33:21 EDT 2014
% Result : Timeout 300.02s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEU897^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n100.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 11:41:21 CDT 2014
% % CPUTime : 300.02
% 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 0x21bbdd0>, <kernel.Type object at 0x21bb440>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x2399440>, <kernel.DependentProduct object at 0x21bb3f8>) of role type named cS
% Using role type
% Declaring cS:(a->Prop)
% FOF formula (<kernel.Constant object at 0x21bb7e8>, <kernel.DependentProduct object at 0x21bb758>) of role type named cR
% Using role type
% Declaring cR:(a->Prop)
% FOF formula ((iff (forall (Xx:a), ((cR Xx)->(cS Xx)))) (forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))) of role conjecture named cTHM30_pme
% Conjecture to prove = ((iff (forall (Xx:a), ((cR Xx)->(cS Xx)))) (forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['((iff (forall (Xx:a), ((cR Xx)->(cS Xx)))) (forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))))']
% Parameter a:Type.
% Parameter cS:(a->Prop).
% Parameter cR:(a->Prop).
% Trying to prove ((iff (forall (Xx:a), ((cR Xx)->(cS Xx)))) (forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))))
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))
% Found (eq_ref00 (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eq_ref0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (x:a)=> ((and (cR x)) (((eq a) Xx) (F x))))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eta_expansion00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion0 Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (x:a)=> ((and (cR x)) (((eq a) Xx) (F x))))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eta_expansion00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion0 Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (x:a)=> ((and (cR x)) (((eq a) Xx) (F x))))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eta_expansion00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion0 Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (x:a)=> ((and (cR x)) (((eq a) Xx) (F x))))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eta_expansion00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion0 Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (x:a)=> ((and (cR x)) (((eq a) Xx) (F x))))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eta_expansion00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion0 Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Instantiate: b:=(fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))):(a->Prop)
% Found x0 as proof of (P b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))):(((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) (fun (x:a)=> ((and (cS x)) (((eq a) Xx) (F x)))))
% Found (eta_expansion00 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found ((eta_expansion0 Prop) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Instantiate: f:=(fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))):(a->Prop)
% Found x0 as proof of (P f)
% Found x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Instantiate: f:=(fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))):(a->Prop)
% Found x0 as proof of (P f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))):(((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (fun (x:a)=> ((and (cR x)) (((eq a) Xx) (F x)))))
% Found (eta_expansion_dep00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (F x)))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (F x)))))
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))
% Found (eq_ref00 (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eq_ref0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))):(((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (fun (x:a)=> ((and (cR x)) (((eq a) Xx) (F x)))))
% Found (eta_expansion_dep00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Instantiate: b:=(fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))):(a->Prop)
% Found x0 as proof of (P b)
% Found eq_ref00:=(eq_ref0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))):(((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (eq_ref0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Instantiate: f:=(fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))):(a->Prop)
% Found x0 as proof of (P f)
% Found x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Instantiate: f:=(fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))):(a->Prop)
% Found x0 as proof of (P f)
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))):(((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (fun (x:a)=> ((and (cR x)) (((eq a) Xx) (F x)))))
% Found (eta_expansion00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found ((eta_expansion0 Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (F x)))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (F x)))))
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))
% Found (eq_ref00 (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eq_ref0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))):(((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (fun (x:a)=> ((and (cR x)) (((eq a) Xx) (F x)))))
% Found (eta_expansion_dep00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found x4:(((eq a) Xx) (F x1))
% Instantiate: x5:=x1:a
% Found x4 as proof of (((eq a) Xx) (F x5))
% Found x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Instantiate: b:=(fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))):(a->Prop)
% Found x0 as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))):(((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) (fun (x:a)=> ((and (cS x)) (((eq a) Xx) (F x)))))
% Found (eta_expansion_dep00 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found x5:(((eq a) Xx) (F x2))
% Instantiate: x1:=x2:a
% Found x5 as proof of (((eq a) Xx) (F x1))
% Found x5:(((eq a) Xx) (F x1))
% Instantiate: x3:=x1:a
% Found x5 as proof of (((eq a) Xx) (F x3))
% Found x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Instantiate: f:=(fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))):(a->Prop)
% Found x0 as proof of (P f)
% Found x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Instantiate: f:=(fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))):(a->Prop)
% Found x0 as proof of (P f)
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))->(P0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eq_ref0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))->(P0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eq_ref0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found x60:=(x6 x3):(cS x5)
% Found (x6 x3) as proof of (cS x5)
% Found ((x x5) x3) as proof of (cS x5)
% Found ((x x5) x3) as proof of (cS x5)
% Found ((conj10 ((x x5) x3)) x4) as proof of ((and (cS x5)) (((eq a) Xx) (F x5)))
% Found (((conj1 (((eq a) Xx) (F x5))) ((x x5) x3)) x4) as proof of ((and (cS x5)) (((eq a) Xx) (F x5)))
% Found ((((conj (cS x5)) (((eq a) Xx) (F x5))) ((x x5) x3)) x4) as proof of ((and (cS x5)) (((eq a) Xx) (F x5)))
