TSTP Solution File: SEU972^5 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEU972^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n115.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 13:33:27 EDT 2014
% Result : Timeout 300.06s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEU972^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n115.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 11:46:56 CDT 2014
% % CPUTime : 300.06
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x26f3e18>, <kernel.Type object at 0x26f3488>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x26f3f38>, <kernel.DependentProduct object at 0x2aced88>) of role type named cR
% Using role type
% Declaring cR:(a->a)
% FOF formula (<kernel.Constant object at 0x25b90e0>, <kernel.DependentProduct object at 0x26f3ef0>) of role type named cP
% Using role type
% Declaring cP:(a->(a->a))
% FOF formula (<kernel.Constant object at 0x26f3e18>, <kernel.DependentProduct object at 0x2aceef0>) of role type named cL
% Using role type
% Declaring cL:(a->a)
% FOF formula (<kernel.Constant object at 0x26f3f38>, <kernel.Constant object at 0x2acee60>) of role type named cZ
% Using role type
% Declaring cZ:a
% FOF formula (((and ((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))) (forall (X:(a->Prop)), ((forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((iff (((eq a) Xt) cZ)) (((eq a) Xu) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))->(forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->(((eq a) Xt) Xu))))))->(forall (Xt:a), ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))))) of role conjecture named cPU_LEM2E_pme
% Conjecture to prove = (((and ((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))) (forall (X:(a->Prop)), ((forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((iff (((eq a) Xt) cZ)) (((eq a) Xu) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))->(forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->(((eq a) Xt) Xu))))))->(forall (Xt:a), ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))))):Prop
% We need to prove ['(((and ((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))) (forall (X:(a->Prop)), ((forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((iff (((eq a) Xt) cZ)) (((eq a) Xu) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))->(forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->(((eq a) Xt) Xu))))))->(forall (Xt:a), ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu)))))))))))']
% Parameter a:Type.
% Parameter cR:(a->a).
% Parameter cP:(a->(a->a)).
% Parameter cL:(a->a).
% Parameter cZ:a.
% Trying to prove (((and ((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))) (forall (X:(a->Prop)), ((forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((iff (((eq a) Xt) cZ)) (((eq a) Xu) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))->(forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->(((eq a) Xt) Xu))))))->(forall (Xt:a), ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu)))))))))))
% Found eq_substitution0000:=(fun (a0:a)=> ((eq_substitution000 a0) (cP cZ))):(forall (a0:a), ((((eq a) a0) Xt)->(((eq a) ((cP cZ) a0)) ((cP cZ) Xt))))
% Instantiate: x0:=(fun (x1:a)=> (forall (a0:a), ((((eq a) a0) Xt)->(((eq a) ((cP cZ) a0)) x1)))):(a->Prop)
% Found (fun (a0:a)=> ((eq_substitution000 a0) (cP cZ))) as proof of (x0 ((cP cZ) Xt))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((eq_substitution00 a0) Xt)) a0) (cP cZ))) as proof of (x0 ((cP cZ) Xt))
% Found (fun (a0:a)=> (((fun (a0:a)=> (((eq_substitution0 a) a0) Xt)) a0) (cP cZ))) as proof of (x0 ((cP cZ) Xt))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((((eq_substitution a) a) a0) Xt)) a0) (cP cZ))) as proof of (x0 ((cP cZ) Xt))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((((eq_substitution a) a) a0) Xt)) a0) (cP cZ))) as proof of (x0 ((cP cZ) Xt))
% Found eta_expansion000:=(eta_expansion00 (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))):(((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) (fun (x:(a->Prop))=> ((and (x ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((x ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x ((cP (cL Xt0)) (cL Xu))))) (x ((cP (cR Xt0)) (cR Xu)))))))))
% Found (eta_expansion00 (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) b)
% Found ((eta_expansion0 Prop) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) b)
% Found conj0100:=(conj010 x00):((and (x0 ((cP cZ) Xt))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found (conj010 x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found ((conj01 (x0 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found x00:(x0 ((cP Xt0) Xu))
% Found x00 as proof of (x0 ((cP (cR Xt0)) (cR Xu)))
% Found ((conj10 (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00)) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu))))
% Found (((conj1 (x0 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00)) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu))))
% Found ((((conj ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00)) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu))))
% Found (fun (x00:(x0 ((cP Xt0) Xu)))=> ((((conj ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00)) x00)) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu))))
% Found eq_substitution0000:=(fun (a0:a)=> ((eq_substitution000 a0) (cP cZ))):(forall (a0:a), ((((eq a) a0) Xt)->(((eq a) ((cP cZ) a0)) ((cP cZ) Xt))))
% Instantiate: x2:=(fun (x3:a)=> (forall (a0:a), ((((eq a) a0) Xt)->(((eq a) ((cP cZ) a0)) x3)))):(a->Prop)
% Found (fun (a0:a)=> ((eq_substitution000 a0) (cP cZ))) as proof of (x2 ((cP cZ) Xt))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((eq_substitution00 a0) Xt)) a0) (cP cZ))) as proof of (x2 ((cP cZ) Xt))
% Found (fun (a0:a)=> (((fun (a0:a)=> (((eq_substitution0 a) a0) Xt)) a0) (cP cZ))) as proof of (x2 ((cP cZ) Xt))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((((eq_substitution a) a) a0) Xt)) a0) (cP cZ))) as proof of (x2 ((cP cZ) Xt))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((((eq_substitution a) a) a0) Xt)) a0) (cP cZ))) as proof of (x2 ((cP cZ) Xt))
% Found eq_substitution0000:=(fun (a0:a)=> ((eq_substitution000 a0) (cP cZ))):(forall (a0:a), ((((eq a) a0) Xt)->(((eq a) ((cP cZ) a0)) ((cP cZ) Xt))))
% Instantiate: x0:=(fun (x3:a)=> (forall (a0:a), ((((eq a) a0) Xt)->(((eq a) ((cP cZ) a0)) x3)))):(a->Prop)
% Found (fun (a0:a)=> ((eq_substitution000 a0) (cP cZ))) as proof of (x0 ((cP cZ) Xt))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((eq_substitution00 a0) Xt)) a0) (cP cZ))) as proof of (x0 ((cP cZ) Xt))
% Found (fun (a0:a)=> (((fun (a0:a)=> (((eq_substitution0 a) a0) Xt)) a0) (cP cZ))) as proof of (x0 ((cP cZ) Xt))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((((eq_substitution a) a) a0) Xt)) a0) (cP cZ))) as proof of (x0 ((cP cZ) Xt))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((((eq_substitution a) a) a0) Xt)) a0) (cP cZ))) as proof of (x0 ((cP cZ) Xt))
% Found x50:=(fun (Xx:a)=> ((x5 Xx) ((cP cZ) Xt))):(forall (Xx:a), (((eq a) (cR ((cP Xx) ((cP cZ) Xt)))) ((cP cZ) Xt)))
% Instantiate: x6:=(fun (x7:a)=> (forall (Xx:a), (((eq a) (cR ((cP Xx) x7))) x7))):(a->Prop)
% Found (fun (Xx:a)=> ((x5 Xx) ((cP cZ) Xt))) as proof of (x6 ((cP cZ) Xt))
% Found (fun (Xx:a)=> ((x5 Xx) ((cP cZ) Xt))) as proof of (x6 ((cP cZ) Xt))
% Found eq_ref00:=(eq_ref0 (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))):(((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu)))))))))
% Found (eq_ref0 (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) b)
% Found x60:=(fun (Xx:a)=> ((x6 Xx) ((cP cZ) Xt))):(forall (Xx:a), (((eq a) (cR ((cP Xx) ((cP cZ) Xt)))) ((cP cZ) Xt)))
% Instantiate: x0:=(fun (x7:a)=> (forall (Xx:a), (((eq a) (cR ((cP Xx) x7))) x7))):(a->Prop)
% Found (fun (Xx:a)=> ((x6 Xx) ((cP cZ) Xt))) as proof of (x0 ((cP cZ) Xt))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP cZ) Xt))) as proof of (x0 ((cP cZ) Xt))
% Found x60:=(fun (Xx:a)=> ((x6 Xx) ((cP cZ) Xt))):(forall (Xx:a), (((eq a) (cR ((cP Xx) ((cP cZ) Xt)))) ((cP cZ) Xt)))
% Instantiate: x2:=(fun (x7:a)=> (forall (Xx:a), (((eq a) (cR ((cP Xx) x7))) x7))):(a->Prop)
% Found (fun (Xx:a)=> ((x6 Xx) ((cP cZ) Xt))) as proof of (x2 ((cP cZ) Xt))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP cZ) Xt))) as proof of (x2 ((cP cZ) Xt))
% Found x60:=(fun (Xx:a)=> ((x6 Xx) ((cP cZ) Xt))):(forall (Xx:a), (((eq a) (cR ((cP Xx) ((cP cZ) Xt)))) ((cP cZ) Xt)))
% Instantiate: x4:=(fun (x7:a)=> (forall (Xx:a), (((eq a) (cR ((cP Xx) x7))) x7))):(a->Prop)
% Found (fun (Xx:a)=> ((x6 Xx) ((cP cZ) Xt))) as proof of (x4 ((cP cZ) Xt))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP cZ) Xt))) as proof of (x4 ((cP cZ) Xt))
% Found conj0100:=(conj010 x00):((and (x2 ((cP cZ) Xt))) (x2 ((cP (cL Xt0)) (cL Xu))))
% Found (conj010 x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))
% Found ((conj01 (x2 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x2 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x2 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x2 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))
% Found x00:(x2 ((cP Xt0) Xu))
% Found x00 as proof of (x2 ((cP (cR Xt0)) (cR Xu)))
% Found ((conj10 (((fun (B:Prop)=> ((conj0 B) x00)) (x2 ((cP (cL Xt0)) (cL Xu)))) x00)) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))) (x2 ((cP (cR Xt0)) (cR Xu))))
% Found (((conj1 (x2 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x2 ((cP (cL Xt0)) (cL Xu)))) x00)) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))) (x2 ((cP (cR Xt0)) (cR Xu))))
% Found ((((conj ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))) (x2 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x2 ((cP (cL Xt0)) (cL Xu)))) x00)) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))) (x2 ((cP (cR Xt0)) (cR Xu))))
% Found (fun (x00:(x2 ((cP Xt0) Xu)))=> ((((conj ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))) (x2 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x2 ((cP (cL Xt0)) (cL Xu)))) x00)) x00)) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))) (x2 ((cP (cR Xt0)) (cR Xu))))
% Found eq_substitution0000:=(fun (a0:a)=> ((eq_substitution000 a0) (cP cZ))):(forall (a0:a), ((((eq a) a0) Xt)->(((eq a) ((cP cZ) a0)) ((cP cZ) Xt))))
% Instantiate: x4:=(fun (x5:a)=> (forall (a0:a), ((((eq a) a0) Xt)->(((eq a) ((cP cZ) a0)) x5)))):(a->Prop)
