%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEU868^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n105.star.cs.uiowa.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory   : 32286.75MB
% OS       : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:33:18 EDT 2014

% Result   : Theorem 280.91s
% Output   : Proof 280.91s
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----ERROR: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEU868^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n105.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 11:38:06 CDT 2014
% % CPUTime  : 280.91 
% 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 0x242b440>, <kernel.Type object at 0x242b248>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x1ff1488>, <kernel.DependentProduct object at 0x21b4560>) of role type named cC
% Using role type
% Declaring cC:(a->Prop)
% FOF formula ((forall (Xw:((a->Prop)->Prop)), (((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))->(Xw cC)))->(forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P cC)))) of role conjecture named cTHM551_pme
% Conjecture to prove = ((forall (Xw:((a->Prop)->Prop)), (((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))->(Xw cC)))->(forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P cC)))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['((forall (Xw:((a->Prop)->Prop)), (((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))->(Xw cC)))->(forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P cC))))']
% Parameter a:Type.
% Parameter cC:(a->Prop).
% Trying to prove ((forall (Xw:((a->Prop)->Prop)), (((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))->(Xw cC)))->(forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P cC))))
% Found x6:False
% Found (fun (x6:False)=> x6) as proof of False
% Found (fun (x5:a) (x6:False)=> x6) as proof of (False->False)
% Found (fun (x5:a) (x6:False)=> x6) as proof of (a->(False->False))
% Found (ex_ind00 (fun (x5:a) (x6:False)=> x6)) as proof of False
% Found ((ex_ind0 False) (fun (x5:a) (x6:False)=> x6)) as proof of False
% Found (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6)) as proof of False
% Found (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6))) as proof of False
% Found (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6))) as proof of (not ((ex a) (fun (Xt:a)=> False)))
% Found eq_ref000:=(eq_ref00 P):((P Xr)->(P Xr))
% Found (eq_ref00 P) as proof of (P0 Xr)
% Found ((eq_ref0 Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found x6:False
% Found (fun (x6:False)=> x6) as proof of False
% Found (fun (x5:a) (x6:False)=> x6) as proof of (False->False)
% Found (fun (x5:a) (x6:False)=> x6) as proof of (a->(False->False))
% Found (ex_ind00 (fun (x5:a) (x6:False)=> x6)) as proof of False
% Found ((ex_ind0 False) (fun (x5:a) (x6:False)=> x6)) as proof of False
% Found (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6)) as proof of False
% Found (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6))) as proof of False
% Found (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6))) as proof of (((ex a) (fun (Xt:a)=> False))->False)
% Found (x10 (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6)))) as proof of (P (fun (Xx:a)=> False))
% Found ((x1 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6)))) as proof of (P (fun (Xx:a)=> False))
% Found ((x1 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6)))) as proof of (P (fun (Xx:a)=> False))
% Found eq_ref000:=(eq_ref00 P):((P Xr)->(P Xr))
% Found (eq_ref00 P) as proof of (P0 Xr)
% Found ((eq_ref0 Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found eq_ref000:=(eq_ref00 P):((P Xr)->(P Xr))
% Found (eq_ref00 P) as proof of (P0 Xr)
% Found ((eq_ref0 Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found eq_ref000:=(eq_ref00 P):((P Xr)->(P Xr))
% Found (eq_ref00 P) as proof of (P0 Xr)
% Found ((eq_ref0 Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found x6:False
% Found (fun (x6:False)=> x6) as proof of False
% Found (fun (x5:a) (x6:False)=> x6) as proof of (False->False)
% Found (fun (x5:a) (x6:False)=> x6) as proof of (a->(False->False))
% Found (ex_ind00 (fun (x5:a) (x6:False)=> x6)) as proof of False
% Found ((ex_ind0 False) (fun (x5:a) (x6:False)=> x6)) as proof of False
% Found (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6)) as proof of False
% Found (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6))) as proof of False
% Found (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6))) as proof of (((ex a) (fun (Xt:a)=> False))->False)
% Found (x20 (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6)))) as proof of (P (fun (Xx:a)=> False))
% Found ((x2 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6)))) as proof of (P (fun (Xx:a)=> False))
% Found ((x2 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6)))) as proof of (P (fun (Xx:a)=> False))
% Found eq_ref000:=(eq_ref00 P):((P Xr)->(P Xr))
% Found (eq_ref00 P) as proof of (P0 Xr)
% Found ((eq_ref0 Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found eq_ref000:=(eq_ref00 P):((P Xr)->(P Xr))
% Found (eq_ref00 P) as proof of (P0 Xr)
% Found ((eq_ref0 Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found eq_ref000:=(eq_ref00 P):((P Xr)->(P Xr))
% Found (eq_ref00 P) as proof of (P0 Xr)
% Found ((eq_ref0 Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found x5:False
% Found (fun (x5:False)=> x5) as proof of False
% Found (fun (x4:a) (x5:False)=> x5) as proof of (False->False)
% Found (fun (x4:a) (x5:False)=> x5) as proof of (a->(False->False))
% Found (ex_ind00 (fun (x4:a) (x5:False)=> x5)) as proof of False
% Found ((ex_ind0 False) (fun (x4:a) (x5:False)=> x5)) as proof of False
% Found (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5)) as proof of False
% Found (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5))) as proof of False
% Found (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5))) as proof of (((ex a) (fun (Xt:a)=> False))->False)
% Found (x10 (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5)))) as proof of (P (fun (Xx:a)=> False))
% Found ((x1 (fun (Xx:a)=> False)) (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5)))) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x20:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 (fun (Xx:a)=> False)) (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5))))) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x20:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 (fun (Xx:a)=> False)) (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5))))) as proof of ((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P (fun (Xx:a)=> False)))
% Found eq_ref000:=(eq_ref00 P):((P Xr)->(P Xr))
% Found (eq_ref00 P) as proof of (P0 Xr)
% Found ((eq_ref0 Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found eq_ref00:=(eq_ref0 cC):(((eq (a->Prop)) cC) cC)
% Found (eq_ref0 cC) as proof of (((eq (a->Prop)) cC) b)
% Found ((eq_ref (a->Prop)) cC) as proof of (((eq (a->Prop)) cC) b)
% Found ((eq_ref (a->Prop)) cC) as proof of (((eq (a->Prop)) cC) b)
% Found ((eq_ref (a->Prop)) cC) as proof of (((eq (a->Prop)) cC) b)
% Found eq_ref000:=(eq_ref00 P):((P Xr)->(P Xr))
% Found (eq_ref00 P) as proof of (P0 Xr)
% Found ((eq_ref0 Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found eq_ref000:=(eq_ref00 P):((P Xr)->(P Xr))
% Found (eq_ref00 P) as proof of (P0 Xr)
% Found ((eq_ref0 Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found x4:False
% Found (fun (x4:False)=> x4) as proof of False
% Found (fun (x2:a) (x4:False)=> x4) as proof of (False->False)
% Found (fun (x2:a) (x4:False)=> x4) as proof of (a->(False->False))
% Found (ex_ind00 (fun (x2:a) (x4:False)=> x4)) as proof of False
% Found ((ex_ind0 False) (fun (x2:a) (x4:False)=> x4)) as proof of False
% Found (((fun (P0:Prop) (x2:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x2) x3)) False) (fun (x2:a) (x4:False)=> x4)) as proof of False
% Found (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x2:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x2) x3)) False) (fun (x2:a) (x4:False)=> x4))) as proof of False
% Found (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x2:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x2) x3)) False) (fun (x2:a) (x4:False)=> x4))) as proof of (((ex a) (fun (Xt:a)=> False))->False)
% Found (x100 (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x2:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x2) x3)) False) (fun (x2:a) (x4:False)=> x4)))) as proof of (P (fun (Xx:a)=> False))
% Found ((x10 (fun (Xx:a)=> False)) (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x2:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x2) x3)) False) (fun (x2:a) (x4:False)=> x4)))) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x20:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x10 (fun (Xx:a)=> False)) (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x2:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x2) x3)) False) (fun (x2:a) (x4:False)=> x4))))) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x10:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x20:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x10 (fun (Xx:a)=> False)) (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x2:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x2) x3)) False) (fun (x2:a) (x4:False)=> x4))))) as proof of ((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P (fun (Xx:a)=> False)))
% Found (fun (x10:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x20:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x10 (fun (Xx:a)=> False)) (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x2:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x2) x3)) False) (fun (x2:a) (x4:False)=> x4))))) as proof of ((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P (fun (Xx:a)=> False))))
% Found x6:False
% Found (fun (x6:False)=> x6) as proof of False
% Found (fun (x5:a) (x6:False)=> x6) as proof of (False->False)
% Found (fun (x5:a) (x6:False)=> x6) as proof of (a->(False->False))
% Found (ex_ind00 (fun (x5:a) (x6:False)=> x6)) as proof of False
% Found ((ex_ind0 False) (fun (x5:a) (x6:False)=> x6)) as proof of False
% Found (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6)) as proof of False
% Found (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6))) as proof of False
% Found (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6))) as proof of (not ((ex a) (fun (Xt:a)=> False)))
% Found (x10 (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6)))) as proof of (P (fun (Xx:a)=> False))
% Found ((x1 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6)))) as proof of (P (fun (Xx:a)=> False))
% Found ((x1 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6)))) as proof of (P (fun (Xx:a)=> False))
% Found eq_ref000:=(eq_ref00 P):((P Xr)->(P Xr))
% Found (eq_ref00 P) as proof of (P0 Xr)
% Found ((eq_ref0 Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found eq_ref000:=(eq_ref00 P):((P Xr)->(P Xr))
% Found (eq_ref00 P) as proof of (P0 Xr)
% Found ((eq_ref0 Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found eq_ref000:=(eq_ref00 P):((P Xr)->(P Xr))
% Found (eq_ref00 P) as proof of (P0 Xr)
% Found ((eq_ref0 Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found eq_ref00:=(eq_ref0 (forall (Xr:(a->Prop)) (Xx:a), ((forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr)))->(forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))):(((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr)))->(forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))) (forall (Xr:(a->Prop)) (Xx:a), ((forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr)))->(forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))))
% Found (eq_ref0 (forall (Xr:(a->Prop)) (Xx:a), ((forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr)))->(forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr)))->(forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), ((forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr)))->(forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr)))->(forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), ((forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr)))->(forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr)))->(forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), ((forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr)))->(forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr)))->(forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xr:(a->Prop)) (Xx:a), ((((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr))->(((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))):(((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr))->(((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) (forall (Xr:(a->Prop)) (Xx:a), ((((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr))->(((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))
% Found (eq_ref0 (forall (Xr:(a->Prop)) (Xx:a), ((((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr))->(((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr))->(((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), ((((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr))->(((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr))->(((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), ((((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr))->(((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr))->(((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), ((((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr))->(((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr))->(((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) b)
% Found x6:False
% Found (fun (x6:False)=> x6) as proof of False
% Found (fun (x5:a) (x6:False)=> x6) as proof of (False->False)
% Found (fun (x5:a) (x6:False)=> x6) as proof of (a->(False->False))
% Found (ex_ind00 (fun (x5:a) (x6:False)=> x6)) as proof of False
% Found ((ex_ind0 False) (fun (x5:a) (x6:False)=> x6)) as proof of False
% Found (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6)) as proof of False
% Found (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6))) as proof of False
% Found (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6))) as proof of (not ((ex a) (fun (Xt:a)=> False)))
% Found (x20 (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6)))) as proof of (P (fun (Xx:a)=> False))
% Found ((x2 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6)))) as proof of (P (fun (Xx:a)=> False))
% Found ((x2 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6)))) as proof of (P (fun (Xx:a)=> False))
% Found eq_ref000:=(eq_ref00 P):((P Xr)->(P Xr))
% Found (eq_ref00 P) as proof of (P0 Xr)
% Found ((eq_ref0 Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found eq_ref000:=(eq_ref00 P):((P Xr)->(P Xr))
% Found (eq_ref00 P) as proof of (P0 Xr)
% Found ((eq_ref0 Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found eq_ref000:=(eq_ref00 P):((P Xr)->(P Xr))
% Found (eq_ref00 P) as proof of (P0 Xr)
% Found ((eq_ref0 Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found x5:False
% Found (fun (x5:False)=> x5) as proof of False
% Found (fun (x4:a) (x5:False)=> x5) as proof of (False->False)
% Found (fun (x4:a) (x5:False)=> x5) as proof of (a->(False->False))
% Found (ex_ind00 (fun (x4:a) (x5:False)=> x5)) as proof of False
% Found ((ex_ind0 False) (fun (x4:a) (x5:False)=> x5)) as proof of False
% Found (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5)) as proof of False
% Found (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5))) as proof of False
% Found (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5))) as proof of (not ((ex a) (fun (Xt:a)=> False)))