% Found ((((conj (cS x5)) (((eq a) Xx) (F x5))) ((x x5) x3)) x4) as proof of ((and (cS x5)) (((eq a) Xx) (F x5)))
% Found (ex_intro000 ((((conj (cS x5)) (((eq a) Xx) (F x5))) ((x x5) x3)) x4)) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((ex_intro00 x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((ex_intro0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (fun (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))) as proof of ((((eq a) Xx) (F x1))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))
% Found (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))) as proof of ((cR x1)->((((eq a) Xx) (F x1))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))
% Found (and_rect00 (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((and_rect0 ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (fun (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))) as proof of (((and (cR x1)) (((eq a) Xx) (F x1)))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))
% Found (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))) as proof of (forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))
% Found (ex_ind00 (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)))))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((ex_ind0 ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)))))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((fun (P:Prop) (x1:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)))))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (fun (x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))=> (((fun (P:Prop) (x1:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (fun (Xx:a) (x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))=> (((fun (P:Prop) (x1:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))))) as proof of (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))
% Found (fun (F:(a->a)) (Xx:a) (x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))=> (((fun (P:Prop) (x1:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))))) as proof of (forall (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))
% Found (fun (x:(forall (Xx:a), ((cR Xx)->(cS Xx)))) (F:(a->a)) (Xx:a) (x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))=> (((fun (P:Prop) (x1:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))))) as proof of (forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))
% Found (fun (x:(forall (Xx:a), ((cR Xx)->(cS Xx)))) (F:(a->a)) (Xx:a) (x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))=> (((fun (P:Prop) (x1:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))))) as proof of ((forall (Xx:a), ((cR Xx)->(cS Xx)))->(forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))))
% Found x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Instantiate: b:=(fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))):(a->Prop)
% Found x0 as proof of (P b)
% Found x4:(((eq a) Xx) (F x1))
% Instantiate: x5:=x1:a
% Found x4 as proof of (((eq a) Xx) (F x5))
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))):(((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) (fun (x:a)=> ((and (cS x)) (((eq a) Xx) (F x)))))
% Found (eta_expansion00 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found ((eta_expansion0 Prop) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found x5:(((eq a) Xx) (F x2))
% Instantiate: x1:=x2:a
% Found x5 as proof of (((eq a) Xx) (F x1))
% Found x5:(((eq a) Xx) (F x1))
% Instantiate: x3:=x1:a
% Found x5 as proof of (((eq a) Xx) (F x3))
% Found x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Instantiate: f:=(fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))):(a->Prop)
% Found x0 as proof of (P f)
% Found x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Instantiate: f:=(fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))):(a->Prop)
% Found x0 as proof of (P f)
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))->(P0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eq_ref0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))->(P0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eq_ref0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found x60:=(x6 x3):(cS x5)
% Found (x6 x3) as proof of (cS x5)
% Found ((x x5) x3) as proof of (cS x5)
% Found ((x x5) x3) as proof of (cS x5)
% Found ((conj10 ((x x5) x3)) x4) as proof of ((and (cS x5)) (((eq a) Xx) (F x5)))
% Found (((conj1 (((eq a) Xx) (F x5))) ((x x5) x3)) x4) as proof of ((and (cS x5)) (((eq a) Xx) (F x5)))
% Found ((((conj (cS x5)) (((eq a) Xx) (F x5))) ((x x5) x3)) x4) as proof of ((and (cS x5)) (((eq a) Xx) (F x5)))
% Found ((((conj (cS x5)) (((eq a) Xx) (F x5))) ((x x5) x3)) x4) as proof of ((and (cS x5)) (((eq a) Xx) (F x5)))
% Found (ex_intro000 ((((conj (cS x5)) (((eq a) Xx) (F x5))) ((x x5) x3)) x4)) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((ex_intro00 x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((ex_intro0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (fun (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))) as proof of ((((eq a) Xx) (F x1))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))
% Found (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))) as proof of ((cR x1)->((((eq a) Xx) (F x1))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))
% Found (and_rect00 (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((and_rect0 ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (fun (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))) as proof of (((and (cR x1)) (((eq a) Xx) (F x1)))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))
% Found (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))) as proof of (forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))
% Found (ex_ind00 (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)))))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((ex_ind0 ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)))))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((fun (P:Prop) (x1:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)))))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (fun (x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))=> (((fun (P:Prop) (x1:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (fun (Xx:a) (x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))=> (((fun (P:Prop) (x1:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))))) as proof of (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))
% Found (fun (F:(a->a)) (Xx:a) (x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))=> (((fun (P:Prop) (x1:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))))) as proof of (forall (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))
% Found (fun (x:(forall (Xx:a), ((cR Xx)->(cS Xx)))) (F:(a->a)) (Xx:a) (x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))=> (((fun (P:Prop) (x1:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))))) as proof of (forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))