% Found (fun (a0:a)=> ((eq_substitution000 a0) (cP cZ))) as proof of (x4 ((cP cZ) Xt))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((eq_substitution00 a0) Xt)) a0) (cP cZ))) as proof of (x4 ((cP cZ) Xt))
% Found (fun (a0:a)=> (((fun (a0:a)=> (((eq_substitution0 a) a0) Xt)) a0) (cP cZ))) as proof of (x4 ((cP cZ) Xt))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((((eq_substitution a) a) a0) Xt)) a0) (cP cZ))) as proof of (x4 ((cP cZ) Xt))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((((eq_substitution a) a) a0) Xt)) a0) (cP cZ))) as proof of (x4 ((cP cZ) Xt))
% Found conj0100:=(conj010 x00):((and (x0 ((cP cZ) Xt))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found (conj010 x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found ((conj01 (x0 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found x00:(x0 ((cP Xt0) Xu))
% Found x00 as proof of (x0 ((cP (cR Xt0)) (cR Xu)))
% Found ((conj10 (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00)) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu))))
% Found (((conj1 (x0 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00)) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu))))
% Found ((((conj ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00)) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu))))
% Found (fun (x00:(x0 ((cP Xt0) Xu)))=> ((((conj ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00)) x00)) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))):(((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) (fun (x:(a->Prop))=> ((and (x ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((x ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x ((cP (cL Xt0)) (cL Xu))))) (x ((cP (cR Xt0)) (cR Xu)))))))))
% Found (eta_expansion_dep00 (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) b)
% Found ((eta_expansion_dep0 (fun (x5:(a->Prop))=> Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) b)
% Found eq_substitution0000:=(fun (a0:a)=> ((eq_substitution000 a0) (cP cZ))):(forall (a0:a), ((((eq a) a0) Xt)->(((eq a) ((cP cZ) a0)) ((cP cZ) Xt))))
% Instantiate: x0:=(fun (x5:a)=> (forall (a0:a), ((((eq a) a0) Xt)->(((eq a) ((cP cZ) a0)) x5)))):(a->Prop)
% Found (fun (a0:a)=> ((eq_substitution000 a0) (cP cZ))) as proof of (x0 ((cP cZ) Xt))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((eq_substitution00 a0) Xt)) a0) (cP cZ))) as proof of (x0 ((cP cZ) Xt))
% Found (fun (a0:a)=> (((fun (a0:a)=> (((eq_substitution0 a) a0) Xt)) a0) (cP cZ))) as proof of (x0 ((cP cZ) Xt))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((((eq_substitution a) a) a0) Xt)) a0) (cP cZ))) as proof of (x0 ((cP cZ) Xt))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((((eq_substitution a) a) a0) Xt)) a0) (cP cZ))) as proof of (x0 ((cP cZ) Xt))
% Found eq_substitution0000:=(fun (a0:a)=> ((eq_substitution000 a0) (cP cZ))):(forall (a0:a), ((((eq a) a0) Xt)->(((eq a) ((cP cZ) a0)) ((cP cZ) Xt))))
% Instantiate: x2:=(fun (x5:a)=> (forall (a0:a), ((((eq a) a0) Xt)->(((eq a) ((cP cZ) a0)) x5)))):(a->Prop)
% Found (fun (a0:a)=> ((eq_substitution000 a0) (cP cZ))) as proof of (x2 ((cP cZ) Xt))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((eq_substitution00 a0) Xt)) a0) (cP cZ))) as proof of (x2 ((cP cZ) Xt))
% Found (fun (a0:a)=> (((fun (a0:a)=> (((eq_substitution0 a) a0) Xt)) a0) (cP cZ))) as proof of (x2 ((cP cZ) Xt))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((((eq_substitution a) a) a0) Xt)) a0) (cP cZ))) as proof of (x2 ((cP cZ) Xt))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((((eq_substitution a) a) a0) Xt)) a0) (cP cZ))) as proof of (x2 ((cP cZ) Xt))
% Found conj0100:=(conj010 x00):((and (x0 ((cP cZ) Xt))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found (conj010 x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found ((conj01 (x0 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found x00:(x0 ((cP Xt0) Xu))
% Found x00 as proof of (x0 ((cP (cR Xt0)) (cR Xu)))
% Found ((conj10 (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00)) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu))))
% Found (((conj1 (x0 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00)) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu))))
% Found ((((conj ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00)) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu))))
% Found (fun (x2:(forall (X:(a->Prop)), ((forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((iff (((eq a) Xt) cZ)) (((eq a) Xu0) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))->(forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->(((eq a) Xt) Xu0))))))=> ((((conj ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00)) x00)) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu))))
% Found (fun (x1:((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))) (x2:(forall (X:(a->Prop)), ((forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((iff (((eq a) Xt) cZ)) (((eq a) Xu0) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))->(forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->(((eq a) Xt) Xu0))))))=> ((((conj ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00)) x00)) as proof of ((forall (X:(a->Prop)), ((forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((iff (((eq a) Xt) cZ)) (((eq a) Xu0) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))->(forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->(((eq a) Xt) Xu0)))))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu)))))
% Found (fun (x1:((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))) (x2:(forall (X:(a->Prop)), ((forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((iff (((eq a) Xt) cZ)) (((eq a) Xu0) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))->(forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->(((eq a) Xt) Xu0))))))=> ((((conj ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00)) x00)) as proof of (((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))->((forall (X:(a->Prop)), ((forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((iff (((eq a) Xt) cZ)) (((eq a) Xu0) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))->(forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->(((eq a) Xt) Xu0)))))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu))))))
% Found (and_rect00 (fun (x1:((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))) (x2:(forall (X:(a->Prop)), ((forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((iff (((eq a) Xt) cZ)) (((eq a) Xu0) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))->(forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->(((eq a) Xt) Xu0))))))=> ((((conj ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00)) x00))) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu))))
% Found ((and_rect0 ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu))))) (fun (x1:((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))) (x2:(forall (X:(a->Prop)), ((forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((iff (((eq a) Xt) cZ)) (((eq a) Xu0) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))->(forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->(((eq a) Xt) Xu0))))))=> ((((conj ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00)) x00))) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu))))
% Found (((fun (P:Type) (x1:(((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))->((forall (X:(a->Prop)), ((forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((iff (((eq a) Xt) cZ)) (((eq a) Xu) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))->(forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->(((eq a) Xt) Xu)))))->P)))=> (((((and_rect ((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))) (forall (X:(a->Prop)), ((forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((iff (((eq a) Xt) cZ)) (((eq a) Xu) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))->(forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->(((eq a) Xt) Xu)))))) P) x1) x)) ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu))))) (fun (x1:((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))) (x2:(forall (X:(a->Prop)), ((forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((iff (((eq a) Xt) cZ)) (((eq a) Xu0) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))->(forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->(((eq a) Xt) Xu0))))))=> ((((conj ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00)) x00))) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu))))
% Found (fun (x00:(x0 ((cP Xt0) Xu)))=> (((fun (P:Type) (x1:(((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))->((forall (X:(a->Prop)), ((forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((iff (((eq a) Xt) cZ)) (((eq a) Xu) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))->(forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->(((eq a) Xt) Xu)))))->P)))=> (((((and_rect ((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))) (forall (X:(a->Prop)), ((forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((iff (((eq a) Xt) cZ)) (((eq a) Xu) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))->(forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->(((eq a) Xt) Xu)))))) P) x1) x)) ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu))))) (fun (x1:((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))) (x2:(forall (X:(a->Prop)), ((forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((iff (((eq a) Xt) cZ)) (((eq a) Xu0) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))->(forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->(((eq a) Xt) Xu0))))))=> ((((conj ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00)) x00)))) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu))))
% Found x50:=(fun (Xx:a)=> ((x5 Xx) ((cP cZ) Xt))):(forall (Xx:a), (((eq a) (cR ((cP Xx) ((cP cZ) Xt)))) ((cP cZ) Xt)))
% Instantiate: x8:=(fun (x9:a)=> (forall (Xx:a), (((eq a) (cR ((cP Xx) x9))) x9))):(a->Prop)
% Found (fun (Xx:a)=> ((x5 Xx) ((cP cZ) Xt))) as proof of (x8 ((cP cZ) Xt))
% Found (fun (Xx:a)=> ((x5 Xx) ((cP cZ) Xt))) as proof of (x8 ((cP cZ) Xt))
% Found x500:=(x50 ((cP (cR Xt0)) (cR Xu))):(((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (x50 ((cP (cR Xt0)) (cR Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found ((x5 Xx) ((cP (cR Xt0)) (cR Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x5 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x5 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x6 ((cP (cR Xt0)) (cR Xu)))