% Found (x10 (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5)))) as proof of (P (fun (Xx:a)=> False))
% Found ((x1 (fun (Xx:a)=> False)) (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5)))) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x20:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 (fun (Xx:a)=> False)) (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5))))) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x20:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 (fun (Xx:a)=> False)) (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5))))) as proof of ((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P (fun (Xx:a)=> False)))
% Found eq_ref00:=(eq_ref0 cC):(((eq (a->Prop)) cC) cC)
% Found (eq_ref0 cC) as proof of (((eq (a->Prop)) cC) b)
% Found ((eq_ref (a->Prop)) cC) as proof of (((eq (a->Prop)) cC) b)
% Found ((eq_ref (a->Prop)) cC) as proof of (((eq (a->Prop)) cC) b)
% Found ((eq_ref (a->Prop)) cC) as proof of (((eq (a->Prop)) cC) b)
% Found eq_ref00:=(eq_ref0 (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))):(((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))
% Found (eq_ref0 (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) b)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (cC x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (cC x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (cC x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (cC x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (cC x)))
% Found eq_ref000:=(eq_ref00 P):((P Xr)->(P Xr))
% Found (eq_ref00 P) as proof of (P0 Xr)
% Found ((eq_ref0 Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found eq_ref000:=(eq_ref00 P):((P Xr)->(P Xr))
% Found (eq_ref00 P) as proof of (P0 Xr)
% Found ((eq_ref0 Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (cC x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (cC x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (cC x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (cC x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (cC x)))
% Found eq_ref00:=(eq_ref0 (forall (Xr:(a->Prop)) (Xx:a), ((forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr)))->(forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))):(((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr)))->(forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))) (forall (Xr:(a->Prop)) (Xx:a), ((forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr)))->(forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))))
% Found (eq_ref0 (forall (Xr:(a->Prop)) (Xx:a), ((forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr)))->(forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr)))->(forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), ((forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr)))->(forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr)))->(forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), ((forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr)))->(forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr)))->(forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), ((forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr)))->(forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr)))->(forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))) b)
% Found x4:False
% Found (fun (x4:False)=> x4) as proof of False
% Found (fun (x3:a) (x4:False)=> x4) as proof of (False->False)
% Found (fun (x3:a) (x4:False)=> x4) as proof of (a->(False->False))
% Found (ex_ind00 (fun (x3:a) (x4:False)=> x4)) as proof of False
% Found ((ex_ind0 False) (fun (x3:a) (x4:False)=> x4)) as proof of False
% Found (((fun (P0:Prop) (x3:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x3) x2)) False) (fun (x3:a) (x4:False)=> x4)) as proof of False
% Found (fun (x2:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x3:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x3) x2)) False) (fun (x3:a) (x4:False)=> x4))) as proof of False
% Found (fun (x2:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x3:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x3) x2)) False) (fun (x3:a) (x4:False)=> x4))) as proof of (not ((ex a) (fun (Xt:a)=> False)))
% Found (x100 (fun (x2:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x3:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x3) x2)) False) (fun (x3:a) (x4:False)=> x4)))) as proof of (P (fun (Xx:a)=> False))
% Found ((x10 (fun (Xx:a)=> False)) (fun (x2:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x3:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x3) x2)) False) (fun (x3:a) (x4:False)=> x4)))) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x20:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x10 (fun (Xx:a)=> False)) (fun (x2:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x3:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x3) x2)) False) (fun (x3:a) (x4:False)=> x4))))) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x10:(forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (x20:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x10 (fun (Xx:a)=> False)) (fun (x2:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x3:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x3) x2)) False) (fun (x3:a) (x4:False)=> x4))))) as proof of ((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P (fun (Xx:a)=> False)))
% Found (fun (x10:(forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (x20:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x10 (fun (Xx:a)=> False)) (fun (x2:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x3:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x3) x2)) False) (fun (x3:a) (x4:False)=> x4))))) as proof of ((forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P (fun (Xx:a)=> False))))
% Found eq_ref00:=(eq_ref0 (forall (Xr:(a->Prop)) (Xx:a), ((((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr))->(((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))):(((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr))->(((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) (forall (Xr:(a->Prop)) (Xx:a), ((((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr))->(((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))
% Found (eq_ref0 (forall (Xr:(a->Prop)) (Xx:a), ((((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr))->(((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr))->(((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), ((((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr))->(((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr))->(((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), ((((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr))->(((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr))->(((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), ((((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr))->(((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr))->(((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))):(((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))
% Found (eq_ref0 (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) b)
% Found ex_ind:(forall (A:Type) (F:(A->Prop)) (P:Prop), ((forall (x:A), ((F x)->P))->(((ex A) F)->P)))
% Instantiate: b:=(forall (A:Type) (F:(A->Prop)) (P:Prop), ((forall (x:A), ((F x)->P))->(((ex A) F)->P))):Prop
% Found ex_ind as proof of b
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (cC x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (cC x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (cC x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (cC x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (cC x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (cC x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (cC x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (cC x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (cC x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (cC x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 cC):(((eq (a->Prop)) cC) (fun (x:a)=> (cC x)))
% Found (eta_expansion_dep00 cC) as proof of (((eq (a->Prop)) cC) b)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) cC) as proof of (((eq (a->Prop)) cC) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) cC) as proof of (((eq (a->Prop)) cC) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) cC) as proof of (((eq (a->Prop)) cC) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) cC) as proof of (((eq (a->Prop)) cC) b)
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b
% Found eq_ref00:=(eq_ref0 (forall (Xr:(a->Prop)) (Xx:a), (((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr)))->((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))):(((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), (((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr)))->((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))) (forall (Xr:(a->Prop)) (Xx:a), (((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr)))->((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))))
% Found (eq_ref0 (forall (Xr:(a->Prop)) (Xx:a), (((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr)))->((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), (((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr)))->((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), (((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr)))->((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), (((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr)))->((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), (((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr)))->((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), (((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr)))->((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), (((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr)))->((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), (((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr)))->((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))) b)
% Found x4:False
% Found (fun (x4:False)=> x4) as proof of False
% Found (fun (x3:a) (x4:False)=> x4) as proof of (False->False)
% Found (fun (x3:a) (x4:False)=> x4) as proof of (a->(False->False))
% Found (ex_ind00 (fun (x3:a) (x4:False)=> x4)) as proof of False
% Found ((ex_ind0 False) (fun (x3:a) (x4:False)=> x4)) as proof of False
% Found (((fun (P0:Prop) (x3:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x3) x20)) False) (fun (x3:a) (x4:False)=> x4)) as proof of False
% Found (fun (x20:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x3:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x3) x20)) False) (fun (x3:a) (x4:False)=> x4))) as proof of False
% Found (fun (x20:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x3:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x3) x20)) False) (fun (x3:a) (x4:False)=> x4))) as proof of (((ex a) (fun (Xt:a)=> False))->False)
% Found (x10 (fun (x20:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x3:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x3) x20)) False) (fun (x3:a) (x4:False)=> x4)))) as proof of (P (fun (Xx:a)=> False))
% Found ((x1 (fun (Xx:a)=> False)) (fun (x20:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x3:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x3) x20)) False) (fun (x3:a) (x4:False)=> x4)))) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 (fun (Xx:a)=> False)) (fun (x20:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x3:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x3) x20)) False) (fun (x3:a) (x4:False)=> x4))))) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x1:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 (fun (Xx:a)=> False)) (fun (x20:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x3:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x3) x20)) False) (fun (x3:a) (x4:False)=> x4))))) as proof of ((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P (fun (Xx:a)=> False)))
% Found (fun (x1:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 (fun (Xx:a)=> False)) (fun (x20:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x3:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x3) x20)) False) (fun (x3:a) (x4:False)=> x4))))) as proof of ((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P (fun (Xx:a)=> False))))
% Found (and_rect00 (fun (x1:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 (fun (Xx:a)=> False)) (fun (x20:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x3:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x3) x20)) False) (fun (x3:a) (x4:False)=> x4)))))) as proof of (P (fun (Xx:a)=> False))
% Found ((and_rect0 (P (fun (Xx:a)=> False))) (fun (x1:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 (fun (Xx:a)=> False)) (fun (x20:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x3:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x3) x20)) False) (fun (x3:a) (x4:False)=> x4)))))) as proof of (P (fun (Xx:a)=> False))
% Found (((fun (P0:Type) (x1:((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->P0)))=> (((((and_rect (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))) P0) x1) x00)) (P (fun (Xx:a)=> False))) (fun (x1:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 (fun (Xx:a)=> False)) (fun (x20:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x3:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x3) x20)) False) (fun (x3:a) (x4:False)=> x4)))))) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x00:((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))))=> (((fun (P0:Type) (x1:((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->P0)))=> (((((and_rect (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))) P0) x1) x00)) (P (fun (Xx:a)=> False))) (fun (x1:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 (fun (Xx:a)=> False)) (fun (x20:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x3:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x3) x20)) False) (fun (x3:a) (x4:False)=> x4))))))) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x00:((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))))=> (((fun (P0:Type) (x1:((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->P0)))=> (((((and_rect (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))) P0) x1) x00)) (P (fun (Xx:a)=> False))) (fun (x1:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 (fun (Xx:a)=> False)) (fun (x20:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x3:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x3) x20)) False) (fun (x3:a) (x4:False)=> x4))))))) as proof of (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P (fun (Xx:a)=> False)))
% Found eq_sym0:=(eq_sym Prop):(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a)))
% Instantiate: b:=(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))):Prop
% Found eq_sym0 as proof of b
% Found eq_ref00:=(eq_ref0 (forall (Xr:(a->Prop)) (Xx:a), (((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))):(((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), (((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) (forall (Xr:(a->Prop)) (Xx:a), (((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))
% Found (eq_ref0 (forall (Xr:(a->Prop)) (Xx:a), (((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), (((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), (((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), (((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), (((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), (((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), (((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), (((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) b)