% Found (fun (x:(forall (Xx:a), ((cR Xx)->(cS Xx)))) (F:(a->a)) (Xx:a) (x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))=> (((fun (P:Prop) (x1:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))))) as proof of ((forall (Xx:a), ((cR Xx)->(cS Xx)))->(forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))
% Found (eq_ref00 (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eq_ref0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (x:a)=> ((and (cR x)) (((eq a) Xx) (F x))))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eta_expansion00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion0 Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (x:a)=> ((and (cR x)) (((eq a) Xx) (F x))))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eta_expansion00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion0 Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found eq_ref00:=(eq_ref0 ((forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))->(forall (Xx:a), ((cR Xx)->(cS Xx))))):(((eq Prop) ((forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))->(forall (Xx:a), ((cR Xx)->(cS Xx))))) ((forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))->(forall (Xx:a), ((cR Xx)->(cS Xx)))))
% Found (eq_ref0 ((forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))->(forall (Xx:a), ((cR Xx)->(cS Xx))))) as proof of (((eq Prop) ((forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))->(forall (Xx:a), ((cR Xx)->(cS Xx))))) b)
% Found ((eq_ref Prop) ((forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))->(forall (Xx:a), ((cR Xx)->(cS Xx))))) as proof of (((eq Prop) ((forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))->(forall (Xx:a), ((cR Xx)->(cS Xx))))) b)
% Found ((eq_ref Prop) ((forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))->(forall (Xx:a), ((cR Xx)->(cS Xx))))) as proof of (((eq Prop) ((forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))->(forall (Xx:a), ((cR Xx)->(cS Xx))))) b)
% Found ((eq_ref Prop) ((forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))->(forall (Xx:a), ((cR Xx)->(cS Xx))))) as proof of (((eq Prop) ((forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))->(forall (Xx:a), ((cR Xx)->(cS Xx))))) b)
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (x:a)=> ((and (cR x)) (((eq a) Xx) (F x))))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eta_expansion00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion0 Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (x:a)=> ((and (cR x)) (((eq a) Xx) (F x))))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eta_expansion00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion0 Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found eq_ref00:=(eq_ref0 a00):(((eq a) a00) a00)
% Found (eq_ref0 a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found eq_ref00:=(eq_ref0 a00):(((eq a) a00) a00)
% Found (eq_ref0 a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (x:a)=> ((and (cR x)) (((eq a) Xx) (F x))))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eta_expansion00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion0 Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (x:a)=> ((and (cR x)) (((eq a) Xx) (F x))))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eta_expansion00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion0 Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (x:a)=> ((and (cR x)) (((eq a) Xx) (F x))))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eta_expansion00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion0 Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (x:a)=> ((and (cR x)) (((eq a) Xx) (F x))))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eta_expansion00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion0 Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found eq_ref00:=(eq_ref0 ((forall (Xx:a), ((cR Xx)->(cS Xx)))->(forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))))):(((eq Prop) ((forall (Xx:a), ((cR Xx)->(cS Xx)))->(forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))))) ((forall (Xx:a), ((cR Xx)->(cS Xx)))->(forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))))
% Found (eq_ref0 ((forall (Xx:a), ((cR Xx)->(cS Xx)))->(forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))))) as proof of (((eq Prop) ((forall (Xx:a), ((cR Xx)->(cS Xx)))->(forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))))) b)
% Found ((eq_ref Prop) ((forall (Xx:a), ((cR Xx)->(cS Xx)))->(forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))))) as proof of (((eq Prop) ((forall (Xx:a), ((cR Xx)->(cS Xx)))->(forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))))) b)
% Found ((eq_ref Prop) ((forall (Xx:a), ((cR Xx)->(cS Xx)))->(forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))))) as proof of (((eq Prop) ((forall (Xx:a), ((cR Xx)->(cS Xx)))->(forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))))) b)
% Found ((eq_ref Prop) ((forall (Xx:a), ((cR Xx)->(cS Xx)))->(forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))))) as proof of (((eq Prop) ((forall (Xx:a), ((cR Xx)->(cS Xx)))->(forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))))) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found eq_ref00:=(eq_ref0 a00):(((eq a) a00) a00)
% Found (eq_ref0 a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found eq_ref00:=(eq_ref0 a00):(((eq a) a00) a00)
% Found (eq_ref0 a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 a00):(((eq a) a00) a00)
% Found (eq_ref0 a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Instantiate: b:=(fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))):(a->Prop)
% Found x0 as proof of (P b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))):(((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) (fun (x:a)=> ((and (cS x)) (((eq a) Xx) (F x)))))
% Found (eta_expansion00 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found ((eta_expansion0 Prop) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 a00):(((eq a) a00) a00)
% Found (eq_ref0 a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Instantiate: f:=(fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))):(a->Prop)
% Found x0 as proof of (P f)
% Found x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Instantiate: f:=(fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))):(a->Prop)
% Found x0 as proof of (P f)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))):(((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (fun (x:a)=> ((and (cR x)) (((eq a) Xx) (F x)))))
% Found (eta_expansion_dep00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (F x)))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (F x)))))
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))
% Found (eq_ref00 (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eq_ref0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))):(((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (fun (x:a)=> ((and (cR x)) (((eq a) Xx) (F x)))))
% Found (eta_expansion_dep00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Instantiate: b:=(fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))):(a->Prop)