% Found x60:=(fun (Xx:a)=> ((x6 Xx) ((cP cZ) Xt))):(forall (Xx:a), (((eq a) (cR ((cP Xx) ((cP cZ) Xt)))) ((cP cZ) Xt)))
% Instantiate: x0:=(fun (x9:a)=> (forall (Xx:a), (((eq a) (cR ((cP Xx) x9))) x9))):(a->Prop)
% Found (fun (Xx:a)=> ((x6 Xx) ((cP cZ) Xt))) as proof of (x0 ((cP cZ) Xt))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP cZ) Xt))) as proof of (x0 ((cP cZ) Xt))
% Found x60:=(fun (Xx:a)=> ((x6 Xx) ((cP cZ) Xt))):(forall (Xx:a), (((eq a) (cR ((cP Xx) ((cP cZ) Xt)))) ((cP cZ) Xt)))
% Instantiate: x2:=(fun (x9:a)=> (forall (Xx:a), (((eq a) (cR ((cP Xx) x9))) x9))):(a->Prop)
% Found (fun (Xx:a)=> ((x6 Xx) ((cP cZ) Xt))) as proof of (x2 ((cP cZ) Xt))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP cZ) Xt))) as proof of (x2 ((cP cZ) Xt))
% Found x60:=(fun (Xx:a)=> ((x6 Xx) ((cP cZ) Xt))):(forall (Xx:a), (((eq a) (cR ((cP Xx) ((cP cZ) Xt)))) ((cP cZ) Xt)))
% Instantiate: x4:=(fun (x9:a)=> (forall (Xx:a), (((eq a) (cR ((cP Xx) x9))) x9))):(a->Prop)
% Found (fun (Xx:a)=> ((x6 Xx) ((cP cZ) Xt))) as proof of (x4 ((cP cZ) Xt))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP cZ) Xt))) as proof of (x4 ((cP cZ) Xt))
% Found x50:=(fun (Xx:a)=> ((x5 Xx) ((cP cZ) Xt))):(forall (Xx:a), (((eq a) (cR ((cP Xx) ((cP cZ) Xt)))) ((cP cZ) Xt)))
% Instantiate: x6:=(fun (x9:a)=> (forall (Xx:a), (((eq a) (cR ((cP Xx) x9))) x9))):(a->Prop)
% Found (fun (Xx:a)=> ((x5 Xx) ((cP cZ) Xt))) as proof of (x6 ((cP cZ) Xt))
% Found (fun (Xx:a)=> ((x5 Xx) ((cP cZ) Xt))) as proof of (x6 ((cP cZ) Xt))
% Found x600:=(x60 ((cP (cR Xt0)) (cR Xu))):(((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (x60 ((cP (cR Xt0)) (cR Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found ((x6 Xx) ((cP (cR Xt0)) (cR Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x0 ((cP (cR Xt0)) (cR Xu)))
% Found x600:=(x60 ((cP (cR Xt0)) (cR Xu))):(((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (x60 ((cP (cR Xt0)) (cR Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found ((x6 Xx) ((cP (cR Xt0)) (cR Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x2 ((cP (cR Xt0)) (cR Xu)))
% Found x600:=(x60 ((cP (cR Xt0)) (cR Xu))):(((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (x60 ((cP (cR Xt0)) (cR Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found ((x6 Xx) ((cP (cR Xt0)) (cR Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x4 ((cP (cR Xt0)) (cR Xu)))
% Found x6:(((eq a) Xt0) cZ)
% Found x6 as proof of (((eq a) Xt0) cZ)
% Found conj0100:=(conj010 x00):((and (x4 ((cP cZ) Xt))) (x4 ((cP (cL Xt0)) (cL Xu))))
% Found (conj010 x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x4 ((cP (cL Xt0)) (cL Xu))))
% Found ((conj01 (x4 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x4 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x4 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x4 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x4 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x4 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x4 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x4 ((cP (cL Xt0)) (cL Xu))))
% Found x00:(x4 ((cP Xt0) Xu))
% Found x00 as proof of (x4 ((cP (cR Xt0)) (cR Xu)))
% Found ((conj10 (((fun (B:Prop)=> ((conj0 B) x00)) (x4 ((cP (cL Xt0)) (cL Xu)))) x00)) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x4 ((cP (cL Xt0)) (cL Xu))))) (x4 ((cP (cR Xt0)) (cR Xu))))
% Found (((conj1 (x4 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x4 ((cP (cL Xt0)) (cL Xu)))) x00)) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x4 ((cP (cL Xt0)) (cL Xu))))) (x4 ((cP (cR Xt0)) (cR Xu))))
% Found ((((conj ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x4 ((cP (cL Xt0)) (cL Xu))))) (x4 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x4 ((cP (cL Xt0)) (cL Xu)))) x00)) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x4 ((cP (cL Xt0)) (cL Xu))))) (x4 ((cP (cR Xt0)) (cR Xu))))
% Found (fun (x00:(x4 ((cP Xt0) Xu)))=> ((((conj ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x4 ((cP (cL Xt0)) (cL Xu))))) (x4 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x4 ((cP (cL Xt0)) (cL Xu)))) x00)) x00)) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x4 ((cP (cL Xt0)) (cL Xu))))) (x4 ((cP (cR Xt0)) (cR Xu))))
% Found x50:=(fun (Xx:a)=> ((x5 Xx) ((cP (cR Xt0)) (cR Xu)))):(forall (Xx:a), (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu))))
% Instantiate: x6:=(fun (x7:a)=> (forall (Xx:a), (((eq a) (cR ((cP Xx) x7))) x7))):(a->Prop)
% Found (fun (Xx:a)=> ((x5 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x6 ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x5 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x6 ((cP (cR Xt0)) (cR Xu)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))):(((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) (fun (x:(a->Prop))=> ((and (x ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((x ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x ((cP (cL Xt0)) (cL Xu))))) (x ((cP (cR Xt0)) (cR Xu)))))))))
% Found (eta_expansion_dep00 (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) b)
% Found ((eta_expansion_dep0 (fun (x7:(a->Prop))=> Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x7:(a->Prop))=> Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x7:(a->Prop))=> Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x7:(a->Prop))=> Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) b)
% Found conj0100:=(conj010 x00):((and (x0 ((cP cZ) Xt))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found (conj010 x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found ((conj01 (x0 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found x00:(x0 ((cP Xt0) Xu))
% Found x00 as proof of (x0 ((cP (cR Xt0)) (cR Xu)))
% Found ((conj10 (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00)) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu))))
% Found (((conj1 (x0 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00)) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu))))
% Found ((((conj ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00)) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu))))
% Found (fun (x00:(x0 ((cP Xt0) Xu)))=> ((((conj ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00)) x00)) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu))))
% Found conj0100:=(conj010 x00):((and (x2 ((cP cZ) Xt))) (x2 ((cP (cL Xt0)) (cL Xu))))
% Found (conj010 x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))
% Found ((conj01 (x2 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x2 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x2 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x2 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))
% Found x00:(x2 ((cP Xt0) Xu))
% Found x00 as proof of (x2 ((cP (cR Xt0)) (cR Xu)))
% Found ((conj10 (((fun (B:Prop)=> ((conj0 B) x00)) (x2 ((cP (cL Xt0)) (cL Xu)))) x00)) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))) (x2 ((cP (cR Xt0)) (cR Xu))))
% Found (((conj1 (x2 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x2 ((cP (cL Xt0)) (cL Xu)))) x00)) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))) (x2 ((cP (cR Xt0)) (cR Xu))))
% Found ((((conj ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))) (x2 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x2 ((cP (cL Xt0)) (cL Xu)))) x00)) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))) (x2 ((cP (cR Xt0)) (cR Xu))))
% Found (fun (x00:(x2 ((cP Xt0) Xu)))=> ((((conj ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))) (x2 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x2 ((cP (cL Xt0)) (cL Xu)))) x00)) x00)) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))) (x2 ((cP (cR Xt0)) (cR Xu))))
% Found x60:=(fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))):(forall (Xx:a), (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu))))
% Instantiate: x0:=(fun (x7:a)=> (forall (Xx:a), (((eq a) (cR ((cP Xx) x7))) x7))):(a->Prop)
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x0 ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x0 ((cP (cR Xt0)) (cR Xu)))
% Found x60:=(fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))):(forall (Xx:a), (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu))))
% Instantiate: x2:=(fun (x7:a)=> (forall (Xx:a), (((eq a) (cR ((cP Xx) x7))) x7))):(a->Prop)
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x2 ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x2 ((cP (cR Xt0)) (cR Xu)))
% Found x60:=(fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))):(forall (Xx:a), (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu))))
% Instantiate: x4:=(fun (x7:a)=> (forall (Xx:a), (((eq a) (cR ((cP Xx) x7))) x7))):(a->Prop)
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x4 ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x4 ((cP (cR Xt0)) (cR Xu)))
% Found conj0100:=(conj010 x00):((and (x6 ((cP cZ) Xt))) (x6 ((cP (cL Xt0)) (cL Xu))))
% Found (conj010 x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x6 ((cP (cL Xt0)) (cL Xu))))
% Found ((conj01 (x6 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x6 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x6 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x6 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x6 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x6 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x6 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x6 ((cP (cL Xt0)) (cL Xu))))
% Found conj0100:=(conj010 x00):((and (x0 ((cP cZ) Xt))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found (conj010 x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found ((conj01 (x0 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found x00:(x0 ((cP Xt0) Xu))
% Found x00 as proof of (x0 ((cP (cR Xt0)) (cR Xu)))
% Found ((conj10 (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00)) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu))))
% Found (((conj1 (x0 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00)) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu))))
% Found ((((conj ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00)) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu))))