% Found ex_ind:(forall (A:Type) (F:(A->Prop)) (P:Prop), ((forall (x:A), ((F x)->P))->(((ex A) F)->P)))
% Instantiate: b:=(forall (A:Type) (F:(A->Prop)) (P:Prop), ((forall (x:A), ((F x)->P))->(((ex A) F)->P))):Prop
% Found ex_ind as proof of b
% Found x4:False
% Found (fun (x4:False)=> x4) as proof of False
% Found (fun (x3:a) (x4:False)=> x4) as proof of (False->False)
% Found (fun (x3:a) (x4:False)=> x4) as proof of (a->(False->False))
% Found (ex_ind00 (fun (x3:a) (x4:False)=> x4)) as proof of False
% Found ((ex_ind0 False) (fun (x3:a) (x4:False)=> x4)) as proof of False
% Found (((fun (P0:Prop) (x3:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x3) x20)) False) (fun (x3:a) (x4:False)=> x4)) as proof of False
% Found (fun (x20:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x3:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x3) x20)) False) (fun (x3:a) (x4:False)=> x4))) as proof of False
% Found (fun (x20:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x3:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x3) x20)) False) (fun (x3:a) (x4:False)=> x4))) as proof of (((ex a) (fun (Xt:a)=> False))->False)
% Found (x10 (fun (x20:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x3:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x3) x20)) False) (fun (x3:a) (x4:False)=> x4)))) as proof of (P (fun (Xx:a)=> False))
% Found ((x1 (fun (Xx:a)=> False)) (fun (x20:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x3:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x3) x20)) False) (fun (x3:a) (x4:False)=> x4)))) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 (fun (Xx:a)=> False)) (fun (x20:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x3:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x3) x20)) False) (fun (x3:a) (x4:False)=> x4))))) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x1:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 (fun (Xx:a)=> False)) (fun (x20:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x3:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x3) x20)) False) (fun (x3:a) (x4:False)=> x4))))) as proof of ((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P (fun (Xx:a)=> False)))
% Found (fun (x1:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 (fun (Xx:a)=> False)) (fun (x20:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x3:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x3) x20)) False) (fun (x3:a) (x4:False)=> x4))))) as proof of ((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P (fun (Xx:a)=> False))))
% Found (and_rect00 (fun (x1:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 (fun (Xx:a)=> False)) (fun (x20:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x3:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x3) x20)) False) (fun (x3:a) (x4:False)=> x4)))))) as proof of (P (fun (Xx:a)=> False))
% Found ((and_rect0 (P (fun (Xx:a)=> False))) (fun (x1:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 (fun (Xx:a)=> False)) (fun (x20:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x3:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x3) x20)) False) (fun (x3:a) (x4:False)=> x4)))))) as proof of (P (fun (Xx:a)=> False))
% Found (((fun (P0:Type) (x1:((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->P0)))=> (((((and_rect (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))) P0) x1) x00)) (P (fun (Xx:a)=> False))) (fun (x1:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 (fun (Xx:a)=> False)) (fun (x20:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x3:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x3) x20)) False) (fun (x3:a) (x4:False)=> x4)))))) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x00:((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))))=> (((fun (P0:Type) (x1:((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->P0)))=> (((((and_rect (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))) P0) x1) x00)) (P (fun (Xx:a)=> False))) (fun (x1:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 (fun (Xx:a)=> False)) (fun (x20:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x3:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x3) x20)) False) (fun (x3:a) (x4:False)=> x4))))))) as proof of (P (fun (Xx:a)=> False))
% Found (fun (P:((a->Prop)->Prop)) (x00:((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))))=> (((fun (P0:Type) (x1:((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->P0)))=> (((((and_rect (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))) P0) x1) x00)) (P (fun (Xx:a)=> False))) (fun (x1:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 (fun (Xx:a)=> False)) (fun (x20:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x3:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x3) x20)) False) (fun (x3:a) (x4:False)=> x4))))))) as proof of (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P (fun (Xx:a)=> False)))
% Found (fun (P:((a->Prop)->Prop)) (x00:((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))))=> (((fun (P0:Type) (x1:((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->P0)))=> (((((and_rect (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))) P0) x1) x00)) (P (fun (Xx:a)=> False))) (fun (x1:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 (fun (Xx:a)=> False)) (fun (x20:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x3:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x3) x20)) False) (fun (x3:a) (x4:False)=> x4))))))) as proof of (forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P (fun (Xx:a)=> False))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 cC):(((eq (a->Prop)) cC) (fun (x:a)=> (cC x)))
% Found (eta_expansion_dep00 cC) as proof of (((eq (a->Prop)) cC) b)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) cC) as proof of (((eq (a->Prop)) cC) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) cC) as proof of (((eq (a->Prop)) cC) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) cC) as proof of (((eq (a->Prop)) cC) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) cC) as proof of (((eq (a->Prop)) cC) b)
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b
% Found eq_ref00:=(eq_ref0 (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))):(((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))
% Found (eq_ref0 (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) b)
% Found x6:False
% Found (fun (x6:False)=> x6) as proof of False
% Found (fun (x5:a) (x6:False)=> x6) as proof of (False->False)
% Found (fun (x5:a) (x6:False)=> x6) as proof of (a->(False->False))
% Found (ex_ind00 (fun (x5:a) (x6:False)=> x6)) as proof of False
% Found ((ex_ind0 False) (fun (x5:a) (x6:False)=> x6)) as proof of False
% Found (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6)) as proof of False
% Found (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6))) as proof of False
% Found (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6))) as proof of (((ex a) (fun (Xt:a)=> False))->False)
% Found (x20 (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6)))) as proof of (P (fun (Xx:a)=> False))
% Found ((x2 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6)))) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x3:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x2 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6))))) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x2:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x3:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x2 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6))))) as proof of ((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P (fun (Xx:a)=> False)))
% Found (fun (x2:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x3:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x2 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6))))) as proof of ((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P (fun (Xx:a)=> False))))
% Found (and_rect00 (fun (x2:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x3:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x2 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6)))))) as proof of (P (fun (Xx:a)=> False))
% Found ((and_rect0 (P (fun (Xx:a)=> False))) (fun (x2:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x3:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x2 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6)))))) as proof of (P (fun (Xx:a)=> False))
% Found (((fun (P0:Type) (x2:((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->P0)))=> (((((and_rect (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))) P0) x2) x0)) (P (fun (Xx:a)=> False))) (fun (x2:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x3:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x2 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6)))))) as proof of (P (fun (Xx:a)=> False))
% Found (((fun (P0:Type) (x2:((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->P0)))=> (((((and_rect (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))) P0) x2) x0)) (P (fun (Xx:a)=> False))) (fun (x2:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x3:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x2 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6)))))) as proof of (P (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (cC x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (cC x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (cC x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (cC x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (cC x)))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (cC x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (cC x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (cC x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (cC x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (cC x)))
% Found eq_ref00:=(eq_ref0 (forall (Xr:(a->Prop)) (Xx:a), (((forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr)))->((forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))):(((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), (((forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr)))->((forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))) (forall (Xr:(a->Prop)) (Xx:a), (((forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr)))->((forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))))
% Found (eq_ref0 (forall (Xr:(a->Prop)) (Xx:a), (((forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr)))->((forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), (((forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr)))->((forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), (((forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr)))->((forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), (((forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr)))->((forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), (((forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr)))->((forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), (((forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr)))->((forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), (((forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr)))->((forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), (((forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr)))->((forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))) b)
% Found eq_sym0:=(eq_sym Prop):(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a)))
% Instantiate: b:=(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))):Prop
% Found eq_sym0 as proof of b
% Found x2:(P Xr)
% Instantiate: b:=Xr:(a->Prop)
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))):(((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found (eq_ref0 (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))):(((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))
% Found (eq_ref0 (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) b)
% Found x5:False
% Found (fun (x5:False)=> x5) as proof of False
% Found (fun (x4:a) (x5:False)=> x5) as proof of (False->False)
% Found (fun (x4:a) (x5:False)=> x5) as proof of (a->(False->False))
% Found (ex_ind00 (fun (x4:a) (x5:False)=> x5)) as proof of False
% Found ((ex_ind0 False) (fun (x4:a) (x5:False)=> x5)) as proof of False
% Found (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5)) as proof of False
% Found (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5))) as proof of False
% Found (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5))) as proof of (not ((ex a) (fun (Xt:a)=> False)))
% Found (x10 (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5)))) as proof of (P (fun (Xx:a)=> False))
% Found ((x1 (fun (Xx:a)=> False)) (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5)))) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 (fun (Xx:a)=> False)) (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5))))) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x1:(forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 (fun (Xx:a)=> False)) (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5))))) as proof of ((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P (fun (Xx:a)=> False)))
% Found (fun (x1:(forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 (fun (Xx:a)=> False)) (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5))))) as proof of ((forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P (fun (Xx:a)=> False))))
% Found (and_rect00 (fun (x1:(forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 (fun (Xx:a)=> False)) (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5)))))) as proof of (P (fun (Xx:a)=> False))
% Found ((and_rect0 (P (fun (Xx:a)=> False))) (fun (x1:(forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 (fun (Xx:a)=> False)) (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5)))))) as proof of (P (fun (Xx:a)=> False))
% Found (((fun (P0:Type) (x1:((forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->P0)))=> (((((and_rect (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))) P0) x1) x00)) (P (fun (Xx:a)=> False))) (fun (x1:(forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 (fun (Xx:a)=> False)) (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5)))))) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x00:((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))))=> (((fun (P0:Type) (x1:((forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->P0)))=> (((((and_rect (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))) P0) x1) x00)) (P (fun (Xx:a)=> False))) (fun (x1:(forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 (fun (Xx:a)=> False)) (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5))))))) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x00:((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))))=> (((fun (P0:Type) (x1:((forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->P0)))=> (((((and_rect (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))) P0) x1) x00)) (P (fun (Xx:a)=> False))) (fun (x1:(forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 (fun (Xx:a)=> False)) (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5))))))) as proof of (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P (fun (Xx:a)=> False)))
% Found eq_ref00:=(eq_ref0 (forall (Xr:(a->Prop)) (Xx:a), (((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))):(((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), (((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) (forall (Xr:(a->Prop)) (Xx:a), (((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))