% Found x0 as proof of (P b)
% Found x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Instantiate: b:=(fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))):(a->Prop)
% Found x0 as proof of (P b)
% Found eq_ref00:=(eq_ref0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))):(((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (eq_ref0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found eq_ref00:=(eq_ref0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))):(((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (eq_ref0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Instantiate: f:=(fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))):(a->Prop)
% Found x0 as proof of (P f)
% Found x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Instantiate: f:=(fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))):(a->Prop)
% Found x0 as proof of (P f)
% Found x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Instantiate: f:=(fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))):(a->Prop)
% Found x0 as proof of (P f)
% Found x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Instantiate: f:=(fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))):(a->Prop)
% Found x0 as proof of (P f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))):(((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (fun (x:a)=> ((and (cR x)) (((eq a) Xx) (F x)))))
% Found (eta_expansion_dep00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))):(((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (fun (x:a)=> ((and (cR x)) (((eq a) Xx) (F x)))))
% Found (eta_expansion00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found ((eta_expansion0 Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (F x)))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (F x)))))
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))
% Found (eq_ref00 (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eq_ref0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))):(((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (fun (x:a)=> ((and (cR x)) (((eq a) Xx) (F x)))))
% Found (eta_expansion_dep00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (F x)))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cS x1)) (((eq a) Xx) (F x1))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (F x)))))
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))
% Found (eq_ref00 (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eq_ref0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))):(((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (fun (x:a)=> ((and (cR x)) (((eq a) Xx) (F x)))))
% Found (eta_expansion_dep00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found x4:(((eq a) Xx) (F x1))
% Instantiate: x5:=x1:a
% Found x4 as proof of (((eq a) Xx) (F x5))
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Instantiate: b:=(fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))):(a->Prop)
% Found x0 as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))):(((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) (fun (x:a)=> ((and (cS x)) (((eq a) Xx) (F x)))))
% Found (eta_expansion_dep00 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found x5:(((eq a) Xx) (F x2))
% Instantiate: x1:=x2:a
% Found x5 as proof of (((eq a) Xx) (F x1))
% Found x5:(((eq a) Xx) (F x1))
% Instantiate: x3:=x1:a
% Found x5 as proof of (((eq a) Xx) (F x3))
% Found x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Instantiate: f:=(fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))):(a->Prop)
% Found x0 as proof of (P f)
% Found x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Instantiate: f:=(fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))):(a->Prop)
% Found x0 as proof of (P f)
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))->(P0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eq_ref0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))->(P0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eq_ref0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found x60:=(x6 x3):(cS x5)
% Found (x6 x3) as proof of (cS x5)
% Found ((x x5) x3) as proof of (cS x5)
% Found ((x x5) x3) as proof of (cS x5)
% Found ((conj10 ((x x5) x3)) x4) as proof of ((and (cS x5)) (((eq a) Xx) (F x5)))
% Found (((conj1 (((eq a) Xx) (F x5))) ((x x5) x3)) x4) as proof of ((and (cS x5)) (((eq a) Xx) (F x5)))
% Found ((((conj (cS x5)) (((eq a) Xx) (F x5))) ((x x5) x3)) x4) as proof of ((and (cS x5)) (((eq a) Xx) (F x5)))
% Found ((((conj (cS x5)) (((eq a) Xx) (F x5))) ((x x5) x3)) x4) as proof of ((and (cS x5)) (((eq a) Xx) (F x5)))
% Found (ex_intro000 ((((conj (cS x5)) (((eq a) Xx) (F x5))) ((x x5) x3)) x4)) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((ex_intro00 x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((ex_intro0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (fun (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))) as proof of ((((eq a) Xx) (F x1))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))
% Found (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))) as proof of ((cR x1)->((((eq a) Xx) (F x1))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))
% Found (and_rect00 (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((and_rect0 ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (fun (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))) as proof of (((and (cR x1)) (((eq a) Xx) (F x1)))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))
% Found (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))) as proof of (forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))
% Found (ex_ind00 (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)))))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((ex_ind0 ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)))))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((fun (P:Prop) (x1:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)))))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (fun (x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))=> (((fun (P:Prop) (x1:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (fun (Xx:a) (x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))=> (((fun (P:Prop) (x1:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))))) as proof of (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))
% Found (fun (F:(a->a)) (Xx:a) (x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))=> (((fun (P:Prop) (x1:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))))) as proof of (forall (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))
% Found (fun (x:(forall (Xx:a), ((cR Xx)->(cS Xx)))) (F:(a->a)) (Xx:a) (x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))=> (((fun (P:Prop) (x1:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))))) as proof of (forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))