% Found (fun (x4:(forall (Xt00:a), ((iff (not (((eq a) Xt00) cZ))) (((eq a) Xt00) ((cP (cL Xt00)) (cR Xt00))))))=> ((((conj ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00)) x00)) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu))))
% Found (fun (x3:((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (x4:(forall (Xt00:a), ((iff (not (((eq a) Xt00) cZ))) (((eq a) Xt00) ((cP (cL Xt00)) (cR Xt00))))))=> ((((conj ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00)) x00)) as proof of ((forall (Xt00:a), ((iff (not (((eq a) Xt00) cZ))) (((eq a) Xt00) ((cP (cL Xt00)) (cR Xt00)))))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu)))))
% Found (fun (x3:((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (x4:(forall (Xt00:a), ((iff (not (((eq a) Xt00) cZ))) (((eq a) Xt00) ((cP (cL Xt00)) (cR Xt00))))))=> ((((conj ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00)) x00)) as proof of (((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))->((forall (Xt00:a), ((iff (not (((eq a) Xt00) cZ))) (((eq a) Xt00) ((cP (cL Xt00)) (cR Xt00)))))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu))))))
% Found (and_rect10 (fun (x3:((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (x4:(forall (Xt00:a), ((iff (not (((eq a) Xt00) cZ))) (((eq a) Xt00) ((cP (cL Xt00)) (cR Xt00))))))=> ((((conj ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00)) x00))) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu))))
% Found ((and_rect1 ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu))))) (fun (x3:((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (x4:(forall (Xt00:a), ((iff (not (((eq a) Xt00) cZ))) (((eq a) Xt00) ((cP (cL Xt00)) (cR Xt00))))))=> ((((conj ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00)) x00))) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu))))
% Found (((fun (P:Type) (x3:(((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))->((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->P)))=> (((((and_rect ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))) P) x3) x1)) ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu))))) (fun (x3:((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (x4:(forall (Xt00:a), ((iff (not (((eq a) Xt00) cZ))) (((eq a) Xt00) ((cP (cL Xt00)) (cR Xt00))))))=> ((((conj ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00)) x00))) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu))))
% Found (fun (x00:(x0 ((cP Xt0) Xu)))=> (((fun (P:Type) (x3:(((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))->((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->P)))=> (((((and_rect ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))) P) x3) x1)) ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu))))) (fun (x3:((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (x4:(forall (Xt00:a), ((iff (not (((eq a) Xt00) cZ))) (((eq a) Xt00) ((cP (cL Xt00)) (cR Xt00))))))=> ((((conj ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00)) x00)))) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))) (x0 ((cP (cR Xt0)) (cR Xu))))
% Found conj0100:=(conj010 x00):((and (x2 ((cP cZ) Xt))) (x2 ((cP (cL Xt0)) (cL Xu))))
% Found (conj010 x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))
% Found ((conj01 (x2 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x2 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x2 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x2 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))
% Found x00:(x2 ((cP Xt0) Xu))
% Found x00 as proof of (x2 ((cP (cR Xt0)) (cR Xu)))
% Found ((conj10 (((fun (B:Prop)=> ((conj0 B) x00)) (x2 ((cP (cL Xt0)) (cL Xu)))) x00)) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))) (x2 ((cP (cR Xt0)) (cR Xu))))
% Found (((conj1 (x2 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x2 ((cP (cL Xt0)) (cL Xu)))) x00)) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))) (x2 ((cP (cR Xt0)) (cR Xu))))
% Found ((((conj ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))) (x2 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x2 ((cP (cL Xt0)) (cL Xu)))) x00)) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))) (x2 ((cP (cR Xt0)) (cR Xu))))
% Found (fun (x4:(forall (Xt00:a), ((iff (not (((eq a) Xt00) cZ))) (((eq a) Xt00) ((cP (cL Xt00)) (cR Xt00))))))=> ((((conj ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))) (x2 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x2 ((cP (cL Xt0)) (cL Xu)))) x00)) x00)) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))) (x2 ((cP (cR Xt0)) (cR Xu))))
% Found (fun (x3:((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (x4:(forall (Xt00:a), ((iff (not (((eq a) Xt00) cZ))) (((eq a) Xt00) ((cP (cL Xt00)) (cR Xt00))))))=> ((((conj ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))) (x2 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x2 ((cP (cL Xt0)) (cL Xu)))) x00)) x00)) as proof of ((forall (Xt00:a), ((iff (not (((eq a) Xt00) cZ))) (((eq a) Xt00) ((cP (cL Xt00)) (cR Xt00)))))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))) (x2 ((cP (cR Xt0)) (cR Xu)))))
% Found (fun (x3:((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (x4:(forall (Xt00:a), ((iff (not (((eq a) Xt00) cZ))) (((eq a) Xt00) ((cP (cL Xt00)) (cR Xt00))))))=> ((((conj ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))) (x2 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x2 ((cP (cL Xt0)) (cL Xu)))) x00)) x00)) as proof of (((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))->((forall (Xt00:a), ((iff (not (((eq a) Xt00) cZ))) (((eq a) Xt00) ((cP (cL Xt00)) (cR Xt00)))))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))) (x2 ((cP (cR Xt0)) (cR Xu))))))
% Found (and_rect10 (fun (x3:((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (x4:(forall (Xt00:a), ((iff (not (((eq a) Xt00) cZ))) (((eq a) Xt00) ((cP (cL Xt00)) (cR Xt00))))))=> ((((conj ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))) (x2 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x2 ((cP (cL Xt0)) (cL Xu)))) x00)) x00))) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))) (x2 ((cP (cR Xt0)) (cR Xu))))
% Found ((and_rect1 ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))) (x2 ((cP (cR Xt0)) (cR Xu))))) (fun (x3:((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (x4:(forall (Xt00:a), ((iff (not (((eq a) Xt00) cZ))) (((eq a) Xt00) ((cP (cL Xt00)) (cR Xt00))))))=> ((((conj ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))) (x2 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x2 ((cP (cL Xt0)) (cL Xu)))) x00)) x00))) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))) (x2 ((cP (cR Xt0)) (cR Xu))))
% Found (((fun (P:Type) (x3:(((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))->((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->P)))=> (((((and_rect ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))) P) x3) x0)) ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))) (x2 ((cP (cR Xt0)) (cR Xu))))) (fun (x3:((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (x4:(forall (Xt00:a), ((iff (not (((eq a) Xt00) cZ))) (((eq a) Xt00) ((cP (cL Xt00)) (cR Xt00))))))=> ((((conj ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))) (x2 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x2 ((cP (cL Xt0)) (cL Xu)))) x00)) x00))) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))) (x2 ((cP (cR Xt0)) (cR Xu))))
% Found (fun (x00:(x2 ((cP Xt0) Xu)))=> (((fun (P:Type) (x3:(((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))->((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->P)))=> (((((and_rect ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))) P) x3) x0)) ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))) (x2 ((cP (cR Xt0)) (cR Xu))))) (fun (x3:((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (x4:(forall (Xt00:a), ((iff (not (((eq a) Xt00) cZ))) (((eq a) Xt00) ((cP (cL Xt00)) (cR Xt00))))))=> ((((conj ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))) (x2 ((cP (cR Xt0)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x2 ((cP (cL Xt0)) (cL Xu)))) x00)) x00)))) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))) (x2 ((cP (cR Xt0)) (cR Xu))))
% Found x60:=(fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))):(forall (Xx:a), (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu))))
% Instantiate: x0:=(fun (x7:a)=> (forall (Xx:a), (((eq a) (cR ((cP Xx) x7))) x7))):(a->Prop)
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x0 ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x0 ((cP (cR Xt0)) (cR Xu)))
% Found x60:=(fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))):(forall (Xx:a), (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu))))
% Instantiate: x2:=(fun (x7:a)=> (forall (Xx:a), (((eq a) (cR ((cP Xx) x7))) x7))):(a->Prop)
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x2 ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x2 ((cP (cR Xt0)) (cR Xu)))
% Found x60:=(fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))):(forall (Xx:a), (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu))))
% Instantiate: x4:=(fun (x7:a)=> (forall (Xx:a), (((eq a) (cR ((cP Xx) x7))) x7))):(a->Prop)
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x4 ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x4 ((cP (cR Xt0)) (cR Xu)))
% Found conj0100:=(conj010 x00):((and (x0 ((cP cZ) Xt))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found (conj010 x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found ((conj01 (x0 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found conj0100:=(conj010 x00):((and (x2 ((cP cZ) Xt))) (x2 ((cP (cL Xt0)) (cL Xu))))
% Found (conj010 x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))
% Found ((conj01 (x2 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x2 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x2 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x2 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))
% Found conj0100:=(conj010 x00):((and (x4 ((cP cZ) Xt))) (x4 ((cP (cL Xt0)) (cL Xu))))
% Found (conj010 x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x4 ((cP (cL Xt0)) (cL Xu))))
% Found ((conj01 (x4 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x4 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x4 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x4 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x4 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x4 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x4 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x4 ((cP (cL Xt0)) (cL Xu))))