% Found (eq_ref0 (forall (Xr:(a->Prop)) (Xx:a), (((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), (((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), (((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), (((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), (((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), (((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), (((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), (((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P Xr))->((forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) b)
% Found x5:False
% Found (fun (x5:False)=> x5) as proof of False
% Found (fun (x4:a) (x5:False)=> x5) as proof of (False->False)
% Found (fun (x4:a) (x5:False)=> x5) as proof of (a->(False->False))
% Found (ex_ind00 (fun (x4:a) (x5:False)=> x5)) as proof of False
% Found ((ex_ind0 False) (fun (x4:a) (x5:False)=> x5)) as proof of False
% Found (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5)) as proof of False
% Found (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5))) as proof of False
% Found (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5))) as proof of (not ((ex a) (fun (Xt:a)=> False)))
% Found (x10 (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5)))) as proof of (P (fun (Xx:a)=> False))
% Found ((x1 (fun (Xx:a)=> False)) (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5)))) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 (fun (Xx:a)=> False)) (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5))))) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x1:(forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 (fun (Xx:a)=> False)) (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5))))) as proof of ((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P (fun (Xx:a)=> False)))
% Found (fun (x1:(forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 (fun (Xx:a)=> False)) (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5))))) as proof of ((forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P (fun (Xx:a)=> False))))
% Found (and_rect00 (fun (x1:(forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 (fun (Xx:a)=> False)) (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5)))))) as proof of (P (fun (Xx:a)=> False))
% Found ((and_rect0 (P (fun (Xx:a)=> False))) (fun (x1:(forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 (fun (Xx:a)=> False)) (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5)))))) as proof of (P (fun (Xx:a)=> False))
% Found (((fun (P0:Type) (x1:((forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->P0)))=> (((((and_rect (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))) P0) x1) x00)) (P (fun (Xx:a)=> False))) (fun (x1:(forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 (fun (Xx:a)=> False)) (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5)))))) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x00:((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))))=> (((fun (P0:Type) (x1:((forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->P0)))=> (((((and_rect (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))) P0) x1) x00)) (P (fun (Xx:a)=> False))) (fun (x1:(forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 (fun (Xx:a)=> False)) (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5))))))) as proof of (P (fun (Xx:a)=> False))
% Found (fun (P:((a->Prop)->Prop)) (x00:((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))))=> (((fun (P0:Type) (x1:((forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->P0)))=> (((((and_rect (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))) P0) x1) x00)) (P (fun (Xx:a)=> False))) (fun (x1:(forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 (fun (Xx:a)=> False)) (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5))))))) as proof of (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P (fun (Xx:a)=> False)))
% Found (fun (P:((a->Prop)->Prop)) (x00:((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))))=> (((fun (P0:Type) (x1:((forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->P0)))=> (((((and_rect (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))) P0) x1) x00)) (P (fun (Xx:a)=> False))) (fun (x1:(forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 (fun (Xx:a)=> False)) (fun (x3:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x4:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x4) x3)) False) (fun (x4:a) (x5:False)=> x5))))))) as proof of (forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P (fun (Xx:a)=> False))))
% Found x2:(P Xr)
% Instantiate: f:=Xr:(a->Prop)
% Found x2 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((or (Xr x3)) (((eq a) x3) Xx)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((or (Xr x3)) (((eq a) x3) Xx)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((or (Xr x3)) (((eq a) x3) Xx)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((or (Xr x3)) (((eq a) x3) Xx)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (Xr x)) (((eq a) x) Xx))))
% Found x2:(P Xr)
% Instantiate: f:=Xr:(a->Prop)
% Found x2 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((or (Xr x3)) (((eq a) x3) Xx)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((or (Xr x3)) (((eq a) x3) Xx)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((or (Xr x3)) (((eq a) x3) Xx)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((or (Xr x3)) (((eq a) x3) Xx)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (Xr x)) (((eq a) x) Xx))))
% Found eq_ref00:=(eq_ref0 (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))):(((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))
% Found (eq_ref0 (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) b)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (cC x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (cC x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (cC x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (cC x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (cC x)))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (cC x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (cC x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (cC x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (cC x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (cC x)))
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b
% Found x6:False
% Found (fun (x6:False)=> x6) as proof of False
% Found (fun (x5:a) (x6:False)=> x6) as proof of (False->False)
% Found (fun (x5:a) (x6:False)=> x6) as proof of (a->(False->False))
% Found (ex_ind00 (fun (x5:a) (x6:False)=> x6)) as proof of False
% Found ((ex_ind0 False) (fun (x5:a) (x6:False)=> x6)) as proof of False
% Found (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6)) as proof of False
% Found (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6))) as proof of False
% Found (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6))) as proof of (not ((ex a) (fun (Xt:a)=> False)))
% Found (x20 (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6)))) as proof of (P (fun (Xx:a)=> False))
% Found ((x2 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6)))) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x3:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x2 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6))))) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x2:(forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (x3:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x2 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6))))) as proof of ((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P (fun (Xx:a)=> False)))
% Found (fun (x2:(forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (x3:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x2 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6))))) as proof of ((forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P (fun (Xx:a)=> False))))
% Found (and_rect00 (fun (x2:(forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (x3:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x2 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6)))))) as proof of (P (fun (Xx:a)=> False))
% Found ((and_rect0 (P (fun (Xx:a)=> False))) (fun (x2:(forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (x3:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x2 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6)))))) as proof of (P (fun (Xx:a)=> False))
% Found (((fun (P0:Type) (x2:((forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->P0)))=> (((((and_rect (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))) P0) x2) x0)) (P (fun (Xx:a)=> False))) (fun (x2:(forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (x3:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x2 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6)))))) as proof of (P (fun (Xx:a)=> False))
% Found (((fun (P0:Type) (x2:((forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->P0)))=> (((((and_rect (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))) P0) x2) x0)) (P (fun (Xx:a)=> False))) (fun (x2:(forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (x3:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x2 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6)))))) as proof of (P (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))):(((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found (eq_ref0 (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found x10:=(x1 x00):(P Xr)
% Instantiate: b:=Xr:(a->Prop)
% Found (x1 x00) as proof of (P0 b)
% Found (x1 x00) as proof of (P0 b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))):(((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) (fun (x:a)=> ((or (Xr x)) (((eq a) x) Xx))))
% Found (eta_expansion00 (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))):(((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))
% Found (eq_ref0 (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) b)
% Found x2:(P Xr)
% Instantiate: b:=Xr:(a->Prop)
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))):(((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found (eq_ref0 (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% 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))
% Instantiate: b:=(forall (P:Prop), ((iff P) P)):Prop
% Found iff_refl as proof of b
% Found eq_ref000:=(eq_ref00 P):((P Xr)->(P Xr))
% Found (eq_ref00 P) as proof of (P0 Xr)
% Found ((eq_ref0 Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found eq_ref00:=(eq_ref0 Xr):(((eq (a->Prop)) Xr) Xr)
% Found (eq_ref0 Xr) as proof of (((eq (a->Prop)) Xr) b)
% Found ((eq_ref (a->Prop)) Xr) as proof of (((eq (a->Prop)) Xr) b)
% Found ((eq_ref (a->Prop)) Xr) as proof of (((eq (a->Prop)) Xr) b)
% Found ((eq_ref (a->Prop)) Xr) as proof of (((eq (a->Prop)) Xr) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xr):(((eq (a->Prop)) Xr) (fun (x:a)=> (Xr x)))
% Found (eta_expansion_dep00 Xr) as proof of (((eq (a->Prop)) Xr) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) Xr) as proof of (((eq (a->Prop)) Xr) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xr) as proof of (((eq (a->Prop)) Xr) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xr) as proof of (((eq (a->Prop)) Xr) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xr) as proof of (((eq (a->Prop)) Xr) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->(False->False)))):(((eq Prop) ((a->Prop)->(a->(False->False)))) ((a->Prop)->(a->(False->False))))
% Found (eq_ref0 ((a->Prop)->(a->(False->False)))) as proof of (((eq Prop) ((a->Prop)->(a->(False->False)))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(False->False)))) as proof of (((eq Prop) ((a->Prop)->(a->(False->False)))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(False->False)))) as proof of (((eq Prop) ((a->Prop)->(a->(False->False)))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(False->False)))) as proof of (((eq Prop) ((a->Prop)->(a->(False->False)))) b)
% Found eq_ref00:=(eq_ref0 (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P0 E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P0 Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z))))->(P0 (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P0 E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P0 Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z0))))->(P0 (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))))):(((eq Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P0 E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P0 Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z))))->(P0 (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P0 E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P0 Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z0))))->(P0 (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P0 E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P0 Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z))))->(P0 (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P0 E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P0 Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z0))))->(P0 (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx)))))))))
% Found (eq_ref0 (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P0 E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P0 Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z))))->(P0 (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P0 E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P0 Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z0))))->(P0 (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P0 E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P0 Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z))))->(P0 (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P0 E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P0 Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z0))))->(P0 (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P0 E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P0 Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z))))->(P0 (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P0 E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P0 Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z0))))->(P0 (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P0 E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P0 Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z))))->(P0 (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P0 E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P0 Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z0))))->(P0 (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P0 E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P0 Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z))))->(P0 (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P0 E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P0 Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z0))))->(P0 (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P0 E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P0 Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z))))->(P0 (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P0 E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P0 Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z0))))->(P0 (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P0 E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P0 Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z))))->(P0 (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P0 E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P0 Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z0))))->(P0 (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P0 E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P0 Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z))))->(P0 (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P0 E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P0 Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z0))))->(P0 (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))))) b)