% Found (fun (x:(forall (Xx:a), ((cR Xx)->(cS Xx)))) (F:(a->a)) (Xx:a) (x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))=> (((fun (P:Prop) (x1:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))))) as proof of ((forall (Xx:a), ((cR Xx)->(cS Xx)))->(forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Instantiate: b:=(fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))):(a->Prop)
% Found x0 as proof of (P b)
% Found x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Instantiate: b:=(fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))):(a->Prop)
% Found x0 as proof of (P b)
% Found x4:(((eq a) Xx) (F x1))
% Instantiate: x5:=x1:a
% Found x4 as proof of (((eq a) Xx) (F x5))
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))):(((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) (fun (x:a)=> ((and (cS x)) (((eq a) Xx) (F x)))))
% Found (eta_expansion00 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found ((eta_expansion0 Prop) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found x4:(((eq a) Xx) (F x1))
% Instantiate: x5:=x1:a
% Found x4 as proof of (((eq a) Xx) (F x5))
% Found eq_ref00:=(eq_ref0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))):(((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (eq_ref0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found x5:(((eq a) Xx) (F x2))
% Instantiate: x1:=x2:a
% Found x5 as proof of (((eq a) Xx) (F x1))
% Found x5:(((eq a) Xx) (F x2))
% Instantiate: x1:=x2:a
% Found x5 as proof of (((eq a) Xx) (F x1))
% Found x5:(((eq a) Xx) (F x1))
% Instantiate: x3:=x1:a
% Found x5 as proof of (((eq a) Xx) (F x3))
% Found x5:(((eq a) Xx) (F x1))
% Instantiate: x3:=x1:a
% Found x5 as proof of (((eq a) Xx) (F x3))
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found eq_ref00:=(eq_ref0 a00):(((eq a) a00) a00)
% Found (eq_ref0 a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Instantiate: f:=(fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))):(a->Prop)
% Found x0 as proof of (P f)
% Found x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Instantiate: f:=(fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))):(a->Prop)
% Found x0 as proof of (P f)
% Found x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Instantiate: f:=(fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))):(a->Prop)
% Found x0 as proof of (P f)
% Found x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Instantiate: f:=(fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))):(a->Prop)
% Found x0 as proof of (P f)
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))->(P0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eq_ref0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))->(P0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eq_ref0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))->(P0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eq_ref0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))->(P0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eq_ref0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) P0) as proof of (P1 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found eq_ref00:=(eq_ref0 a00):(((eq a) a00) a00)
% Found (eq_ref0 a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found eq_ref00:=(eq_ref0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))):(((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (eq_ref0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found eq_ref00:=(eq_ref0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))):(((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (eq_ref0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found eq_ref00:=(eq_ref0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))):(((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (eq_ref0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))):(((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) (fun (x:a)=> ((and (cS x)) (((eq a) Xx) (F x)))))
% Found (eta_expansion00 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found ((eta_expansion0 Prop) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found x60:=(x6 x3):(cS x5)
% Found (x6 x3) as proof of (cS x5)
% Found ((x x5) x3) as proof of (cS x5)
% Found ((x x5) x3) as proof of (cS x5)
% Found ((conj10 ((x x5) x3)) x4) as proof of ((and (cS x5)) (((eq a) Xx) (F x5)))
% Found (((conj1 (((eq a) Xx) (F x5))) ((x x5) x3)) x4) as proof of ((and (cS x5)) (((eq a) Xx) (F x5)))
% Found ((((conj (cS x5)) (((eq a) Xx) (F x5))) ((x x5) x3)) x4) as proof of ((and (cS x5)) (((eq a) Xx) (F x5)))
% Found ((((conj (cS x5)) (((eq a) Xx) (F x5))) ((x x5) x3)) x4) as proof of ((and (cS x5)) (((eq a) Xx) (F x5)))
% Found (ex_intro000 ((((conj (cS x5)) (((eq a) Xx) (F x5))) ((x x5) x3)) x4)) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((ex_intro00 x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((ex_intro0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (fun (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))) as proof of ((((eq a) Xx) (F x1))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))
% Found (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))) as proof of ((cR x1)->((((eq a) Xx) (F x1))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))
% Found (and_rect00 (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((and_rect0 ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (fun (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))) as proof of (((and (cR x1)) (((eq a) Xx) (F x1)))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))
% Found (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))) as proof of (forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))
% Found (ex_ind00 (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)))))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((ex_ind0 ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)))))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((fun (P:Prop) (x1:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)))))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (fun (x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))=> (((fun (P:Prop) (x1:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (fun (Xx:a) (x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))=> (((fun (P:Prop) (x1:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))))) as proof of (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))
% Found (fun (F:(a->a)) (Xx:a) (x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))=> (((fun (P:Prop) (x1:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))))) as proof of (forall (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))
% Found (fun (x:(forall (Xx:a), ((cR Xx)->(cS Xx)))) (F:(a->a)) (Xx:a) (x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))=> (((fun (P:Prop) (x1:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))))) as proof of (forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))