% Found x500:=(x50 ((cP (cL Xt0)) (cL Xu))):(((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found (x50 ((cP (cL Xt0)) (cL Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found ((x5 Xx) ((cP (cL Xt0)) (cL Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found (fun (Xx:a)=> ((x5 Xx) ((cP (cL Xt0)) (cL Xu)))) as proof of (((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found (fun (Xx:a)=> ((x5 Xx) ((cP (cL Xt0)) (cL Xu)))) as proof of (x6 ((cP (cL Xt0)) (cL Xu)))
% Found x500:=(x50 ((cP (cL Xt0)) (cL Xu))):(((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found (x50 ((cP (cL Xt0)) (cL Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found ((x5 Xx) ((cP (cL Xt0)) (cL Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found (fun (Xx:a)=> ((x5 Xx) ((cP (cL Xt0)) (cL Xu)))) as proof of (((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found (fun (Xx:a)=> ((x5 Xx) ((cP (cL Xt0)) (cL Xu)))) as proof of (x6 ((cP (cL Xt0)) (cL Xu)))
% Found x50:=(fun (Xx:a)=> ((x5 Xx) ((cP cZ) Xt))):(forall (Xx:a), (((eq a) (cR ((cP Xx) ((cP cZ) Xt)))) ((cP cZ) Xt)))
% Instantiate: x10:=(fun (x11:a)=> (forall (Xx:a), (((eq a) (cR ((cP Xx) x11))) x11))):(a->Prop)
% Found (fun (Xx:a)=> ((x5 Xx) ((cP cZ) Xt))) as proof of (x10 ((cP cZ) Xt))
% Found (fun (Xx:a)=> ((x5 Xx) ((cP cZ) Xt))) as proof of (x10 ((cP cZ) Xt))
% Found conj0100:=(conj010 x00):((and (x0 ((cP cZ) Xt))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found (conj010 x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found ((conj01 (x0 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found conj0100:=(conj010 x00):((and (x2 ((cP cZ) Xt))) (x2 ((cP (cL Xt0)) (cL Xu))))
% Found (conj010 x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))
% Found ((conj01 (x2 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x2 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x2 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x2 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))
% Found conj0100:=(conj010 x00):((and (x4 ((cP cZ) Xt))) (x4 ((cP (cL Xt0)) (cL Xu))))
% Found (conj010 x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x4 ((cP (cL Xt0)) (cL Xu))))
% Found ((conj01 (x4 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x4 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x4 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x4 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x4 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x4 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x4 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x4 ((cP (cL Xt0)) (cL Xu))))
% Found x6:(((eq a) Xt0) cZ)
% Found x6 as proof of (((eq a) Xt0) cZ)
% Found x500:=(x50 ((cP (cR Xt0)) (cR Xu))):(((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (x50 ((cP (cR Xt0)) (cR Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found ((x5 Xx) ((cP (cR Xt0)) (cR Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x5 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x5 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x8 ((cP (cR Xt0)) (cR Xu)))
% Found x600:=(x60 ((cP (cL Xt0)) (cL Xu))):(((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found (x60 ((cP (cL Xt0)) (cL Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found ((x6 Xx) ((cP (cL Xt0)) (cL Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cL Xt0)) (cL Xu)))) as proof of (((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cL Xt0)) (cL Xu)))) as proof of (x0 ((cP (cL Xt0)) (cL Xu)))
% Found x600:=(x60 ((cP (cL Xt0)) (cL Xu))):(((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found (x60 ((cP (cL Xt0)) (cL Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found ((x6 Xx) ((cP (cL Xt0)) (cL Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cL Xt0)) (cL Xu)))) as proof of (((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cL Xt0)) (cL Xu)))) as proof of (x0 ((cP (cL Xt0)) (cL Xu)))
% Found x600:=(x60 ((cP (cL Xt0)) (cL Xu))):(((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found (x60 ((cP (cL Xt0)) (cL Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found ((x6 Xx) ((cP (cL Xt0)) (cL Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cL Xt0)) (cL Xu)))) as proof of (((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cL Xt0)) (cL Xu)))) as proof of (x2 ((cP (cL Xt0)) (cL Xu)))
% Found x600:=(x60 ((cP (cL Xt0)) (cL Xu))):(((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found (x60 ((cP (cL Xt0)) (cL Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found ((x6 Xx) ((cP (cL Xt0)) (cL Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cL Xt0)) (cL Xu)))) as proof of (((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cL Xt0)) (cL Xu)))) as proof of (x2 ((cP (cL Xt0)) (cL Xu)))
% Found x600:=(x60 ((cP (cL Xt0)) (cL Xu))):(((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found (x60 ((cP (cL Xt0)) (cL Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found ((x6 Xx) ((cP (cL Xt0)) (cL Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cL Xt0)) (cL Xu)))) as proof of (((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cL Xt0)) (cL Xu)))) as proof of (x4 ((cP (cL Xt0)) (cL Xu)))
% Found x600:=(x60 ((cP (cL Xt0)) (cL Xu))):(((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found (x60 ((cP (cL Xt0)) (cL Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found ((x6 Xx) ((cP (cL Xt0)) (cL Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cL Xt0)) (cL Xu)))) as proof of (((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cL Xt0)) (cL Xu)))) as proof of (x4 ((cP (cL Xt0)) (cL Xu)))
% Found x60:=(fun (Xx:a)=> ((x6 Xx) ((cP cZ) Xt))):(forall (Xx:a), (((eq a) (cR ((cP Xx) ((cP cZ) Xt)))) ((cP cZ) Xt)))
% Instantiate: x0:=(fun (x11:a)=> (forall (Xx:a), (((eq a) (cR ((cP Xx) x11))) x11))):(a->Prop)
% Found (fun (Xx:a)=> ((x6 Xx) ((cP cZ) Xt))) as proof of (x0 ((cP cZ) Xt))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP cZ) Xt))) as proof of (x0 ((cP cZ) Xt))
% Found x60:=(fun (Xx:a)=> ((x6 Xx) ((cP cZ) Xt))):(forall (Xx:a), (((eq a) (cR ((cP Xx) ((cP cZ) Xt)))) ((cP cZ) Xt)))
% Instantiate: x2:=(fun (x11:a)=> (forall (Xx:a), (((eq a) (cR ((cP Xx) x11))) x11))):(a->Prop)
% Found (fun (Xx:a)=> ((x6 Xx) ((cP cZ) Xt))) as proof of (x2 ((cP cZ) Xt))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP cZ) Xt))) as proof of (x2 ((cP cZ) Xt))
% Found x60:=(fun (Xx:a)=> ((x6 Xx) ((cP cZ) Xt))):(forall (Xx:a), (((eq a) (cR ((cP Xx) ((cP cZ) Xt)))) ((cP cZ) Xt)))
% Instantiate: x4:=(fun (x11:a)=> (forall (Xx:a), (((eq a) (cR ((cP Xx) x11))) x11))):(a->Prop)
% Found (fun (Xx:a)=> ((x6 Xx) ((cP cZ) Xt))) as proof of (x4 ((cP cZ) Xt))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP cZ) Xt))) as proof of (x4 ((cP cZ) Xt))
% Found x50:=(fun (Xx:a)=> ((x5 Xx) ((cP cZ) Xt))):(forall (Xx:a), (((eq a) (cR ((cP Xx) ((cP cZ) Xt)))) ((cP cZ) Xt)))
% Instantiate: x6:=(fun (x11:a)=> (forall (Xx:a), (((eq a) (cR ((cP Xx) x11))) x11))):(a->Prop)
% Found (fun (Xx:a)=> ((x5 Xx) ((cP cZ) Xt))) as proof of (x6 ((cP cZ) Xt))
% Found (fun (Xx:a)=> ((x5 Xx) ((cP cZ) Xt))) as proof of (x6 ((cP cZ) Xt))
% Found x50:=(fun (Xx:a)=> ((x5 Xx) ((cP cZ) Xt))):(forall (Xx:a), (((eq a) (cR ((cP Xx) ((cP cZ) Xt)))) ((cP cZ) Xt)))
% Instantiate: x8:=(fun (x11:a)=> (forall (Xx:a), (((eq a) (cR ((cP Xx) x11))) x11))):(a->Prop)
% Found (fun (Xx:a)=> ((x5 Xx) ((cP cZ) Xt))) as proof of (x8 ((cP cZ) Xt))
% Found (fun (Xx:a)=> ((x5 Xx) ((cP cZ) Xt))) as proof of (x8 ((cP cZ) Xt))
% Found x7:(((eq a) Xt0) cZ)
% Found x7 as proof of (((eq a) Xt0) cZ)
% Found x600:=(x60 ((cP (cR Xt0)) (cR Xu))):(((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (x60 ((cP (cR Xt0)) (cR Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found ((x6 Xx) ((cP (cR Xt0)) (cR Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x0 ((cP (cR Xt0)) (cR Xu)))
% Found x600:=(x60 ((cP (cR Xt0)) (cR Xu))):(((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (x60 ((cP (cR Xt0)) (cR Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found ((x6 Xx) ((cP (cR Xt0)) (cR Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x2 ((cP (cR Xt0)) (cR Xu)))
% Found x600:=(x60 ((cP (cR Xt0)) (cR Xu))):(((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (x60 ((cP (cR Xt0)) (cR Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found ((x6 Xx) ((cP (cR Xt0)) (cR Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x4 ((cP (cR Xt0)) (cR Xu)))
% Found x500:=(x50 ((cP (cR Xt0)) (cR Xu))):(((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (x50 ((cP (cR Xt0)) (cR Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found ((x5 Xx) ((cP (cR Xt0)) (cR Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x5 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x5 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x6 ((cP (cR Xt0)) (cR Xu)))
% Found eq_ref00:=(eq_ref0 (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))):(((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu)))))))))
% Found (eq_ref0 (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP cZ) Xt))) (forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))))) b)
% Found x8:(((eq a) Xt0) cZ)
% Found x8 as proof of (((eq a) Xt0) cZ)
% Found x8:(((eq a) Xt0) cZ)
% Found x8 as proof of (((eq a) Xt0) cZ)
% Found eq_sym:=(fun (T:Type) (a:T) (b:T) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq T) x) a))) ((eq_ref T) a))):(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a)))
% Found eq_sym as proof of (forall (T:Type) (a00:T) (b:T), ((((eq T) a00) b)->(((eq T) b) a00)))
% Found eq_ref:=(fun (T:Type) (a:T) (P:(T->Prop)) (x:(P a))=> x):(forall (T:Type) (a:T), (((eq T) a) a))
% Found eq_ref as proof of (forall (T:Type) (a00:T), (((eq T) a00) a00))
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Found conj as proof of (forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Found eq_stepl:=(fun (T:Type) (a:T) (b:T) (c:T) (X:(((eq T) a) b)) (Y:(((eq T) a) c))=> ((((((eq_trans T) c) a) b) ((((eq_sym T) a) c) Y)) X)):(forall (T:Type) (a:T) (b:T) (c:T), ((((eq T) a) b)->((((eq T) a) c)->(((eq T) c) b))))
% Found eq_stepl as proof of (forall (T:Type) (a00:T) (b:T) (c:T), ((((eq T) a00) b)->((((eq T) a00) c)->(((eq T) c) b))))
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Found proj1 as proof of (forall (A:Prop) (B:Prop), (((and A) B)->A))