% Found x2:(P Xr)
% Instantiate: f:=Xr:(a->Prop)
% Found x2 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((or (Xr x3)) (((eq a) x3) Xx)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((or (Xr x3)) (((eq a) x3) Xx)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((or (Xr x3)) (((eq a) x3) Xx)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((or (Xr x3)) (((eq a) x3) Xx)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (Xr x)) (((eq a) x) Xx))))
% Found x2:(P Xr)
% Instantiate: f:=Xr:(a->Prop)
% Found x2 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((or (Xr x3)) (((eq a) x3) Xx)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((or (Xr x3)) (((eq a) x3) Xx)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((or (Xr x3)) (((eq a) x3) Xx)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((or (Xr x3)) (((eq a) x3) Xx)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (Xr x)) (((eq a) x) Xx))))
% Found x1:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))
% Instantiate: b:=(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E))):Prop
% Found x1 as proof of b
% Found x10:=(x1 x00):(P Xr)
% Instantiate: f:=Xr:(a->Prop)
% Found (x1 x00) as proof of (P0 f)
% Found (x1 x00) as proof of (P0 f)
% 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)) ((or (Xr x2)) (((eq a) x2) Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((or (Xr x2)) (((eq a) x2) Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((or (Xr x2)) (((eq a) x2) Xx)))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((or (Xr x2)) (((eq a) x2) Xx)))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (Xr x)) (((eq a) x) Xx))))
% Found x10:=(x1 x00):(P Xr)
% Instantiate: f:=Xr:(a->Prop)
% Found (x1 x00) as proof of (P0 f)
% Found (x1 x00) as proof of (P0 f)
% 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)) ((or (Xr x2)) (((eq a) x2) Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((or (Xr x2)) (((eq a) x2) Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((or (Xr x2)) (((eq a) x2) Xx)))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((or (Xr x2)) (((eq a) x2) Xx)))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (Xr x)) (((eq a) x) Xx))))
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b
% Found eq_ref00:=(eq_ref0 (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx)))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z0))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx)))))))):(((eq Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx)))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z0))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx)))))))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx)))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z0))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))))
% Found (eq_ref0 (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx)))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z0))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx)))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx)))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z0))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx)))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx)))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z0))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx)))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx)))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z0))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx)))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx)))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z0))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx)))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx)))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z0))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx)))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx)))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z0))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx)))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx)))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z0))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx)))))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))):(((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) (fun (x:a)=> ((or (Xr x)) (((eq a) x) Xx))))
% Found (eta_expansion_dep00 (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found x10:=(x1 x00):(P Xr)
% Instantiate: b:=Xr:(a->Prop)
% Found (x1 x00) as proof of (P0 b)
% Found (x1 x00) as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))):(((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) (fun (x:a)=> ((or (Xr x)) (((eq a) x) Xx))))
% Found (eta_expansion_dep00 (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found x1:=(x P):(((and (P (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))->(P cC))
% Instantiate: b:=(((and (P (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))->(P cC)):Prop
% Found x1 as proof of b
% Found eq_ref00:=(eq_ref0 (forall (Xr:(a->Prop)) (Xx:a), ((not ((ex a) (fun (Xt:a)=> (Xr Xt))))->(not ((ex a) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))):(((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((not ((ex a) (fun (Xt:a)=> (Xr Xt))))->(not ((ex a) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) (forall (Xr:(a->Prop)) (Xx:a), ((not ((ex a) (fun (Xt:a)=> (Xr Xt))))->(not ((ex a) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))))
% Found (eq_ref0 (forall (Xr:(a->Prop)) (Xx:a), ((not ((ex a) (fun (Xt:a)=> (Xr Xt))))->(not ((ex a) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((not ((ex a) (fun (Xt:a)=> (Xr Xt))))->(not ((ex a) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), ((not ((ex a) (fun (Xt:a)=> (Xr Xt))))->(not ((ex a) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((not ((ex a) (fun (Xt:a)=> (Xr Xt))))->(not ((ex a) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), ((not ((ex a) (fun (Xt:a)=> (Xr Xt))))->(not ((ex a) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((not ((ex a) (fun (Xt:a)=> (Xr Xt))))->(not ((ex a) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(a->Prop)) (Xx:a), ((not ((ex a) (fun (Xt:a)=> (Xr Xt))))->(not ((ex a) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) as proof of (((eq Prop) (forall (Xr:(a->Prop)) (Xx:a), ((not ((ex a) (fun (Xt:a)=> (Xr Xt))))->(not ((ex a) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))) b)
% Found eq_ref000:=(eq_ref00 P):((P Xr)->(P Xr))
% Found (eq_ref00 P) as proof of (P0 Xr)
% Found ((eq_ref0 Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% 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))
% Instantiate: b:=(forall (P:Prop), ((iff P) P)):Prop
% Found iff_refl as proof of b
% Found eta_expansion000:=(eta_expansion00 Xr):(((eq (a->Prop)) Xr) (fun (x:a)=> (Xr x)))
% Found (eta_expansion00 Xr) as proof of (((eq (a->Prop)) Xr) b)
% Found ((eta_expansion0 Prop) Xr) as proof of (((eq (a->Prop)) Xr) b)
% Found (((eta_expansion a) Prop) Xr) as proof of (((eq (a->Prop)) Xr) b)
% Found (((eta_expansion a) Prop) Xr) as proof of (((eq (a->Prop)) Xr) b)
% Found (((eta_expansion a) Prop) Xr) as proof of (((eq (a->Prop)) Xr) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found eq_ref00:=(eq_ref0 Xr):(((eq (a->Prop)) Xr) Xr)
% Found (eq_ref0 Xr) as proof of (((eq (a->Prop)) Xr) b)
% Found ((eq_ref (a->Prop)) Xr) as proof of (((eq (a->Prop)) Xr) b)
% Found ((eq_ref (a->Prop)) Xr) as proof of (((eq (a->Prop)) Xr) b)
% Found ((eq_ref (a->Prop)) Xr) as proof of (((eq (a->Prop)) Xr) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% 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)) ((or (Xr x2)) (((eq a) x2) Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((or (Xr x2)) (((eq a) x2) Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((or (Xr x2)) (((eq a) x2) Xx)))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((or (Xr x2)) (((eq a) x2) Xx)))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (Xr x)) (((eq a) x) Xx))))
% 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)) ((or (Xr x2)) (((eq a) x2) Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((or (Xr x2)) (((eq a) x2) Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((or (Xr x2)) (((eq a) x2) Xx)))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((or (Xr x2)) (((eq a) x2) Xx)))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (Xr x)) (((eq a) x) Xx))))
% Found eq_ref00:=(eq_ref0 (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))):(((eq Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx)))))))
% Found (eq_ref0 (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P0 E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P0 Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z))))->(P0 (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P0 E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P0 Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z0))))->(P0 (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))))):(((eq Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P0 E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P0 Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z))))->(P0 (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P0 E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P0 Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z0))))->(P0 (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P0 E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P0 Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z))))->(P0 (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P0 E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P0 Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z0))))->(P0 (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx)))))))))
% Found (eq_ref0 (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P0 E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P0 Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z))))->(P0 (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P0 E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P0 Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z0))))->(P0 (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P0 E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P0 Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z))))->(P0 (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P0 E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P0 Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z0))))->(P0 (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P0 E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P0 Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z))))->(P0 (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P0 E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P0 Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z0))))->(P0 (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P0 E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P0 Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z))))->(P0 (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P0 E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P0 Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z0))))->(P0 (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P0 E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P0 Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z))))->(P0 (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P0 E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P0 Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z0))))->(P0 (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P0 E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P0 Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z))))->(P0 (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P0 E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P0 Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z0))))->(P0 (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P0 E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P0 Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z))))->(P0 (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P0 E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P0 Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z0))))->(P0 (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P0 E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P0 Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z))))->(P0 (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(forall (P0:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P0 E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P0 Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P0 Z0))))->(P0 (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))))) b)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->(False->False)))):(((eq Prop) ((a->Prop)->(a->(False->False)))) ((a->Prop)->(a->(False->False))))
% Found (eq_ref0 ((a->Prop)->(a->(False->False)))) as proof of (((eq Prop) ((a->Prop)->(a->(False->False)))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(False->False)))) as proof of (((eq Prop) ((a->Prop)->(a->(False->False)))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(False->False)))) as proof of (((eq Prop) ((a->Prop)->(a->(False->False)))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(False->False)))) as proof of (((eq Prop) ((a->Prop)->(a->(False->False)))) b)
% Found x1:(forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))
% Instantiate: b:=(forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E))):Prop
% Found x1 as proof of b
% Found x10:=(x1 x00):(P Xr)
% Instantiate: f:=Xr:(a->Prop)
% Found (x1 x00) as proof of (P0 f)
% Found (x1 x00) as proof of (P0 f)
% 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)) ((or (Xr x2)) (((eq a) x2) Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((or (Xr x2)) (((eq a) x2) Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((or (Xr x2)) (((eq a) x2) Xx)))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((or (Xr x2)) (((eq a) x2) Xx)))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (Xr x)) (((eq a) x) Xx))))
% Found x10:=(x1 x00):(P Xr)
% Instantiate: f:=Xr:(a->Prop)
% Found (x1 x00) as proof of (P0 f)
% Found (x1 x00) as proof of (P0 f)
% 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)) ((or (Xr x2)) (((eq a) x2) Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((or (Xr x2)) (((eq a) x2) Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((or (Xr x2)) (((eq a) x2) Xx)))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((or (Xr x2)) (((eq a) x2) Xx)))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (Xr x)) (((eq a) x) Xx))))
% Found eq_ref00:=(eq_ref0 (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx)))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z0))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx)))))))):(((eq Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx)))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z0))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx)))))))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx)))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z0))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))))