% Found (fun (x:(forall (Xx:a), ((cR Xx)->(cS Xx)))) (F:(a->a)) (Xx:a) (x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))=> (((fun (P:Prop) (x1:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))))) as proof of ((forall (Xx:a), ((cR Xx)->(cS Xx)))->(forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))->(forall (Xx:a), ((cR Xx)->(cS Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))->(forall (Xx:a), ((cR Xx)->(cS Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))->(forall (Xx:a), ((cR Xx)->(cS Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))->(forall (Xx:a), ((cR Xx)->(cS Xx)))))
% Found x60:=(x6 x3):(cS x5)
% Found (x6 x3) as proof of (cS x5)
% Found ((x x5) x3) as proof of (cS x5)
% Found ((x x5) x3) as proof of (cS x5)
% Found ((conj10 ((x x5) x3)) x4) as proof of ((and (cS x5)) (((eq a) Xx) (F x5)))
% Found (((conj1 (((eq a) Xx) (F x5))) ((x x5) x3)) x4) as proof of ((and (cS x5)) (((eq a) Xx) (F x5)))
% Found ((((conj (cS x5)) (((eq a) Xx) (F x5))) ((x x5) x3)) x4) as proof of ((and (cS x5)) (((eq a) Xx) (F x5)))
% Found ((((conj (cS x5)) (((eq a) Xx) (F x5))) ((x x5) x3)) x4) as proof of ((and (cS x5)) (((eq a) Xx) (F x5)))
% Found (ex_intro000 ((((conj (cS x5)) (((eq a) Xx) (F x5))) ((x x5) x3)) x4)) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((ex_intro00 x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((ex_intro0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (fun (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))) as proof of ((((eq a) Xx) (F x1))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))
% Found (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))) as proof of ((cR x1)->((((eq a) Xx) (F x1))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))
% Found (and_rect00 (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((and_rect0 ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (fun (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))) as proof of (((and (cR x1)) (((eq a) Xx) (F x1)))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))
% Found (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))) as proof of (forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))
% Found (ex_ind00 (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)))))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((ex_ind0 ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)))))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((fun (P:Prop) (x1:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4)))))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (fun (x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))=> (((fun (P:Prop) (x1:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (fun (Xx:a) (x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))=> (((fun (P:Prop) (x1:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))))) as proof of (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))
% Found (fun (F:(a->a)) (Xx:a) (x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))=> (((fun (P:Prop) (x1:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))))) as proof of (forall (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))
% Found (fun (x:(forall (Xx:a), ((cR Xx)->(cS Xx)))) (F:(a->a)) (Xx:a) (x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))=> (((fun (P:Prop) (x1:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))))) as proof of (forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))
% Found (fun (x:(forall (Xx:a), ((cR Xx)->(cS Xx)))) (F:(a->a)) (Xx:a) (x0:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))=> (((fun (P:Prop) (x1:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x1:a) (x2:((and (cR x1)) (((eq a) Xx) (F x1))))=> (((fun (P:Type) (x3:((cR x1)->((((eq a) Xx) (F x1))->P)))=> (((((and_rect (cR x1)) (((eq a) Xx) (F x1))) P) x3) x2)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x3:(cR x1)) (x4:(((eq a) Xx) (F x1)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x1) ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x x1) x3)) x4))))))) as proof of ((forall (Xx:a), ((cR Xx)->(cS Xx)))->(forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found eq_ref00:=(eq_ref0 a00):(((eq a) a00) a00)
% Found (eq_ref0 a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((forall (Xx:a), ((cR Xx)->(cS Xx)))->(forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((forall (Xx:a), ((cR Xx)->(cS Xx)))->(forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((forall (Xx:a), ((cR Xx)->(cS Xx)))->(forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((forall (Xx:a), ((cR Xx)->(cS Xx)))->(forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found eq_ref00:=(eq_ref0 a00):(((eq a) a00) a00)
% Found (eq_ref0 a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 a00):(((eq a) a00) a00)
% Found (eq_ref0 a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 a00):(((eq a) a00) a00)
% Found (eq_ref0 a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of a0
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of a0
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex a)):(((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (x:a)=> ((and (cR x)) (((eq a) Xx) (F x))))))
% Found (eta_expansion_dep000 (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eta_expansion_dep00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex a)):(((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (x:a)=> ((and (cR x)) (((eq a) Xx) (F x))))))
% Found (eta_expansion_dep000 (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eta_expansion_dep00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))
% Found (eq_ref00 (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eq_ref0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex a)):(((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (x:a)=> ((and (cR x)) (((eq a) Xx) (F x))))))
% Found (eta_expansion_dep000 (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eta_expansion_dep00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex a)):(((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (x:a)=> ((and (cR x)) (((eq a) Xx) (F x))))))
% Found (eta_expansion_dep000 (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eta_expansion_dep00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))
% Found (eq_ref00 (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eq_ref0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex a)):(((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (x:a)=> ((and (cR x)) (((eq a) Xx) (F x))))))
% Found (eta_expansion_dep000 (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eta_expansion_dep00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex a)):(((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (x:a)=> ((and (cR x)) (((eq a) Xx) (F x))))))
% Found (eta_expansion_dep000 (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eta_expansion_dep00 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))
% Found (eq_ref00 (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found ((eq_ref0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) (ex a)) as proof of (P0 (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found x01:(((eq a) Xx) (F x))
% Instantiate: x1:=x:a
% Found x01 as proof of (((eq a) Xx) (F x1))
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found x01:(((eq a) Xx) (F x))
