TSTP Solution File: SEV174^5 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEV174^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n189.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 13:33:50 EDT 2014
% Result : Timeout 300.11s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEV174^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n189.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 08:20:41 CDT 2014
% % CPUTime : 300.11
% 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 0x1fb48c0>, <kernel.Type object at 0x1fb4b48>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x238c560>, <kernel.DependentProduct object at 0x1fb45a8>) of role type named cP
% Using role type
% Declaring cP:((a->Prop)->Prop)
% FOF formula (((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))->(((eq (a->Prop)) Xp) Xq))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (S Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc))))))) of role conjecture named cTHM555_pme
% Conjecture to prove = (((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))->(((eq (a->Prop)) Xp) Xq))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (S Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc))))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))->(((eq (a->Prop)) Xp) Xq))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (S Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))))']
% Parameter a:Type.
% Parameter cP:((a->Prop)->Prop).
% Trying to prove (((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))->(((eq (a->Prop)) Xp) Xq))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (S Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))))
% Found x0:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))
% Found x0 as proof of (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))
% Found iff_refl0:=(iff_refl (x0 Xx)):((iff (x0 Xx)) (x0 Xx))
% Found (iff_refl (x0 Xx)) as proof of ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))
% Found (iff_refl (x0 Xx)) as proof of ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))
% Found (fun (Xx:a)=> (iff_refl (x0 Xx))) as proof of ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))
% Found (fun (Xx:a)=> (iff_refl (x0 Xx))) as proof of (forall (Xx:a), ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx))))))
% Found x1:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))
% Found x1 as proof of (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))
% Found iff_refl0:=(iff_refl (x2 Xx)):((iff (x2 Xx)) (x2 Xx))
% Found (iff_refl (x2 Xx)) as proof of ((iff (x2 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))
% Found (iff_refl (x2 Xx)) as proof of ((iff (x2 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))
% Found (fun (Xx:a)=> (iff_refl (x2 Xx))) as proof of ((iff (x2 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))
% Found (fun (Xx:a)=> (iff_refl (x2 Xx))) as proof of (forall (Xx:a), ((iff (x2 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx))))))
% Found ((conj10 x0) (fun (Xx:a)=> (iff_refl (x2 Xx)))) as proof of ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x2 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))
% Found (((conj1 (forall (Xx:a), ((iff (x2 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx))))))) x0) (fun (Xx:a)=> (iff_refl (x2 Xx)))) as proof of ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x2 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))
% Found ((((conj (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x2 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx))))))) x0) (fun (Xx:a)=> (iff_refl (x2 Xx)))) as proof of ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x2 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))
% Found ((((conj (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x2 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx))))))) x0) (fun (Xx:a)=> (iff_refl (x2 Xx)))) as proof of ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x2 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (S:(a->Prop))=> ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (S Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (S Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))) (fun (x:(a->Prop))=> ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc))))))
% Found (eta_expansion_dep00 (fun (S:(a->Prop))=> ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (S Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (S Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))) b)
% Found ((eta_expansion_dep0 (fun (x1:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (S Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (S Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x1:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (S Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (S Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x1:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (S Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (S Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x1:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (S Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (S Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))) b)
% Found iff_refl0:=(iff_refl (x0 Xx)):((iff (x0 Xx)) (x0 Xx))
% Found (iff_refl (x0 Xx)) as proof of ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))
% Found (iff_refl (x0 Xx)) as proof of ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))
% Found (fun (Xx:a)=> (iff_refl (x0 Xx))) as proof of ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))
% Found (fun (Xx:a)=> (iff_refl (x0 Xx))) as proof of (forall (Xx:a), ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx))))))
% Found ((conj10 x1) (fun (Xx:a)=> (iff_refl (x0 Xx)))) as proof of ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))
% Found (((conj1 (forall (Xx:a), ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx))))))) x1) (fun (Xx:a)=> (iff_refl (x0 Xx)))) as proof of ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))
% Found ((((conj (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx))))))) x1) (fun (Xx:a)=> (iff_refl (x0 Xx)))) as proof of ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))