% Found x1:(forall (X:(a->Prop)), ((forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((iff (((eq a) Xt0) cZ)) (((eq a) Xu) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))->(forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->(((eq a) Xt0) Xu)))))
% Found x1 as proof of (forall (X0:(a->Prop)), ((forall (Xt00:a) (Xu0:a), ((X0 ((cP Xt00) Xu0))->((and ((and ((iff (((eq a) Xt00) cZ)) (((eq a) Xu0) cZ))) (X0 ((cP (cL Xt00)) (cL Xu0))))) (X0 ((cP (cR Xt00)) (cR Xu0))))))->(forall (Xt00:a) (Xu0:a), ((X0 ((cP Xt00) Xu0))->(((eq a) Xt00) Xu0)))))
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Found eq_substitution as proof of (forall (T:Type) (U:Type) (a00:T) (b:T) (f:(T->U)), ((((eq T) a00) b)->(((eq U) (f a00)) (f b))))
% Found x30:(((eq a) a0) Xt)
% Found x30 as proof of (((eq a) a0) Xt)
% Found eta_expansion_dep:=(fun (A:Type) (B:(A->Type)) (f:(forall (x:A), (B x)))=> (((((functional_extensionality_dep A) (fun (x1:A)=> (B x1))) f) (fun (x:A)=> (f x))) (fun (x:A) (P:((B x)->Prop)) (x0:(P (f x)))=> x0))):(forall (A:Type) (B:(A->Type)) (f:(forall (x:A), (B x))), (((eq (forall (x:A), (B x))) f) (fun (x:A)=> (f x))))
% Found eta_expansion_dep as proof of (forall (A:Type) (B:(A->Type)) (f:(forall (x5:A), (B x5))), (((eq (forall (x5:A), (B x5))) f) (fun (x5:A)=> (f x5))))
% Found functional_extensionality:=(fun (A:Type) (B:Type)=> ((functional_extensionality_dep A) (fun (x1:A)=> B))):(forall (A:Type) (B:Type) (f:(A->B)) (g:(A->B)), ((forall (x:A), (((eq B) (f x)) (g x)))->(((eq (A->B)) f) g)))
% Found functional_extensionality as proof of (forall (A:Type) (B:Type) (f:(A->B)) (g:(A->B)), ((forall (x5:A), (((eq B) (f x5)) (g x5)))->(((eq (A->B)) f) g)))
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Found iff_trans as proof of (forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Found functional_extensionality_dep:(forall (A:Type) (B:(A->Type)) (f:(forall (x:A), (B x))) (g:(forall (x:A), (B x))), ((forall (x:A), (((eq (B x)) (f x)) (g x)))->(((eq (forall (x:A), (B x))) f) g)))
% Found functional_extensionality_dep as proof of (forall (A:Type) (B:(A->Type)) (f:(forall (x5:A), (B x5))) (g:(forall (x5:A), (B x5))), ((forall (x5:A), (((eq (B x5)) (f x5)) (g x5)))->(((eq (forall (x5:A), (B x5))) f) g)))
% Found proj2:(forall (A:Prop) (B:Prop), (((and A) B)->B))
% Found proj2 as proof of (forall (A:Prop) (B:Prop), (((and A) B)->B))
% Found and_comm_i:=(fun (A:Prop) (B:Prop) (H:((and A) B))=> ((((conj B) A) (((proj2 A) B) H)) (((proj1 A) B) H))):(forall (A:Prop) (B:Prop), (((and A) B)->((and B) A)))
% Found and_comm_i as proof of (forall (A:Prop) (B:Prop), (((and A) B)->((and B) A)))
% Found functional_extensionality_double:=(fun (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))) (x:(forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y))))=> (((((functional_extensionality_dep A) (fun (x2:A)=> (B->C))) f) g) (fun (x0:A)=> (((((functional_extensionality_dep B) (fun (x3:B)=> C)) (f x0)) (g x0)) (x x0))))):(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g)))
% Found functional_extensionality_double as proof of (forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x5:A) (y:B), (((eq C) ((f x5) y)) ((g x5) y)))->(((eq (A->(B->C))) f) g)))
% Found x10:=(fun (X:(a->Prop)) (x3:(forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((iff (((eq a) Xt0) cZ)) (((eq a) Xu) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu)))))))=> (((x1 X) x3) ((cP cZ) a0))):(forall (X:(a->Prop)), ((forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((iff (((eq a) Xt0) cZ)) (((eq a) Xu) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))->(forall (Xu:a), ((X ((cP ((cP cZ) a0)) Xu))->(((eq a) ((cP cZ) a0)) Xu)))))
% Found x10 as proof of (forall (X0:(a->Prop)), ((forall (Xt00:a) (Xu0:a), ((X0 ((cP Xt00) Xu0))->((and ((and ((iff (((eq a) Xt00) cZ)) (((eq a) Xu0) cZ))) (X0 ((cP (cL Xt00)) (cL Xu0))))) (X0 ((cP (cR Xt00)) (cR Xu0))))))->(forall (Xu0:a), ((X0 ((cP ((cP cZ) a0)) Xu0))->(((eq a) ((cP cZ) a0)) Xu0)))))
% Found x0:((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0))))))
% Found x0 as proof of ((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt00:a), ((iff (not (((eq a) Xt00) cZ))) (((eq a) Xt00) ((cP (cL Xt00)) (cR Xt00))))))
% Found eta_expansion:=(fun (A:Type) (B:Type)=> ((eta_expansion_dep A) (fun (x1:A)=> B))):(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x))))
% Found eta_expansion as proof of (forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x5:A)=> (f x5))))
% Found NNPP:=(fun (P:Prop) (H:(not (not P)))=> ((fun (C:((or P) (not P)))=> ((((((or_ind P) (not P)) P) (fun (H0:P)=> H0)) (fun (N:(not P))=> ((False_rect P) (H N)))) C)) (classic P))):(forall (P:Prop), ((not (not P))->P))
% Found NNPP as proof of (forall (P:Prop), ((not (not P))->P))
% Found x00:(x2 ((cP Xt0) Xu))
% Found x00 as proof of (x2 ((cP Xt0) Xu))
% Found iff_refl:=(fun (A:Prop)=> ((((conj (A->A)) (A->A)) (fun (H:A)=> H)) (fun (H:A)=> H))):(forall (P:Prop), ((iff P) P))
% Found iff_refl as proof of (forall (P:Prop), ((iff P) P))
% Found eq_trans:=(fun (T:Type) (a:T) (b:T) (c:T) (X:(((eq T) a) b)) (Y:(((eq T) b) c))=> ((Y (fun (t:T)=> (((eq T) a) t))) X)):(forall (T:Type) (a:T) (b:T) (c:T), ((((eq T) a) b)->((((eq T) b) c)->(((eq T) a) c))))
% Found eq_trans as proof of (forall (T:Type) (a00:T) (b:T) (c:T), ((((eq T) a00) b)->((((eq T) b) c)->(((eq T) a00) c))))
% 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)))
% Found iff_sym as proof of (forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Found x:((and ((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))) (forall (X:(a->Prop)), ((forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((iff (((eq a) Xt) cZ)) (((eq a) Xu) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))->(forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->(((eq a) Xt) Xu))))))
% Found x as proof of ((and ((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt1:a), ((iff (not (((eq a) Xt1) cZ))) (((eq a) Xt1) ((cP (cL Xt1)) (cR Xt1))))))) (forall (X0:(a->Prop)), ((forall (Xt1:a) (Xu0:a), ((X0 ((cP Xt1) Xu0))->((and ((and ((iff (((eq a) Xt1) cZ)) (((eq a) Xu0) cZ))) (X0 ((cP (cL Xt1)) (cL Xu0))))) (X0 ((cP (cR Xt1)) (cR Xu0))))))->(forall (Xt1:a) (Xu0:a), ((X0 ((cP Xt1) Xu0))->(((eq a) Xt1) Xu0))))))
% Found x50:=(fun (Xx:a)=> ((x5 Xx) ((cP (cR Xt0)) (cR Xu)))):(forall (Xx:a), (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu))))
% Instantiate: x8:=(fun (x9:a)=> (forall (Xx:a), (((eq a) (cR ((cP Xx) x9))) x9))):(a->Prop)
% Found (fun (Xx:a)=> ((x5 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x8 ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x5 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x8 ((cP (cR Xt0)) (cR Xu)))
% Found x600:=(x60 ((cP (cL Xt0)) (cL Xu))):(((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found (x60 ((cP (cL Xt0)) (cL Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found ((x6 Xx) ((cP (cL Xt0)) (cL Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cL Xt0)) (cL Xu)))) as proof of (((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cL Xt0)) (cL Xu)))) as proof of (x0 ((cP (cL Xt0)) (cL Xu)))
% Found x600:=(x60 ((cP (cL Xt0)) (cL Xu))):(((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found (x60 ((cP (cL Xt0)) (cL Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found ((x6 Xx) ((cP (cL Xt0)) (cL Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cL Xt0)) (cL Xu)))) as proof of (((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cL Xt0)) (cL Xu)))) as proof of (x2 ((cP (cL Xt0)) (cL Xu)))
% Found x600:=(x60 ((cP (cL Xt0)) (cL Xu))):(((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found (x60 ((cP (cL Xt0)) (cL Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found ((x6 Xx) ((cP (cL Xt0)) (cL Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cL Xt0)) (cL Xu)))) as proof of (((eq a) (cR ((cP Xx) ((cP (cL Xt0)) (cL Xu))))) ((cP (cL Xt0)) (cL Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cL Xt0)) (cL Xu)))) as proof of (x4 ((cP (cL Xt0)) (cL Xu)))
% Found x600:=(x60 ((cP (cR Xt0)) (cR Xu))):(((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (x60 ((cP (cR Xt0)) (cR Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found ((x6 Xx) ((cP (cR Xt0)) (cR Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x0 ((cP (cR Xt0)) (cR Xu)))
% Found x600:=(x60 ((cP (cR Xt0)) (cR Xu))):(((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (x60 ((cP (cR Xt0)) (cR Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found ((x6 Xx) ((cP (cR Xt0)) (cR Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x2 ((cP (cR Xt0)) (cR Xu)))
% Found x60:=(fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))):(forall (Xx:a), (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu))))
% Instantiate: x0:=(fun (x9:a)=> (forall (Xx:a), (((eq a) (cR ((cP Xx) x9))) x9))):(a->Prop)
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x0 ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x0 ((cP (cR Xt0)) (cR Xu)))
% Found x600:=(x60 ((cP (cR Xt0)) (cR Xu))):(((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (x60 ((cP (cR Xt0)) (cR Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found ((x6 Xx) ((cP (cR Xt0)) (cR Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x4 ((cP (cR Xt0)) (cR Xu)))
% Found x60:=(fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))):(forall (Xx:a), (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu))))
% Instantiate: x2:=(fun (x9:a)=> (forall (Xx:a), (((eq a) (cR ((cP Xx) x9))) x9))):(a->Prop)
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x2 ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x2 ((cP (cR Xt0)) (cR Xu)))
% Found eta_expansion:=(fun (A:Type) (B:Type)=> ((eta_expansion_dep A) (fun (x1:A)=> B))):(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x))))
% Found eta_expansion as proof of (forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x5:A)=> (f x5))))
% Found eq_trans:=(fun (T:Type) (a:T) (b:T) (c:T) (X:(((eq T) a) b)) (Y:(((eq T) b) c))=> ((Y (fun (t:T)=> (((eq T) a) t))) X)):(forall (T:Type) (a:T) (b:T) (c:T), ((((eq T) a) b)->((((eq T) b) c)->(((eq T) a) c))))
% Found eq_trans as proof of (forall (T:Type) (a00:T) (b:T) (c:T), ((((eq T) a00) b)->((((eq T) b) c)->(((eq T) a00) c))))