% Instantiate: x1:=x:a
% Found x01 as proof of (((eq a) Xx) (F x1))
% Found eq_ref00:=(eq_ref0 ((forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))->(forall (Xx:a), ((cR Xx)->(cS Xx))))):(((eq Prop) ((forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))->(forall (Xx:a), ((cR Xx)->(cS Xx))))) ((forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))->(forall (Xx:a), ((cR Xx)->(cS Xx)))))
% Found (eq_ref0 ((forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))->(forall (Xx:a), ((cR Xx)->(cS Xx))))) as proof of (((eq Prop) ((forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))->(forall (Xx:a), ((cR Xx)->(cS Xx))))) b0)
% Found ((eq_ref Prop) ((forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))->(forall (Xx:a), ((cR Xx)->(cS Xx))))) as proof of (((eq Prop) ((forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))->(forall (Xx:a), ((cR Xx)->(cS Xx))))) b0)
% Found ((eq_ref Prop) ((forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))->(forall (Xx:a), ((cR Xx)->(cS Xx))))) as proof of (((eq Prop) ((forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))->(forall (Xx:a), ((cR Xx)->(cS Xx))))) b0)
% Found ((eq_ref Prop) ((forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))->(forall (Xx:a), ((cR Xx)->(cS Xx))))) as proof of (((eq Prop) ((forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))->(forall (Xx:a), ((cR Xx)->(cS Xx))))) b0)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found x11:(((eq a) Xx) (F x0))
% Instantiate: x:=x0:a
% Found x11 as proof of (((eq a) Xx) (F x))
% Found x01:(((eq a) Xx) (F x))
% Instantiate: x1:=x:a
% Found x01 as proof of (((eq a) Xx) (F x1))
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found x11:(((eq a) Xx) (F x0))
% Instantiate: x:=x0:a
% Found x11 as proof of (((eq a) Xx) (F x))
% Found x01:(((eq a) Xx) (F x))
% Instantiate: x1:=x:a
% Found x01 as proof of (((eq a) Xx) (F x1))
% Found x200:=(x20 x00):(cS x1)
% Found (x20 x00) as proof of (cS x1)
% Found ((x2 x1) x00) as proof of (cS x1)
% Found ((x2 x1) x00) as proof of (cS x1)
% Found ((conj10 ((x2 x1) x00)) x01) as proof of ((and (cS x1)) (((eq a) Xx) (F x1)))
% Found (((conj1 (((eq a) Xx) (F x1))) ((x2 x1) x00)) x01) as proof of ((and (cS x1)) (((eq a) Xx) (F x1)))
% Found ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x2 x1) x00)) x01) as proof of ((and (cS x1)) (((eq a) Xx) (F x1)))
% Found ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x2 x1) x00)) x01) as proof of ((and (cS x1)) (((eq a) Xx) (F x1)))
% Found (ex_intro000 ((((conj (cS x1)) (((eq a) Xx) (F x1))) ((x2 x1) x00)) x01)) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((ex_intro00 x) ((((conj (cS x)) (((eq a) Xx) (F x))) ((x2 x) x00)) x01)) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((ex_intro0 (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x) ((((conj (cS x)) (((eq a) Xx) (F x))) ((x2 x) x00)) x01)) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x) ((((conj (cS x)) (((eq a) Xx) (F x))) ((x2 x) x00)) x01)) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (fun (x01:(((eq a) Xx) (F x)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x) ((((conj (cS x)) (((eq a) Xx) (F x))) ((x2 x) x00)) x01))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (fun (x00:(cR x)) (x01:(((eq a) Xx) (F x)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x) ((((conj (cS x)) (((eq a) Xx) (F x))) ((x2 x) x00)) x01))) as proof of ((((eq a) Xx) (F x))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))
% Found (fun (x00:(cR x)) (x01:(((eq a) Xx) (F x)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x) ((((conj (cS x)) (((eq a) Xx) (F x))) ((x2 x) x00)) x01))) as proof of ((cR x)->((((eq a) Xx) (F x))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))
% Found (and_rect00 (fun (x00:(cR x)) (x01:(((eq a) Xx) (F x)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x) ((((conj (cS x)) (((eq a) Xx) (F x))) ((x2 x) x00)) x01)))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((and_rect0 ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x00:(cR x)) (x01:(((eq a) Xx) (F x)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x) ((((conj (cS x)) (((eq a) Xx) (F x))) ((x2 x) x00)) x01)))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((fun (P0:Type) (x1:((cR x)->((((eq a) Xx) (F x))->P0)))=> (((((and_rect (cR x)) (((eq a) Xx) (F x))) P0) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x00:(cR x)) (x01:(((eq a) Xx) (F x)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x) ((((conj (cS x)) (((eq a) Xx) (F x))) ((x2 x) x00)) x01)))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (fun (x0:((and (cR x)) (((eq a) Xx) (F x))))=> (((fun (P0:Type) (x1:((cR x)->((((eq a) Xx) (F x))->P0)))=> (((((and_rect (cR x)) (((eq a) Xx) (F x))) P0) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x00:(cR x)) (x01:(((eq a) Xx) (F x)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x) ((((conj (cS x)) (((eq a) Xx) (F x))) ((x2 x) x00)) x01))))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (fun (x:a) (x0:((and (cR x)) (((eq a) Xx) (F x))))=> (((fun (P0:Type) (x1:((cR x)->((((eq a) Xx) (F x))->P0)))=> (((((and_rect (cR x)) (((eq a) Xx) (F x))) P0) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x00:(cR x)) (x01:(((eq a) Xx) (F x)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x) ((((conj (cS x)) (((eq a) Xx) (F x))) ((x2 x) x00)) x01))))) as proof of (((and (cR x)) (((eq a) Xx) (F x)))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))
% Found (fun (x:a) (x0:((and (cR x)) (((eq a) Xx) (F x))))=> (((fun (P0:Type) (x1:((cR x)->((((eq a) Xx) (F x))->P0)))=> (((((and_rect (cR x)) (((eq a) Xx) (F x))) P0) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x00:(cR x)) (x01:(((eq a) Xx) (F x)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x) ((((conj (cS x)) (((eq a) Xx) (F x))) ((x2 x) x00)) x01))))) as proof of (forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))
% Found (ex_ind00 (fun (x:a) (x0:((and (cR x)) (((eq a) Xx) (F x))))=> (((fun (P0:Type) (x1:((cR x)->((((eq a) Xx) (F x))->P0)))=> (((((and_rect (cR x)) (((eq a) Xx) (F x))) P0) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x00:(cR x)) (x01:(((eq a) Xx) (F x)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x) ((((conj (cS x)) (((eq a) Xx) (F x))) ((x2 x) x00)) x01)))))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found ((ex_ind0 ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x:a) (x0:((and (cR x)) (((eq a) Xx) (F x))))=> (((fun (P0:Type) (x1:((cR x)->((((eq a) Xx) (F x))->P0)))=> (((((and_rect (cR x)) (((eq a) Xx) (F x))) P0) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x00:(cR x)) (x01:(((eq a) Xx) (F x)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x) ((((conj (cS x)) (((eq a) Xx) (F x))) ((x2 x) x00)) x01)))))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (((fun (P0:Prop) (x:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P0)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P0) x) x3)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x:a) (x0:((and (cR x)) (((eq a) Xx) (F x))))=> (((fun (P0:Type) (x1:((cR x)->((((eq a) Xx) (F x))->P0)))=> (((((and_rect (cR x)) (((eq a) Xx) (F x))) P0) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x00:(cR x)) (x01:(((eq a) Xx) (F x)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x) ((((conj (cS x)) (((eq a) Xx) (F x))) ((x2 x) x00)) x01)))))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (fun (x3:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))=> (((fun (P0:Prop) (x:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P0)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P0) x) x3)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x:a) (x0:((and (cR x)) (((eq a) Xx) (F x))))=> (((fun (P0:Type) (x1:((cR x)->((((eq a) Xx) (F x))->P0)))=> (((((and_rect (cR x)) (((eq a) Xx) (F x))) P0) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x00:(cR x)) (x01:(((eq a) Xx) (F x)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x) ((((conj (cS x)) (((eq a) Xx) (F x))) ((x2 x) x00)) x01))))))) as proof of ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))