% Found ((((conj (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx))))))) x1) (fun (Xx:a)=> (iff_refl (x0 Xx)))) as proof of ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))
% Found (fun (x0:(a->Prop))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))
% Found (fun (x0:(a->Prop))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc))))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))
% Found (fun (x0:(a->Prop))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))
% Found (fun (x0:(a->Prop))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc))))))
% Found eq_ref00:=(eq_ref0 (fun (S:(a->Prop))=> ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (S Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (S Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))) (fun (S:(a->Prop))=> ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (S Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc))))))
% Found (eq_ref0 (fun (S:(a->Prop))=> ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (S Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (S Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (S Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (S Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (S Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (S Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (S Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (S Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))) b)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x2 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x2 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x2 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x2 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x2 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x2 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x2 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x2 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and (cP Xb)) (cP Xc))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc))))))
% Found x1:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))
% Found x1 as proof of (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))
% Found iff_refl0:=(iff_refl (x0 Xx)):((iff (x0 Xx)) (x0 Xx))
% Found (iff_refl (x0 Xx)) as proof of ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))
% Found (iff_refl (x0 Xx)) as proof of ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))
% Found (fun (Xx:a)=> (iff_refl (x0 Xx))) as proof of ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))
% Found (fun (Xx:a)=> (iff_refl (x0 Xx))) as proof of (forall (Xx:a), ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx))))))
% Found ((conj10 x1) (fun (Xx:a)=> (iff_refl (x0 Xx)))) as proof of ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))
% Found (((conj1 (forall (Xx:a), ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx))))))) x1) (fun (Xx:a)=> (iff_refl (x0 Xx)))) as proof of ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))
% Found ((((conj (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx))))))) x1) (fun (Xx:a)=> (iff_refl (x0 Xx)))) as proof of ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))
% Found (fun (x2:(forall (Xx:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))->(((eq (a->Prop)) Xp) Xq))))=> ((((conj (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx))))))) x1) (fun (Xx:a)=> (iff_refl (x0 Xx))))) as proof of ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))
% Found (fun (x1:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (x2:(forall (Xx:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))->(((eq (a->Prop)) Xp) Xq))))=> ((((conj (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx))))))) x1) (fun (Xx:a)=> (iff_refl (x0 Xx))))) as proof of ((forall (Xx:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))->(((eq (a->Prop)) Xp) Xq)))->((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx))))))))
% Found (fun (x1:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (x2:(forall (Xx:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))->(((eq (a->Prop)) Xp) Xq))))=> ((((conj (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx))))))) x1) (fun (Xx:a)=> (iff_refl (x0 Xx))))) as proof of ((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))->(((eq (a->Prop)) Xp) Xq)))->((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))))
% Found (and_rect00 (fun (x1:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (x2:(forall (Xx:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))->(((eq (a->Prop)) Xp) Xq))))=> ((((conj (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx))))))) x1) (fun (Xx:a)=> (iff_refl (x0 Xx)))))) as proof of ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))
% Found ((and_rect0 ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (fun (x1:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (x2:(forall (Xx:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))->(((eq (a->Prop)) Xp) Xq))))=> ((((conj (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx))))))) x1) (fun (Xx:a)=> (iff_refl (x0 Xx)))))) as proof of ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))
% Found (((fun (P:Type) (x1:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))->(((eq (a->Prop)) Xp) Xq)))->P)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))->(((eq (a->Prop)) Xp) Xq)))) P) x1) x)) ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (fun (x1:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (x2:(forall (Xx:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))->(((eq (a->Prop)) Xp) Xq))))=> ((((conj (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx))))))) x1) (fun (Xx:a)=> (iff_refl (x0 Xx)))))) as proof of ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))
% Found (((fun (P:Type) (x1:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))->(((eq (a->Prop)) Xp) Xq)))->P)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))->(((eq (a->Prop)) Xp) Xq)))) P) x1) x)) ((and (forall (Xa:(a->Prop)), ((cP Xa)->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (x0 Xx)) ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (cP S0)) (S0 Xx)))))))) (fun (x1:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (x2:(forall (Xx:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))->
% EOF
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