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Found conj as proof of (forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Found x20:=(fun (X:(a->Prop)) (x3:(forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((iff (((eq a) Xt0) cZ)) (((eq a) Xu) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu)))))))=> (((x2 X) x3) ((cP cZ) a0))):(forall (X:(a->Prop)), ((forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((iff (((eq a) Xt0) cZ)) (((eq a) Xu) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))->(forall (Xu:a), ((X ((cP ((cP cZ) a0)) Xu))->(((eq a) ((cP cZ) a0)) Xu)))))
% Found x20 as proof of (forall (X0:(a->Prop)), ((forall (Xt00:a) (Xu0:a), ((X0 ((cP Xt00) Xu0))->((and ((and ((iff (((eq a) Xt00) cZ)) (((eq a) Xu0) cZ))) (X0 ((cP (cL Xt00)) (cL Xu0))))) (X0 ((cP (cR Xt00)) (cR Xu0))))))->(forall (Xu0:a), ((X0 ((cP ((cP cZ) a0)) Xu0))->(((eq a) ((cP cZ) a0)) Xu0)))))
% Found x2:(forall (X:(a->Prop)), ((forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->((and ((and ((iff (((eq a) Xt0) cZ)) (((eq a) Xu) cZ))) (X ((cP (cL Xt0)) (cL Xu))))) (X ((cP (cR Xt0)) (cR Xu))))))->(forall (Xt0:a) (Xu:a), ((X ((cP Xt0) Xu))->(((eq a) Xt0) Xu)))))
% Found x2 as proof of (forall (X0:(a->Prop)), ((forall (Xt00:a) (Xu0:a), ((X0 ((cP Xt00) Xu0))->((and ((and ((iff (((eq a) Xt00) cZ)) (((eq a) Xu0) cZ))) (X0 ((cP (cL Xt00)) (cL Xu0))))) (X0 ((cP (cR Xt00)) (cR Xu0))))))->(forall (Xt00:a) (Xu0:a), ((X0 ((cP Xt00) Xu0))->(((eq a) Xt00) Xu0)))))
% Found functional_extensionality:=(fun (A:Type) (B:Type)=> ((functional_extensionality_dep A) (fun (x1:A)=> B))):(forall (A:Type) (B:Type) (f:(A->B)) (g:(A->B)), ((forall (x:A), (((eq B) (f x)) (g x)))->(((eq (A->B)) f) g)))
% Found functional_extensionality as proof of (forall (A:Type) (B:Type) (f:(A->B)) (g:(A->B)), ((forall (x5:A), (((eq B) (f x5)) (g x5)))->(((eq (A->B)) f) g)))
% Found x:((and ((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))) (forall (X:(a->Prop)), ((forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((iff (((eq a) Xt) cZ)) (((eq a) Xu) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))->(forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->(((eq a) Xt) Xu))))))
% Found x as proof of ((and ((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt1:a), ((iff (not (((eq a) Xt1) cZ))) (((eq a) Xt1) ((cP (cL Xt1)) (cR Xt1))))))) (forall (X0:(a->Prop)), ((forall (Xt1:a) (Xu0:a), ((X0 ((cP Xt1) Xu0))->((and ((and ((iff (((eq a) Xt1) cZ)) (((eq a) Xu0) cZ))) (X0 ((cP (cL Xt1)) (cL Xu0))))) (X0 ((cP (cR Xt1)) (cR Xu0))))))->(forall (Xt1:a) (Xu0:a), ((X0 ((cP Xt1) Xu0))->(((eq a) Xt1) Xu0))))))
% Found NNPP:=(fun (P:Prop) (H:(not (not P)))=> ((fun (C:((or P) (not P)))=> ((((((or_ind P) (not P)) P) (fun (H0:P)=> H0)) (fun (N:(not P))=> ((False_rect P) (H N)))) C)) (classic P))):(forall (P:Prop), ((not (not P))->P))
% Found NNPP as proof of (forall (P:Prop), ((not (not P))->P))
% Found eq_ref:=(fun (T:Type) (a:T) (P:(T->Prop)) (x:(P a))=> x):(forall (T:Type) (a:T), (((eq T) a) a))
% Found eq_ref as proof of (forall (T:Type) (a00:T), (((eq T) a00) a00))
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Found proj1 as proof of (forall (A:Prop) (B:Prop), (((and A) B)->A))
% Found x500:=(x50 ((cP (cR Xt0)) (cR Xu))):(((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (x50 ((cP (cR Xt0)) (cR Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found ((x5 Xx) ((cP (cR Xt0)) (cR Xu))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x5 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x5 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x6 ((cP (cR Xt0)) (cR Xu)))
% Found proj2:(forall (A:Prop) (B:Prop), (((and A) B)->B))
% Found proj2 as proof of (forall (A:Prop) (B:Prop), (((and A) B)->B))
% Found x00:(x0 ((cP Xt0) Xu))
% Found x00 as proof of (x0 ((cP Xt0) Xu))
% Found x1:((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0))))))
% Found x1 as proof of ((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt00:a), ((iff (not (((eq a) Xt00) cZ))) (((eq a) Xt00) ((cP (cL Xt00)) (cR Xt00))))))
% 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)))
% Found iff_sym as proof of (forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Found eq_substitution as proof of (forall (T:Type) (U:Type) (a00:T) (b:T) (f:(T->U)), ((((eq T) a00) b)->(((eq U) (f a00)) (f b))))
% Found eq_stepl:=(fun (T:Type) (a:T) (b:T) (c:T) (X:(((eq T) a) b)) (Y:(((eq T) a) c))=> ((((((eq_trans T) c) a) b) ((((eq_sym T) a) c) Y)) X)):(forall (T:Type) (a:T) (b:T) (c:T), ((((eq T) a) b)->((((eq T) a) c)->(((eq T) c) b))))
% Found eq_stepl as proof of (forall (T:Type) (a00:T) (b:T) (c:T), ((((eq T) a00) b)->((((eq T) a00) c)->(((eq T) c) b))))
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Found iff_trans as proof of (forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Found and_comm_i:=(fun (A:Prop) (B:Prop) (H:((and A) B))=> ((((conj B) A) (((proj2 A) B) H)) (((proj1 A) B) H))):(forall (A:Prop) (B:Prop), (((and A) B)->((and B) A)))
% Found and_comm_i as proof of (forall (A:Prop) (B:Prop), (((and A) B)->((and B) A)))
% Found functional_extensionality_dep:(forall (A:Type) (B:(A->Type)) (f:(forall (x:A), (B x))) (g:(forall (x:A), (B x))), ((forall (x:A), (((eq (B x)) (f x)) (g x)))->(((eq (forall (x:A), (B x))) f) g)))
% Found functional_extensionality_dep as proof of (forall (A:Type) (B:(A->Type)) (f:(forall (x5:A), (B x5))) (g:(forall (x5:A), (B x5))), ((forall (x5:A), (((eq (B x5)) (f x5)) (g x5)))->(((eq (forall (x5:A), (B x5))) f) g)))
% Found x30:(((eq a) a0) Xt)
% Found x30 as proof of (((eq a) a0) Xt)
% Found eq_sym:=(fun (T:Type) (a:T) (b:T) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq T) x) a))) ((eq_ref T) a))):(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a)))
% Found eq_sym as proof of (forall (T:Type) (a00:T) (b:T), ((((eq T) a00) b)->(((eq T) b) a00)))
% Found iff_refl:=(fun (A:Prop)=> ((((conj (A->A)) (A->A)) (fun (H:A)=> H)) (fun (H:A)=> H))):(forall (P:Prop), ((iff P) P))
% Found iff_refl as proof of (forall (P:Prop), ((iff P) P))
% Found eta_expansion_dep:=(fun (A:Type) (B:(A->Type)) (f:(forall (x:A), (B x)))=> (((((functional_extensionality_dep A) (fun (x1:A)=> (B x1))) f) (fun (x:A)=> (f x))) (fun (x:A) (P:((B x)->Prop)) (x0:(P (f x)))=> x0))):(forall (A:Type) (B:(A->Type)) (f:(forall (x:A), (B x))), (((eq (forall (x:A), (B x))) f) (fun (x:A)=> (f x))))
% Found eta_expansion_dep as proof of (forall (A:Type) (B:(A->Type)) (f:(forall (x5:A), (B x5))), (((eq (forall (x5:A), (B x5))) f) (fun (x5:A)=> (f x5))))
% Found functional_extensionality_double:=(fun (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))) (x:(forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y))))=> (((((functional_extensionality_dep A) (fun (x2:A)=> (B->C))) f) g) (fun (x0:A)=> (((((functional_extensionality_dep B) (fun (x3:B)=> C)) (f x0)) (g x0)) (x x0))))):(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g)))
% Found functional_extensionality_double as proof of (forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x5:A) (y:B), (((eq C) ((f x5) y)) ((g x5) y)))->(((eq (A->(B->C))) f) g)))
% Found x60:=(fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))):(forall (Xx:a), (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu))))
% Instantiate: x4:=(fun (x9:a)=> (forall (Xx:a), (((eq a) (cR ((cP Xx) x9))) x9))):(a->Prop)
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x4 ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x4 ((cP (cR Xt0)) (cR Xu)))
% Found x50:=(fun (Xx:a)=> ((x5 Xx) ((cP (cR Xt0)) (cR Xu)))):(forall (Xx:a), (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu))))
% Instantiate: x6:=(fun (x9:a)=> (forall (Xx:a), (((eq a) (cR ((cP Xx) x9))) x9))):(a->Prop)
% Found (fun (Xx:a)=> ((x5 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x6 ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x5 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x6 ((cP (cR Xt0)) (cR Xu)))
% Found conj0100:=(conj010 x00):((and (x8 ((cP cZ) Xt))) (x8 ((cP (cL Xt0)) (cL Xu))))
% Found (conj010 x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x8 ((cP (cL Xt0)) (cL Xu))))
% Found ((conj01 (x8 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x8 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x8 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x8 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x8 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x8 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x8 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x8 ((cP (cL Xt0)) (cL Xu))))
% Found x60:=(fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))):(forall (Xx:a), (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu))))
% Instantiate: x0:=(fun (x9:a)=> (forall (Xx:a), (((eq a) (cR ((cP Xx) x9))) x9))):(a->Prop)
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x0 ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x0 ((cP (cR Xt0)) (cR Xu)))
% Found x60:=(fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))):(forall (Xx:a), (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu))))
% Instantiate: x0:=(fun (x9:a)=> (forall (Xx:a), (((eq a) (cR ((cP Xx) x9))) x9))):(a->Prop)
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x0 ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x0 ((cP (cR Xt0)) (cR Xu)))
% Found x60:=(fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))):(forall (Xx:a), (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu))))
% Instantiate: x2:=(fun (x9:a)=> (forall (Xx:a), (((eq a) (cR ((cP Xx) x9))) x9))):(a->Prop)
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x2 ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x2 ((cP (cR Xt0)) (cR Xu)))
% Found x60:=(fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))):(forall (Xx:a), (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu))))
% Instantiate: x2:=(fun (x9:a)=> (forall (Xx:a), (((eq a) (cR ((cP Xx) x9))) x9))):(a->Prop)
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x2 ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x2 ((cP (cR Xt0)) (cR Xu)))
% Found eq_sym:=(fun (T:Type) (a:T) (b:T) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq T) x) a))) ((eq_ref T) a))):(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a)))
% Found eq_sym as proof of (forall (T:Type) (a00:T) (b:T), ((((eq T) a00) b)->(((eq T) b) a00)))
% Found eq_stepl:=(fun (T:Type) (a:T) (b:T) (c:T) (X:(((eq T) a) b)) (Y:(((eq T) a) c))=> ((((((eq_trans T) c) a) b) ((((eq_sym T) a) c) Y)) X)):(forall (T:Type) (a:T) (b:T) (c:T), ((((eq T) a) b)->((((eq T) a) c)->(((eq T) c) b))))
% Found eq_stepl as proof of (forall (T:Type) (a00:T) (b:T) (c:T), ((((eq T) a00) b)->((((eq T) a00) c)->(((eq T) c) b))))