% Found (fun (Xx:a) (x3:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))=> (((fun (P0:Prop) (x:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P0)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P0) x) x3)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x:a) (x0:((and (cR x)) (((eq a) Xx) (F x))))=> (((fun (P0:Type) (x1:((cR x)->((((eq a) Xx) (F x))->P0)))=> (((((and_rect (cR x)) (((eq a) Xx) (F x))) P0) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x00:(cR x)) (x01:(((eq a) Xx) (F x)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x) ((((conj (cS x)) (((eq a) Xx) (F x))) ((x2 x) x00)) x01))))))) as proof of (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))
% Found (fun (F:(a->a)) (Xx:a) (x3:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))=> (((fun (P0:Prop) (x:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P0)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P0) x) x3)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x:a) (x0:((and (cR x)) (((eq a) Xx) (F x))))=> (((fun (P0:Type) (x1:((cR x)->((((eq a) Xx) (F x))->P0)))=> (((((and_rect (cR x)) (((eq a) Xx) (F x))) P0) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x00:(cR x)) (x01:(((eq a) Xx) (F x)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x) ((((conj (cS x)) (((eq a) Xx) (F x))) ((x2 x) x00)) x01))))))) as proof of (forall (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))
% Found (fun (x2:(forall (Xx:a), ((cR Xx)->(cS Xx)))) (F:(a->a)) (Xx:a) (x3:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))=> (((fun (P0:Prop) (x:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P0)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P0) x) x3)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x:a) (x0:((and (cR x)) (((eq a) Xx) (F x))))=> (((fun (P0:Type) (x1:((cR x)->((((eq a) Xx) (F x))->P0)))=> (((((and_rect (cR x)) (((eq a) Xx) (F x))) P0) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x00:(cR x)) (x01:(((eq a) Xx) (F x)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x) ((((conj (cS x)) (((eq a) Xx) (F x))) ((x2 x) x00)) x01))))))) as proof of (forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))))
% Found (fun (x2:(forall (Xx:a), ((cR Xx)->(cS Xx)))) (F:(a->a)) (Xx:a) (x3:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))=> (((fun (P0:Prop) (x:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P0)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P0) x) x3)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x:a) (x0:((and (cR x)) (((eq a) Xx) (F x))))=> (((fun (P0:Type) (x1:((cR x)->((((eq a) Xx) (F x))->P0)))=> (((((and_rect (cR x)) (((eq a) Xx) (F x))) P0) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x00:(cR x)) (x01:(((eq a) Xx) (F x)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x) ((((conj (cS x)) (((eq a) Xx) (F x))) ((x2 x) x00)) x01))))))) as proof of ((forall (Xx:a), ((cR Xx)->(cS Xx)))->(forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))))
% Found ((conj00 (fun (x2:(forall (Xx:a), ((cR Xx)->(cS Xx)))) (F:(a->a)) (Xx:a) (x3:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))=> (((fun (P0:Prop) (x:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P0)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P0) x) x3)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x:a) (x0:((and (cR x)) (((eq a) Xx) (F x))))=> (((fun (P0:Type) (x1:((cR x)->((((eq a) Xx) (F x))->P0)))=> (((((and_rect (cR x)) (((eq a) Xx) (F x))) P0) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x00:(cR x)) (x01:(((eq a) Xx) (F x)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x) ((((conj (cS x)) (((eq a) Xx) (F x))) ((x2 x) x00)) x01)))))))) conj) as proof of ((and ((forall (Xx:a), ((cR Xx)->(cS Xx)))->(forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))))) b)
% Found (((conj0 b) (fun (x2:(forall (Xx:a), ((cR Xx)->(cS Xx)))) (F:(a->a)) (Xx:a) (x3:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))=> (((fun (P0:Prop) (x:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P0)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P0) x) x3)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x:a) (x0:((and (cR x)) (((eq a) Xx) (F x))))=> (((fun (P0:Type) (x1:((cR x)->((((eq a) Xx) (F x))->P0)))=> (((((and_rect (cR x)) (((eq a) Xx) (F x))) P0) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x00:(cR x)) (x01:(((eq a) Xx) (F x)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x) ((((conj (cS x)) (((eq a) Xx) (F x))) ((x2 x) x00)) x01)))))))) conj) as proof of ((and ((forall (Xx:a), ((cR Xx)->(cS Xx)))->(forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))))) b)
% Found ((((conj ((forall (Xx:a), ((cR Xx)->(cS Xx)))->(forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))))) b) (fun (x2:(forall (Xx:a), ((cR Xx)->(cS Xx)))) (F:(a->a)) (Xx:a) (x3:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))))=> (((fun (P0:Prop) (x:(forall (x:a), (((and (cR x)) (((eq a) Xx) (F x)))->P0)))=> (((((ex_ind a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt))))) P0) x) x3)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x:a) (x0:((and (cR x)) (((eq a) Xx) (F x))))=> (((fun (P0:Type) (x1:((cR x)->((((eq a) Xx) (F x))->P0)))=> (((((and_rect (cR x)) (((eq a) Xx) (F x))) P0) x1) x0)) ((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt)))))) (fun (x00:(cR x)) (x01:(((eq a) Xx) (F x)))=> ((((ex_intro a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))) x) ((((conj (cS x)) (((eq a) Xx) (F x))) ((x2 x) x00)) x01)))))))) conj) as proof of ((and ((forall (Xx:a), ((cR Xx)->(cS Xx)))->(forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))))) b)
% Found ((((conj ((forall (Xx:a), ((cR Xx)->(cS Xx)))->(forall (F:(a->a)) (Xx:a), (((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq a) Xx) (F Xt)))))->((ex a) (fun (Xt:a)=> ((and (cS Xt)) (((eq a) Xx) (F Xt))))))))) b) (fun (x2:(forall (Xx:a), ((cR Xx)->(cS Xx)))) (F:(a->a)) (Xx:a) (x3:((ex a) (fun (Xt:a)=> ((and (cR Xt)) (((eq
% EOF
%------------------------------------------------------------------------------