% Found (eq_ref0 (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx)))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z0))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx)))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx)))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z0))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx)))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx)))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z0))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx)))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx)))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z0))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx)))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx)))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z0))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx)))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx)))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z0))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx)))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx)))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z0))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx)))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))->(P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx)))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z0))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx)))))))) b)
% Found x4:(P Xr)
% Instantiate: b:=Xr:(a->Prop)
% Found x4 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))):(((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) (fun (x:a)=> ((or (Xr x)) (((eq a) x) Xx))))
% Found (eta_expansion_dep00 (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found x1:=(x P):(((and (P (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))->(P cC))
% Instantiate: b:=(((and (P (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))->(P cC)):Prop
% Found x1 as proof of b
% Found eq_ref000:=(eq_ref00 (fun (x6:Prop)=> (b x5))):((b x5)->(b x5))
% Found (eq_ref00 (fun (x6:Prop)=> (b x5))) as proof of ((b x5)->((and False) ((a->Prop)->(a->(False->False)))))
% Found ((eq_ref0 ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5))) as proof of ((b x5)->((and False) ((a->Prop)->(a->(False->False)))))
% Found (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5))) as proof of ((b x5)->((and False) ((a->Prop)->(a->(False->False)))))
% Found (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5))) as proof of ((b x5)->((and False) ((a->Prop)->(a->(False->False)))))
% Found (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5)))) as proof of ((b x5)->((and False) ((a->Prop)->(a->(False->False)))))
% Found (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5)))) as proof of (forall (x:a), ((b x)->((and False) ((a->Prop)->(a->(False->False))))))
% Found (ex_ind00 (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5))))) as proof of ((and False) ((a->Prop)->(a->(False->False))))
% Found ((ex_ind0 ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5))))) as proof of ((and False) ((a->Prop)->(a->(False->False))))
% Found (((fun (P1:Prop) (x5:(forall (x:a), ((b x)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (b Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5))))) as proof of ((and False) ((a->Prop)->(a->(False->False))))
% Found (((fun (P1:Prop) (x5:(forall (x:a), ((b x)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (b Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5))))) as proof of ((and False) ((a->Prop)->(a->(False->False))))
% Found (x4 (((fun (P1:Prop) (x5:(forall (x:a), ((b x)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (b Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5)))))) as proof of False
% Found ((x (fun (x5:(a->Prop))=> False)) (((fun (P1:Prop) (x5:(forall (x0:a), ((b x0)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (b Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5)))))) as proof of False
% Found (fun (x3:((ex a) (fun (Xt:a)=> (b Xt))))=> ((x (fun (x5:(a->Prop))=> False)) (((fun (P1:Prop) (x5:(forall (x0:a), ((b x0)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (b Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5))))))) as proof of False
% Found (fun (x3:((ex a) (fun (Xt:a)=> (b Xt))))=> ((x (fun (x5:(a->Prop))=> False)) (((fun (P1:Prop) (x5:(forall (x0:a), ((b x0)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (b Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5))))))) as proof of (((ex a) (fun (Xt:a)=> (b Xt)))->False)
% Found (x10 (fun (x3:((ex a) (fun (Xt:a)=> (b Xt))))=> ((x (fun (x5:(a->Prop))=> False)) (((fun (P1:Prop) (x5:(forall (x0:a), ((b x0)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (b Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5)))))))) as proof of (P0 b)
% Found ((x1 b) (fun (x3:((ex a) (fun (Xt:a)=> (b Xt))))=> ((x (fun (x5:(a->Prop))=> False)) (((fun (P1:Prop) (x5:(forall (x0:a), ((b x0)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (b Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5)))))))) as proof of (P0 b)
% Found ((x1 b) (fun (x3:((ex a) (fun (Xt:a)=> (b Xt))))=> ((x (fun (x5:(a->Prop))=> False)) (((fun (P1:Prop) (x5:(forall (x0:a), ((b x0)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (b Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5)))))))) as proof of (P0 b)
% Found x1:=(x P):(((and (P (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))->(P cC))
% Instantiate: b:=(((and (P (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))->(P cC)):Prop
% Found x1 as proof of b
% Found x00:((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))
% Found x00 as proof of ((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))
% Found x4:(P Xr)
% Instantiate: b:=Xr:(a->Prop)
% Found x4 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))):(((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) (fun (x:a)=> ((or (Xr x)) (((eq a) x) Xx))))
% Found (eta_expansion_dep00 (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) 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)) ((or (Xr x2)) (((eq a) x2) Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((or (Xr x2)) (((eq a) x2) Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((or (Xr x2)) (((eq a) x2) Xx)))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((or (Xr x2)) (((eq a) x2) Xx)))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (Xr x)) (((eq a) x) Xx))))
% 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)) ((or (Xr x2)) (((eq a) x2) Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((or (Xr x2)) (((eq a) x2) Xx)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((or (Xr x2)) (((eq a) x2) Xx)))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((or (Xr x2)) (((eq a) x2) Xx)))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (Xr x)) (((eq a) x) Xx))))
% Found eq_ref00:=(eq_ref0 (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))):(((eq Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx)))))))
% Found (eq_ref0 (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))) b)
% Found x1:(forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr)))
% Instantiate: b:=(forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P Xr))):Prop
% Found x1 as proof of b
% Found x4:(P Xr)
% Instantiate: f:=Xr:(a->Prop)
% Found x4 as proof of (P0 f)
% Found x4:(P Xr)
% Instantiate: f:=Xr:(a->Prop)
% Found x4 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((or (Xr x5)) (((eq a) x5) Xx)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or (Xr x5)) (((eq a) x5) Xx)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or (Xr x5)) (((eq a) x5) Xx)))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((or (Xr x5)) (((eq a) x5) Xx)))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (Xr x)) (((eq a) x) Xx))))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((or (Xr x5)) (((eq a) x5) Xx)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or (Xr x5)) (((eq a) x5) Xx)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or (Xr x5)) (((eq a) x5) Xx)))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((or (Xr x5)) (((eq a) x5) Xx)))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (Xr x)) (((eq a) x) Xx))))
% Found eq_ref000:=(eq_ref00 (fun (x6:Prop)=> (f x5))):((f x5)->(f x5))
% Found (eq_ref00 (fun (x6:Prop)=> (f x5))) as proof of ((f x5)->((and False) ((a->Prop)->(a->(False->False)))))
% Found ((eq_ref0 ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (f x5))) as proof of ((f x5)->((and False) ((a->Prop)->(a->(False->False)))))
% Found (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (f x5))) as proof of ((f x5)->((and False) ((a->Prop)->(a->(False->False)))))
% Found (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (f x5))) as proof of ((f x5)->((and False) ((a->Prop)->(a->(False->False)))))
% Found (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (f x5)))) as proof of ((f x5)->((and False) ((a->Prop)->(a->(False->False)))))
% Found (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (f x5)))) as proof of (forall (x:a), ((f x)->((and False) ((a->Prop)->(a->(False->False))))))
% Found (ex_ind00 (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (f x5))))) as proof of ((and False) ((a->Prop)->(a->(False->False))))
% Found ((ex_ind0 ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (f x5))))) as proof of ((and False) ((a->Prop)->(a->(False->False))))
% Found (((fun (P1:Prop) (x5:(forall (x:a), ((f x)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (f Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (f x5))))) as proof of ((and False) ((a->Prop)->(a->(False->False))))
% Found (((fun (P1:Prop) (x5:(forall (x:a), ((f x)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (f Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (f x5))))) as proof of ((and False) ((a->Prop)->(a->(False->False))))
% Found (x4 (((fun (P1:Prop) (x5:(forall (x:a), ((f x)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (f Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (f x5)))))) as proof of False
% Found ((x (fun (x5:(a->Prop))=> False)) (((fun (P1:Prop) (x5:(forall (x0:a), ((f x0)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (f Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (f x5)))))) as proof of False
% Found (fun (x3:((ex a) (fun (Xt:a)=> (f Xt))))=> ((x (fun (x5:(a->Prop))=> False)) (((fun (P1:Prop) (x5:(forall (x0:a), ((f x0)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (f Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (f x5))))))) as proof of False
% Found (fun (x3:((ex a) (fun (Xt:a)=> (f Xt))))=> ((x (fun (x5:(a->Prop))=> False)) (((fun (P1:Prop) (x5:(forall (x0:a), ((f x0)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (f Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (f x5))))))) as proof of (((ex a) (fun (Xt:a)=> (f Xt)))->False)
% Found (x10 (fun (x3:((ex a) (fun (Xt:a)=> (f Xt))))=> ((x (fun (x5:(a->Prop))=> False)) (((fun (P1:Prop) (x5:(forall (x0:a), ((f x0)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (f Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (f x5)))))))) as proof of (P0 f)
% Found ((x1 f) (fun (x3:((ex a) (fun (Xt:a)=> (f Xt))))=> ((x (fun (x5:(a->Prop))=> False)) (((fun (P1:Prop) (x5:(forall (x0:a), ((f x0)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (f Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (f x5)))))))) as proof of (P0 f)
% Found ((x1 f) (fun (x3:((ex a) (fun (Xt:a)=> (f Xt))))=> ((x (fun (x5:(a->Prop))=> False)) (((fun (P1:Prop) (x5:(forall (x0:a), ((f x0)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (f Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (f x5)))))))) as proof of (P0 f)
% Found eq_ref000:=(eq_ref00 (fun (x6:Prop)=> (f x5))):((f x5)->(f x5))
% Found (eq_ref00 (fun (x6:Prop)=> (f x5))) as proof of ((f x5)->((and False) ((a->Prop)->(a->(False->False)))))
% Found ((eq_ref0 ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (f x5))) as proof of ((f x5)->((and False) ((a->Prop)->(a->(False->False)))))
% Found (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (f x5))) as proof of ((f x5)->((and False) ((a->Prop)->(a->(False->False)))))
% Found (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (f x5))) as proof of ((f x5)->((and False) ((a->Prop)->(a->(False->False)))))
% Found (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (f x5)))) as proof of ((f x5)->((and False) ((a->Prop)->(a->(False->False)))))
% Found (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (f x5)))) as proof of (forall (x:a), ((f x)->((and False) ((a->Prop)->(a->(False->False))))))
% Found (ex_ind00 (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (f x5))))) as proof of ((and False) ((a->Prop)->(a->(False->False))))
% Found ((ex_ind0 ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (f x5))))) as proof of ((and False) ((a->Prop)->(a->(False->False))))
% Found (((fun (P1:Prop) (x5:(forall (x:a), ((f x)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (f Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (f x5))))) as proof of ((and False) ((a->Prop)->(a->(False->False))))
% Found (((fun (P1:Prop) (x5:(forall (x:a), ((f x)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (f Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (f x5))))) as proof of ((and False) ((a->Prop)->(a->(False->False))))
% Found (x4 (((fun (P1:Prop) (x5:(forall (x:a), ((f x)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (f Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (f x5)))))) as proof of False
% Found ((x (fun (x5:(a->Prop))=> False)) (((fun (P1:Prop) (x5:(forall (x0:a), ((f x0)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (f Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (f x5)))))) as proof of False
% Found (fun (x3:((ex a) (fun (Xt:a)=> (f Xt))))=> ((x (fun (x5:(a->Prop))=> False)) (((fun (P1:Prop) (x5:(forall (x0:a), ((f x0)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (f Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (f x5))))))) as proof of False
% Found (fun (x3:((ex a) (fun (Xt:a)=> (f Xt))))=> ((x (fun (x5:(a->Prop))=> False)) (((fun (P1:Prop) (x5:(forall (x0:a), ((f x0)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (f Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (f x5))))))) as proof of (((ex a) (fun (Xt:a)=> (f Xt)))->False)
% Found (x10 (fun (x3:((ex a) (fun (Xt:a)=> (f Xt))))=> ((x (fun (x5:(a->Prop))=> False)) (((fun (P1:Prop) (x5:(forall (x0:a), ((f x0)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (f Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (f x5)))))))) as proof of (P0 f)
% Found ((x1 f) (fun (x3:((ex a) (fun (Xt:a)=> (f Xt))))=> ((x (fun (x5:(a->Prop))=> False)) (((fun (P1:Prop) (x5:(forall (x0:a), ((f x0)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (f Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (f x5)))))))) as proof of (P0 f)
% Found ((x1 f) (fun (x3:((ex a) (fun (Xt:a)=> (f Xt))))=> ((x (fun (x5:(a->Prop))=> False)) (((fun (P1:Prop) (x5:(forall (x0:a), ((f x0)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (f Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (f x5)))))))) as proof of (P0 f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))):(((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) (fun (x:a)=> ((or (Xr x)) (((eq a) x) Xx))))
% Found (eta_expansion_dep00 (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found x10:=(x1 x00):(P Xr)
% Instantiate: b:=Xr:(a->Prop)
% Found (x1 x00) as proof of (P0 b)
% Found (x1 x00) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))):(((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found (eq_ref0 (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found x210:=(x21 x20):(P Xr)
% Instantiate: b:=Xr:(a->Prop)
% Found (x21 x20) as proof of (P0 b)
% Found ((x2 x10) x20) as proof of (P0 b)
% Found ((x2 x10) x20) as proof of (P0 b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))):(((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) (fun (x:a)=> ((or (Xr x)) (((eq a) x) Xx))))
% Found (eta_expansion00 (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found x30:=(x3 x20):(P Xr)
% Instantiate: b:=Xr:(a->Prop)
% Found (x3 x20) as proof of (P0 b)