% Found x00:(x0 ((cP Xt0) Xu))
% Found x00 as proof of (x0 ((cP Xt0) Xu))
% Found and_comm_i:=(fun (A:Prop) (B:Prop) (H:((and A) B))=> ((((conj B) A) (((proj2 A) B) H)) (((proj1 A) B) H))):(forall (A:Prop) (B:Prop), (((and A) B)->((and B) A)))
% Found and_comm_i as proof of (forall (A:Prop) (B:Prop), (((and A) B)->((and B) A)))
% Found eta_expansion_dep:=(fun (A:Type) (B:(A->Type)) (f:(forall (x:A), (B x)))=> (((((functional_extensionality_dep A) (fun (x1:A)=> (B x1))) f) (fun (x:A)=> (f x))) (fun (x:A) (P:((B x)->Prop)) (x0:(P (f x)))=> x0))):(forall (A:Type) (B:(A->Type)) (f:(forall (x:A), (B x))), (((eq (forall (x:A), (B x))) f) (fun (x:A)=> (f x))))
% Found eta_expansion_dep as proof of (forall (A:Type) (B:(A->Type)) (f:(forall (x5:A), (B x5))), (((eq (forall (x5:A), (B x5))) f) (fun (x5:A)=> (f x5))))
% Found x20:=(fun (X:(a->Prop)) (x3:(forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((iff (((eq a) Xt) cZ)) (((eq a) Xu0) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))=> (((x2 X) x3) ((cP cZ) a0))):(forall (X:(a->Prop)), ((forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((iff (((eq a) Xt) cZ)) (((eq a) Xu0) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))->(forall (Xu0:a), ((X ((cP ((cP cZ) a0)) Xu0))->(((eq a) ((cP cZ) a0)) Xu0)))))
% Found x20 as proof of (forall (X0:(a->Prop)), ((forall (Xt1:a) (Xu0:a), ((X0 ((cP Xt1) Xu0))->((and ((and ((iff (((eq a) Xt1) cZ)) (((eq a) Xu0) cZ))) (X0 ((cP (cL Xt1)) (cL Xu0))))) (X0 ((cP (cR Xt1)) (cR Xu0))))))->(forall (Xu0:a), ((X0 ((cP ((cP cZ) a0)) Xu0))->(((eq a) ((cP cZ) a0)) Xu0)))))
% 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)))
% Found iff_sym as proof of (forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Found eq_ref:=(fun (T:Type) (a:T) (P:(T->Prop)) (x:(P a))=> x):(forall (T:Type) (a:T), (((eq T) a) a))
% Found eq_ref as proof of (forall (T:Type) (a00:T), (((eq T) a00) a00))
% Found eta_expansion:=(fun (A:Type) (B:Type)=> ((eta_expansion_dep A) (fun (x1:A)=> B))):(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x))))
% Found eta_expansion as proof of (forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x5:A)=> (f x5))))
% Found proj2:(forall (A:Prop) (B:Prop), (((and A) B)->B))
% Found proj2 as proof of (forall (A:Prop) (B:Prop), (((and A) B)->B))
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Found proj1 as proof of (forall (A:Prop) (B:Prop), (((and A) B)->A))
% Found x:((and ((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))) (forall (X:(a->Prop)), ((forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((iff (((eq a) Xt) cZ)) (((eq a) Xu) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))->(forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->(((eq a) Xt) Xu))))))
% Found x as proof of ((and ((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt1:a), ((iff (not (((eq a) Xt1) cZ))) (((eq a) Xt1) ((cP (cL Xt1)) (cR Xt1))))))) (forall (X0:(a->Prop)), ((forall (Xt1:a) (Xu0:a), ((X0 ((cP Xt1) Xu0))->((and ((and ((iff (((eq a) Xt1) cZ)) (((eq a) Xu0) cZ))) (X0 ((cP (cL Xt1)) (cL Xu0))))) (X0 ((cP (cR Xt1)) (cR Xu0))))))->(forall (Xt1:a) (Xu0:a), ((X0 ((cP Xt1) Xu0))->(((eq a) Xt1) Xu0))))))
% Found functional_extensionality_dep:(forall (A:Type) (B:(A->Type)) (f:(forall (x:A), (B x))) (g:(forall (x:A), (B x))), ((forall (x:A), (((eq (B x)) (f x)) (g x)))->(((eq (forall (x:A), (B x))) f) g)))
% Found functional_extensionality_dep as proof of (forall (A:Type) (B:(A->Type)) (f:(forall (x5:A), (B x5))) (g:(forall (x5:A), (B x5))), ((forall (x5:A), (((eq (B x5)) (f x5)) (g x5)))->(((eq (forall (x5:A), (B x5))) f) g)))
% Found x100:(((eq a) a0) Xt)
% Found x100 as proof of (((eq a) a0) Xt)
% Found functional_extensionality:=(fun (A:Type) (B:Type)=> ((functional_extensionality_dep A) (fun (x1:A)=> B))):(forall (A:Type) (B:Type) (f:(A->B)) (g:(A->B)), ((forall (x:A), (((eq B) (f x)) (g x)))->(((eq (A->B)) f) g)))
% Found functional_extensionality as proof of (forall (A:Type) (B:Type) (f:(A->B)) (g:(A->B)), ((forall (x5:A), (((eq B) (f x5)) (g x5)))->(((eq (A->B)) f) g)))
% Found x1:((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))
% Found x1 as proof of ((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt1:a), ((iff (not (((eq a) Xt1) cZ))) (((eq a) Xt1) ((cP (cL Xt1)) (cR Xt1))))))
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Found conj as proof of (forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Found functional_extensionality_double:=(fun (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))) (x:(forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y))))=> (((((functional_extensionality_dep A) (fun (x2:A)=> (B->C))) f) g) (fun (x0:A)=> (((((functional_extensionality_dep B) (fun (x3:B)=> C)) (f x0)) (g x0)) (x x0))))):(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g)))
% Found functional_extensionality_double as proof of (forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x5:A) (y:B), (((eq C) ((f x5) y)) ((g x5) y)))->(((eq (A->(B->C))) f) g)))
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Found iff_trans as proof of (forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Found eq_substitution as proof of (forall (T:Type) (U:Type) (a00:T) (b:T) (f:(T->U)), ((((eq T) a00) b)->(((eq U) (f a00)) (f b))))
% Found eq_trans:=(fun (T:Type) (a:T) (b:T) (c:T) (X:(((eq T) a) b)) (Y:(((eq T) b) c))=> ((Y (fun (t:T)=> (((eq T) a) t))) X)):(forall (T:Type) (a:T) (b:T) (c:T), ((((eq T) a) b)->((((eq T) b) c)->(((eq T) a) c))))
% Found eq_trans as proof of (forall (T:Type) (a00:T) (b:T) (c:T), ((((eq T) a00) b)->((((eq T) b) c)->(((eq T) a00) c))))
% Found iff_refl:=(fun (A:Prop)=> ((((conj (A->A)) (A->A)) (fun (H:A)=> H)) (fun (H:A)=> H))):(forall (P:Prop), ((iff P) P))
% Found iff_refl as proof of (forall (P:Prop), ((iff P) P))
% Found NNPP:=(fun (P:Prop) (H:(not (not P)))=> ((fun (C:((or P) (not P)))=> ((((((or_ind P) (not P)) P) (fun (H0:P)=> H0)) (fun (N:(not P))=> ((False_rect P) (H N)))) C)) (classic P))):(forall (P:Prop), ((not (not P))->P))
% Found NNPP as proof of (forall (P:Prop), ((not (not P))->P))
% Found x2:(forall (X:(a->Prop)), ((forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((iff (((eq a) Xt) cZ)) (((eq a) Xu0) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))->(forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->(((eq a) Xt) Xu0)))))
% Found x2 as proof of (forall (X0:(a->Prop)), ((forall (Xt1:a) (Xu0:a), ((X0 ((cP Xt1) Xu0))->((and ((and ((iff (((eq a) Xt1) cZ)) (((eq a) Xu0) cZ))) (X0 ((cP (cL Xt1)) (cL Xu0))))) (X0 ((cP (cR Xt1)) (cR Xu0))))))->(forall (Xt1:a) (Xu0:a), ((X0 ((cP Xt1) Xu0))->(((eq a) Xt1) Xu0)))))
% Found x60:=(fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))):(forall (Xx:a), (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu))))
% Instantiate: x4:=(fun (x9:a)=> (forall (Xx:a), (((eq a) (cR ((cP Xx) x9))) x9))):(a->Prop)
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x4 ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x4 ((cP (cR Xt0)) (cR Xu)))
% Found x60:=(fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))):(forall (Xx:a), (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu))))
% Instantiate: x4:=(fun (x9:a)=> (forall (Xx:a), (((eq a) (cR ((cP Xx) x9))) x9))):(a->Prop)
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x4 ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x6 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x4 ((cP (cR Xt0)) (cR Xu)))
% Found x50:=(fun (Xx:a)=> ((x5 Xx) ((cP (cR Xt0)) (cR Xu)))):(forall (Xx:a), (((eq a) (cR ((cP Xx) ((cP (cR Xt0)) (cR Xu))))) ((cP (cR Xt0)) (cR Xu))))
% Instantiate: x6:=(fun (x9:a)=> (forall (Xx:a), (((eq a) (cR ((cP Xx) x9))) x9))):(a->Prop)
% Found (fun (Xx:a)=> ((x5 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x6 ((cP (cR Xt0)) (cR Xu)))
% Found (fun (Xx:a)=> ((x5 Xx) ((cP (cR Xt0)) (cR Xu)))) as proof of (x6 ((cP (cR Xt0)) (cR Xu)))
% Found conj0100:=(conj010 x00):((and (x0 ((cP cZ) Xt))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found (conj010 x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found ((conj01 (x0 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x0 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x0 ((cP (cL Xt0)) (cL Xu))))
% Found conj0100:=(conj010 x00):((and (x2 ((cP cZ) Xt))) (x2 ((cP (cL Xt0)) (cL Xu))))
% Found (conj010 x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))
% Found ((conj01 (x2 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x2 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x2 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x2 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu))))
% Found conj0100:=(conj010 x00):((and (x4 ((cP cZ) Xt))) (x4 ((cP (cL Xt0)) (cL Xu))))
% Found (conj010 x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x4 ((cP (cL Xt0)) (cL Xu))))
% Found ((conj01 (x4 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x4 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x4 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x4 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x4 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x4 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x4 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x4 ((cP (cL Xt0)) (cL Xu))))
% Found conj0100:=(conj010 x00):((and (x6 ((cP cZ) Xt))) (x6 ((cP (cL Xt0)) (cL Xu))))
% Found (conj010 x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x6 ((cP (cL Xt0)) (cL Xu))))
% Found ((conj01 (x6 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x6 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x6 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x6 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x6 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x6 ((cP (cL Xt0)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x6 ((cP (cL Xt0)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt0) cZ))) (x6 ((cP (cL Xt0)) (cL Xu))))
% Found x8:(((eq a) Xt0) cZ)
% Found x8 as proof of (((eq a) Xt0) cZ)
% Found x60:=(x6 (fun (x7:a)=> (P ((cP (cL x7)) (cR x7))))):((P ((cP (cL Xt0)) (cR Xt0)))->(P ((cP (cL cZ)) (cR cZ))))
% Found (x6 (fun (x7:a)=> (P ((cP (cL x7)) (cR x7))))) as proof of ((P ((cP (cL Xt0)) (cR Xt0)))->(P Xt0))
% Found (x6 (fun (x7:a)=> (P ((cP (cL x7)) (cR x7))))) as proof of ((P ((cP (cL Xt
% EOF
%------------------------------------------------------------------------------