% Found (x3 x20) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))):(((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found (eq_ref0 (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found x4:(P Xr)
% Instantiate: f:=Xr:(a->Prop)
% Found x4 as proof of (P0 f)
% 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)))
% Instantiate: b:=(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a))):Prop
% Found eq_sym as proof of b
% Found x4:(P Xr)
% Instantiate: f:=Xr:(a->Prop)
% Found x4 as proof of (P0 f)
% Found x4:(P Xr)
% Instantiate: b:=Xr:(a->Prop)
% Found x4 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))):(((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) (fun (x:a)=> ((or (Xr x)) (((eq a) x) Xx))))
% Found (eta_expansion_dep00 (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((or (Xr x5)) (((eq a) x5) Xx)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or (Xr x5)) (((eq a) x5) Xx)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or (Xr x5)) (((eq a) x5) Xx)))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((or (Xr x5)) (((eq a) x5) Xx)))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (Xr x)) (((eq a) x) Xx))))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((or (Xr x5)) (((eq a) x5) Xx)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or (Xr x5)) (((eq a) x5) Xx)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((or (Xr x5)) (((eq a) x5) Xx)))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((or (Xr x5)) (((eq a) x5) Xx)))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (Xr x)) (((eq a) x) Xx))))
% Found eq_ref000:=(eq_ref00 P):((P Xr)->(P Xr))
% Found (eq_ref00 P) as proof of (P0 Xr)
% Found ((eq_ref0 Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found x4:(P Xr)
% Instantiate: Y:=Xr:(a->Prop)
% Found x4 as proof of (P Y)
% Found eq_ref00:=(eq_ref0 Xr):(((eq (a->Prop)) Xr) Xr)
% Found (eq_ref0 Xr) as proof of (((eq (a->Prop)) Xr) b)
% Found ((eq_ref (a->Prop)) Xr) as proof of (((eq (a->Prop)) Xr) b)
% Found ((eq_ref (a->Prop)) Xr) as proof of (((eq (a->Prop)) Xr) b)
% Found ((eq_ref (a->Prop)) Xr) as proof of (((eq (a->Prop)) Xr) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found eq_ref00:=(eq_ref0 Xr):(((eq (a->Prop)) Xr) Xr)
% Found (eq_ref0 Xr) as proof of (((eq (a->Prop)) Xr) b)
% Found ((eq_ref (a->Prop)) Xr) as proof of (((eq (a->Prop)) Xr) b)
% Found ((eq_ref (a->Prop)) Xr) as proof of (((eq (a->Prop)) Xr) b)
% Found ((eq_ref (a->Prop)) Xr) as proof of (((eq (a->Prop)) Xr) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found x00:((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z))))
% Found x00 as proof of ((and (forall (E:(a->Prop)), ((not ((ex a) (fun (Xt:a)=> (E Xt))))->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))
% 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)))
% Instantiate: b:=(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a))):Prop
% Found eq_sym as proof of b
% Found x1:=(x P):(((and (P (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))->(P cC))
% Instantiate: b:=(((and (P (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))->(P cC)):Prop
% Found x1 as proof of b
% Found eq_ref000:=(eq_ref00 (fun (x6:Prop)=> (b x5))):((b x5)->(b x5))
% Found (eq_ref00 (fun (x6:Prop)=> (b x5))) as proof of ((b x5)->((and False) ((a->Prop)->(a->(False->False)))))
% Found ((eq_ref0 ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5))) as proof of ((b x5)->((and False) ((a->Prop)->(a->(False->False)))))
% Found (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5))) as proof of ((b x5)->((and False) ((a->Prop)->(a->(False->False)))))
% Found (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5))) as proof of ((b x5)->((and False) ((a->Prop)->(a->(False->False)))))
% Found (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5)))) as proof of ((b x5)->((and False) ((a->Prop)->(a->(False->False)))))
% Found (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5)))) as proof of (forall (x:a), ((b x)->((and False) ((a->Prop)->(a->(False->False))))))
% Found (ex_ind00 (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5))))) as proof of ((and False) ((a->Prop)->(a->(False->False))))
% Found ((ex_ind0 ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5))))) as proof of ((and False) ((a->Prop)->(a->(False->False))))
% Found (((fun (P1:Prop) (x5:(forall (x:a), ((b x)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (b Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5))))) as proof of ((and False) ((a->Prop)->(a->(False->False))))
% Found (((fun (P1:Prop) (x5:(forall (x:a), ((b x)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (b Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5))))) as proof of ((and False) ((a->Prop)->(a->(False->False))))
% Found (x4 (((fun (P1:Prop) (x5:(forall (x:a), ((b x)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (b Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5)))))) as proof of False
% Found ((x (fun (x5:(a->Prop))=> False)) (((fun (P1:Prop) (x5:(forall (x0:a), ((b x0)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (b Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5)))))) as proof of False
% Found (fun (x3:((ex a) (fun (Xt:a)=> (b Xt))))=> ((x (fun (x5:(a->Prop))=> False)) (((fun (P1:Prop) (x5:(forall (x0:a), ((b x0)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (b Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5))))))) as proof of False
% Found (fun (x3:((ex a) (fun (Xt:a)=> (b Xt))))=> ((x (fun (x5:(a->Prop))=> False)) (((fun (P1:Prop) (x5:(forall (x0:a), ((b x0)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (b Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5))))))) as proof of (((ex a) (fun (Xt:a)=> (b Xt)))->False)
% Found (x10 (fun (x3:((ex a) (fun (Xt:a)=> (b Xt))))=> ((x (fun (x5:(a->Prop))=> False)) (((fun (P1:Prop) (x5:(forall (x0:a), ((b x0)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (b Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5)))))))) as proof of (P0 b)
% Found ((x1 b) (fun (x3:((ex a) (fun (Xt:a)=> (b Xt))))=> ((x (fun (x5:(a->Prop))=> False)) (((fun (P1:Prop) (x5:(forall (x0:a), ((b x0)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (b Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5)))))))) as proof of (P0 b)
% Found (fun (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 b) (fun (x3:((ex a) (fun (Xt:a)=> (b Xt))))=> ((x (fun (x5:(a->Prop))=> False)) (((fun (P1:Prop) (x5:(forall (x0:a), ((b x0)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (b Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5))))))))) as proof of (P0 b)
% Found (fun (x1:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 b) (fun (x3:((ex a) (fun (Xt:a)=> (b Xt))))=> ((x (fun (x5:(a->Prop))=> False)) (((fun (P1:Prop) (x5:(forall (x0:a), ((b x0)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (b Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5))))))))) as proof of ((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P0 b))
% Found (fun (x1:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 b) (fun (x3:((ex a) (fun (Xt:a)=> (b Xt))))=> ((x (fun (x5:(a->Prop))=> False)) (((fun (P1:Prop) (x5:(forall (x0:a), ((b x0)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (b Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5))))))))) as proof of ((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P0 b)))
% Found (and_rect00 (fun (x1:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 b) (fun (x3:((ex a) (fun (Xt:a)=> (b Xt))))=> ((x (fun (x5:(a->Prop))=> False)) (((fun (P1:Prop) (x5:(forall (x0:a), ((b x0)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (b Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5)))))))))) as proof of (P0 b)
% Found ((and_rect0 (P0 b)) (fun (x1:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 b) (fun (x3:((ex a) (fun (Xt:a)=> (b Xt))))=> ((x (fun (x5:(a->Prop))=> False)) (((fun (P1:Prop) (x5:(forall (x0:a), ((b x0)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (b Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5)))))))))) as proof of (P0 b)
% Found (((fun (P1:Type) (x1:((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->P1)))=> (((((and_rect (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))) P1) x1) x0)) (P0 b)) (fun (x1:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 b) (fun (x3:((ex a) (fun (Xt:a)=> (b Xt))))=> ((x (fun (x5:(a->Prop))=> False)) (((fun (P1:Prop) (x5:(forall (x00:a), ((b x00)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (b Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5)))))))))) as proof of (P0 b)
% Found (((fun (P1:Type) (x1:((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->P1)))=> (((((and_rect (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))) P1) x1) x0)) (P0 b)) (fun (x1:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x1 b) (fun (x3:((ex a) (fun (Xt:a)=> (b Xt))))=> ((x (fun (x5:(a->Prop))=> False)) (((fun (P1:Prop) (x5:(forall (x00:a), ((b x00)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (b Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5)))))))))) as proof of (P0 b)
% Found eq_ref000:=(eq_ref00 (fun (x6:Prop)=> (b x5))):((b x5)->(b x5))
% Found (eq_ref00 (fun (x6:Prop)=> (b x5))) as proof of ((b x5)->((and False) ((a->Prop)->(a->(False->False)))))
% Found ((eq_ref0 ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5))) as proof of ((b x5)->((and False) ((a->Prop)->(a->(False->False)))))
% Found (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5))) as proof of ((b x5)->((and False) ((a->Prop)->(a->(False->False)))))
% Found (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5))) as proof of ((b x5)->((and False) ((a->Prop)->(a->(False->False)))))
% Found (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5)))) as proof of ((b x5)->((and False) ((a->Prop)->(a->(False->False)))))
% Found (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5)))) as proof of (forall (x:a), ((b x)->((and False) ((a->Prop)->(a->(False->False))))))
% Found (ex_ind00 (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5))))) as proof of ((and False) ((a->Prop)->(a->(False->False))))
% Found ((ex_ind0 ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5))))) as proof of ((and False) ((a->Prop)->(a->(False->False))))
% Found (((fun (P1:Prop) (x5:(forall (x:a), ((b x)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (b Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5))))) as proof of ((and False) ((a->Prop)->(a->(False->False))))
% Found (((fun (P1:Prop) (x5:(forall (x:a), ((b x)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (b Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5))))) as proof of ((and False) ((a->Prop)->(a->(False->False))))
% Found (x4 (((fun (P1:Prop) (x5:(forall (x:a), ((b x)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (b Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5)))))) as proof of False
% Found ((x (fun (x5:(a->Prop))=> False)) (((fun (P1:Prop) (x5:(forall (x0:a), ((b x0)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (b Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5)))))) as proof of False
% Found (fun (x3:((ex a) (fun (Xt:a)=> (b Xt))))=> ((x (fun (x5:(a->Prop))=> False)) (((fun (P1:Prop) (x5:(forall (x0:a), ((b x0)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (b Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5))))))) as proof of False
% Found (fun (x3:((ex a) (fun (Xt:a)=> (b Xt))))=> ((x (fun (x5:(a->Prop))=> False)) (((fun (P1:Prop) (x5:(forall (x0:a), ((b x0)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (b Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5))))))) as proof of (not ((ex a) (fun (Xt:a)=> (b Xt))))
% Found (x10 (fun (x3:((ex a) (fun (Xt:a)=> (b Xt))))=> ((x (fun (x5:(a->Prop))=> False)) (((fun (P1:Prop) (x5:(forall (x0:a), ((b x0)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (b Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5)))))))) as proof of (P0 b)
% Found ((x1 b) (fun (x3:((ex a) (fun (Xt:a)=> (b Xt))))=> ((x (fun (x5:(a->Prop))=> False)) (((fun (P1:Prop) (x5:(forall (x0:a), ((b x0)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (b Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5)))))))) as proof of (P0 b)
% Found ((x1 b) (fun (x3:((ex a) (fun (Xt:a)=> (b Xt))))=> ((x (fun (x5:(a->Prop))=> False)) (((fun (P1:Prop) (x5:(forall (x0:a), ((b x0)->P1)))=> (((((ex_ind a) (fun (Xt:a)=> (b Xt))) P1) x5) x3)) ((and False) ((a->Prop)->(a->(False->False))))) (fun (x5:a)=> (((eq_ref Prop) ((a->Prop)->(a->(False->False)))) (fun (x6:Prop)=> (b x5)))))))) as proof of (P0 b)
% Found x4:(P Xr)
% Instantiate: b:=Xr:(a->Prop)
% Found x4 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))):(((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) (fun (x:a)=> ((or (Xr x)) (((eq a) x) Xx))))
% Found (eta_expansion_dep00 (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found eq_ref000:=(eq_ref00 P):((P Xr)->(P Xr))
% Found (eq_ref00 P) as proof of (P0 Xr)
% Found ((eq_ref0 Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found (((eq_ref (a->Prop)) Xr) P) as proof of (P0 Xr)
% Found x2:(P Xr)
% Instantiate: b:=Xr:(a->Prop)
% Found x2 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))):(((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) (fun (x:a)=> ((or (Xr x)) (((eq a) x) Xx))))
% Found (eta_expansion_dep00 (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) b)
% Found eq_ref00:=(eq_ref0 (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and ((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z0)))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))))):(((eq Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and ((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z0)))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))))) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and ((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z0)))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx)))))))))
% Found (eq_ref0 (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and ((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z0)))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and ((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z0)))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and ((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z0)))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and ((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z0)))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and ((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z0)))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and ((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z0)))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))))) b)
% Found ((eq_ref Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and ((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z0)))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))))) as proof of (((eq Prop) (forall (Y:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and ((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y0:(a->Prop)) (Xx0:a) (Z:(a->Prop)), (((and (P Y0)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y0 Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z)))->(P (fun (Xt:a)=> ((or (Y Xt)) (((eq a) Xt) Xx))))))) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx0:a) (Z0:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z0 Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))->(P Z0)))->(P (fun (Xt:a)=> ((or (Z Xt)) (((eq a) Xt) Xx))))))))) b)
% Found x4:(P Xr)
% Instantiate: Y:=Xr:(a->Prop)
% Found x4 as proof of (P Y)
% Found eq_ref00:=(eq_ref0 ((a->Prop)->(a->(False->False)))):(((eq Prop) ((a->Prop)->(a->(False->False)))) ((a->Prop)->(a->(False->False))))
% Found (eq_ref0 ((a->Prop)->(a->(False->False)))) as proof of (((eq Prop) ((a->Prop)->(a->(False->False)))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(False->False)))) as proof of (((eq Prop) ((a->Prop)->(a->(False->False)))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(False->False)))) as proof of (((eq Prop) ((a->Prop)->(a->(False->False)))) b)
% Found ((eq_ref Prop) ((a->Prop)->(a->(False->False)))) as proof of (((eq Prop) ((a->Prop)->(a->(False->False)))) b)
% Found x:(forall (Xw:((a->Prop)->Prop)), (((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))->(Xw cC)))
% Instantiate: b:=(forall (Xw:((a->Prop)->Prop)), (((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))->(Xw cC))):Prop
% Found x as proof of b
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xr):(((eq (a->Prop)) Xr) (fun (x:a)=> (Xr x)))
% Found (eta_expansion_dep00 Xr) as proof of (((eq (a->Prop)) Xr) b)
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) Xr) as proof of (((eq (a->Prop)) Xr) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xr) as proof of (((eq (a->Prop)) Xr) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xr) as proof of (((eq (a->Prop)) Xr) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xr) as proof of (((eq (a->Prop)) Xr) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found eta_expansion000:=(eta_expansion00 Xr):(((eq (a->Prop)) Xr) (fun (x:a)=> (Xr x)))
% Found (eta_expansion00 Xr) as proof of (((eq (a->Prop)) Xr) b)
% Found ((eta_expansion0 Prop) Xr) as proof of (((eq (a->Prop)) Xr) b)
% Found (((eta_expansion a) Prop) Xr) as proof of (((eq (a->Prop)) Xr) b)
% Found (((eta_expansion a) Prop) Xr) as proof of (((eq (a->Prop)) Xr) b)
% Found (((eta_expansion a) Prop) Xr) as proof of (((eq (a->Prop)) Xr) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found iff_refl0:=(iff_refl ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))):((iff ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) ((or (Xr Xx_16)) (((eq a) Xx_16) Xx)))
% Found (iff_refl ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) as proof of ((iff ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))
% Found (iff_refl ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) as proof of ((iff ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))
% Found (fun (Xx_16:a)=> (iff_refl ((or (Xr Xx_16)) (((eq a) Xx_16) Xx)))) as proof of ((iff ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))
% Found (fun (Xx_16:a)=> (iff_refl ((or (Xr Xx_16)) (((eq a) Xx_16) Xx)))) as proof of (forall (Xx_16:a), ((iff ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0))))
% Found ((conj10 x4) (fun (Xx_16:a)=> (iff_refl ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))))) as proof of ((and (P Y)) (forall (Xx_16:a), ((iff ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))
% Found (((conj1 (forall (Xx_16:a), ((iff ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0))))) x4) (fun (Xx_16:a)=> (iff_refl ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))))) as proof of ((and (P Y)) (forall (Xx_16:a), ((iff ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))
% Found ((((conj (P Y)) (forall (Xx_16:a), ((iff ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0))))) x4) (fun (Xx_16:a)=> (iff_refl ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))))) as proof of ((and (P Y)) (forall (Xx_16:a), ((iff ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))
% Found ((((conj (P Y)) (forall (Xx_16:a), ((iff ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0))))) x4) (fun (Xx_16:a)=> (iff_refl ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))))) as proof of ((and (P Y)) (forall (Xx_16:a), ((iff ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0)))))
% Found (x2000 ((((conj (P Y)) (forall (Xx_16:a), ((iff ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0))))) x4) (fun (Xx_16:a)=> (iff_refl ((or (Xr Xx_16)) (((eq a) Xx_16) Xx)))))) as proof of (P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found ((x200 (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) ((((conj (P Y)) (forall (Xx_16:a), ((iff ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) ((or (Y Xx_16)) (((eq a) Xx_16) Xx0))))) x4) (fun (Xx_16:a)=> (iff_refl ((or (Xr Xx_16)) (((eq a) Xx_16) Xx)))))) as proof of (P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found (((x20 Xx) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) ((((conj (P Y)) (forall (Xx_16:a), ((iff ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) ((or (Y Xx_16)) (((eq a) Xx_16) Xx))))) x4) (fun (Xx_16:a)=> (iff_refl ((or (Xr Xx_16)) (((eq a) Xx_16) Xx)))))) as proof of (P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found ((((x2 Xr) Xx) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) ((((conj (P Xr)) (forall (Xx_16:a), ((iff ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))))) x4) (fun (Xx_16:a)=> (iff_refl ((or (Xr Xx_16)) (((eq a) Xx_16) Xx)))))) as proof of (P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found (fun (x4:(P Xr))=> ((((x2 Xr) Xx) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) ((((conj (P Xr)) (forall (Xx_16:a), ((iff ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))))) x4) (fun (Xx_16:a)=> (iff_refl ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))))))) as proof of (P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))
% Found (fun (Xx:a) (x4:(P Xr))=> ((((x2 Xr) Xx) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) ((((conj (P Xr)) (forall (Xx_16:a), ((iff ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))))) x4) (fun (Xx_16:a)=> (iff_refl ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))))))) as proof of ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))
% Found (fun (Xr:(a->Prop)) (Xx:a) (x4:(P Xr))=> ((((x2 Xr) Xx) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) ((((conj (P Xr)) (forall (Xx_16:a), ((iff ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))))) x4) (fun (Xx_16:a)=> (iff_refl ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))))))) as proof of (forall (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))
% Found (fun (Xr:(a->Prop)) (Xx:a) (x4:(P Xr))=> ((((x2 Xr) Xx) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) ((((conj (P Xr)) (forall (Xx_16:a), ((iff ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))))) x4) (fun (Xx_16:a)=> (iff_refl ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))))))) as proof of (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))
% Found ((conj00 ((x1 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6))))) (fun (Xr:(a->Prop)) (Xx:a) (x4:(P Xr))=> ((((x2 Xr) Xx) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) ((((conj (P Xr)) (forall (Xx_16:a), ((iff ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))))) x4) (fun (Xx_16:a)=> (iff_refl ((or (Xr Xx_16)) (((eq a) Xx_16) Xx)))))))) as proof of ((and (P (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))
% Found (((conj0 (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) ((x1 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6))))) (fun (Xr:(a->Prop)) (Xx:a) (x4:(P Xr))=> ((((x2 Xr) Xx) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) ((((conj (P Xr)) (forall (Xx_16:a), ((iff ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))))) x4) (fun (Xx_16:a)=> (iff_refl ((or (Xr Xx_16)) (((eq a) Xx_16) Xx)))))))) as proof of ((and (P (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))
% Found ((((conj (P (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) ((x1 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6))))) (fun (Xr:(a->Prop)) (Xx:a) (x4:(P Xr))=> ((((x2 Xr) Xx) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) ((((conj (P Xr)) (forall (Xx_16:a), ((iff ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))))) x4) (fun (Xx_16:a)=> (iff_refl ((or (Xr Xx_16)) (((eq a) Xx_16) Xx)))))))) as proof of ((and (P (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))
% Found ((((conj (P (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) ((x1 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6))))) (fun (Xr:(a->Prop)) (Xx:a) (x4:(P Xr))=> ((((x2 Xr) Xx) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) ((((conj (P Xr)) (forall (Xx_16:a), ((iff ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))))) x4) (fun (Xx_16:a)=> (iff_refl ((or (Xr Xx_16)) (((eq a) Xx_16) Xx)))))))) as proof of ((and (P (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))
% Found (x3 ((((conj (P (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) ((x1 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6))))) (fun (Xr:(a->Prop)) (Xx:a) (x4:(P Xr))=> ((((x2 Xr) Xx) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) ((((conj (P Xr)) (forall (Xx_16:a), ((iff ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))))) x4) (fun (Xx_16:a)=> (iff_refl ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))))))))) as proof of (P cC)
% Found ((x P) ((((conj (P (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) ((x1 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6))))) (fun (Xr:(a->Prop)) (Xx:a) (x4:(P Xr))=> ((((x2 Xr) Xx) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) ((((conj (P Xr)) (forall (Xx_16:a), ((iff ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))))) x4) (fun (Xx_16:a)=> (iff_refl ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))))))))) as proof of (P cC)
% Found (fun (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x P) ((((conj (P (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) ((x1 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6))))) (fun (Xr:(a->Prop)) (Xx:a) (x4:(P Xr))=> ((((x2 Xr) Xx) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) ((((conj (P Xr)) (forall (Xx_16:a), ((iff ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))))) x4) (fun (Xx_16:a)=> (iff_refl ((or (Xr Xx_16)) (((eq a) Xx_16) Xx)))))))))) as proof of (P cC)
% Found (fun (x1:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x P) ((((conj (P (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) ((x1 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6))))) (fun (Xr:(a->Prop)) (Xx:a) (x4:(P Xr))=> ((((x2 Xr) Xx) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) ((((conj (P Xr)) (forall (Xx_16:a), ((iff ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))))) x4) (fun (Xx_16:a)=> (iff_refl ((or (Xr Xx_16)) (((eq a) Xx_16) Xx)))))))))) as proof of ((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P cC))
% Found (fun (x1:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x P) ((((conj (P (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) ((x1 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6))))) (fun (Xr:(a->Prop)) (Xx:a) (x4:(P Xr))=> ((((x2 Xr) Xx) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) ((((conj (P Xr)) (forall (Xx_16:a), ((iff ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))))) x4) (fun (Xx_16:a)=> (iff_refl ((or (Xr Xx_16)) (((eq a) Xx_16) Xx)))))))))) as proof of ((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->(P cC)))
% Found (and_rect00 (fun (x1:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x P) ((((conj (P (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) ((x1 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6))))) (fun (Xr:(a->Prop)) (Xx:a) (x4:(P Xr))=> ((((x2 Xr) Xx) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) ((((conj (P Xr)) (forall (Xx_16:a), ((iff ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))))) x4) (fun (Xx_16:a)=> (iff_refl ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))))))))))) as proof of (P cC)
% Found ((and_rect0 (P cC)) (fun (x1:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x P) ((((conj (P (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) ((x1 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6))))) (fun (Xr:(a->Prop)) (Xx:a) (x4:(P Xr))=> ((((x2 Xr) Xx) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) ((((conj (P Xr)) (forall (Xx_16:a), ((iff ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))))) x4) (fun (Xx_16:a)=> (iff_refl ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))))))))))) as proof of (P cC)
% Found (((fun (P0:Type) (x1:((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->P0)))=> (((((and_rect (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))) P0) x1) x0)) (P cC)) (fun (x1:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x P) ((((conj (P (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) ((x1 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6))))) (fun (Xr:(a->Prop)) (Xx:a) (x4:(P Xr))=> ((((x2 Xr) Xx) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) ((((conj (P Xr)) (forall (Xx_16:a), ((iff ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))))) x4) (fun (Xx_16:a)=> (iff_refl ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))))))))))) as proof of (P cC)
% Found (fun (x0:((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))))=> (((fun (P0:Type) (x1:((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->P0)))=> (((((and_rect (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))) P0) x1) x0)) (P cC)) (fun (x1:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x P) ((((conj (P (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) ((x1 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6))))) (fun (Xr:(a->Prop)) (Xx:a) (x4:(P Xr))=> ((((x2 Xr) Xx) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) ((((conj (P Xr)) (forall (Xx_16:a), ((iff ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))))) x4) (fun (Xx_16:a)=> (iff_refl ((or (Xr Xx_16)) (((eq a) Xx_16) Xx)))))))))))) as proof of (P cC)
% Found (fun (P:((a->Prop)->Prop)) (x0:((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))))=> (((fun (P0:Type) (x1:((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->P0)))=> (((((and_rect (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))) P0) x1) x0)) (P cC)) (fun (x1:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x P) ((((conj (P (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) ((x1 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6))))) (fun (Xr:(a->Prop)) (Xx:a) (x4:(P Xr))=> ((((x2 Xr) Xx) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) ((((conj (P Xr)) (forall (Xx_16:a), ((iff ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))))) x4) (fun (Xx_16:a)=> (iff_refl ((or (Xr Xx_16)) (((eq a) Xx_16) Xx)))))))))))) as proof of (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P cC))
% Found (fun (x:(forall (Xw:((a->Prop)->Prop)), (((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))->(Xw cC)))) (P:((a->Prop)->Prop)) (x0:((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))))=> (((fun (P0:Type) (x1:((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->P0)))=> (((((and_rect (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))) P0) x1) x0)) (P cC)) (fun (x1:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x P) ((((conj (P (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) ((x1 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6))))) (fun (Xr:(a->Prop)) (Xx:a) (x4:(P Xr))=> ((((x2 Xr) Xx) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) ((((conj (P Xr)) (forall (Xx_16:a), ((iff ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))))) x4) (fun (Xx_16:a)=> (iff_refl ((or (Xr Xx_16)) (((eq a) Xx_16) Xx)))))))))))) as proof of (forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P cC)))
% Found (fun (x:(forall (Xw:((a->Prop)->Prop)), (((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))->(Xw cC)))) (P:((a->Prop)->Prop)) (x0:((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))))=> (((fun (P0:Type) (x1:((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->P0)))=> (((((and_rect (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))) P0) x1) x0)) (P cC)) (fun (x1:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x P) ((((conj (P (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) ((x1 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6))))) (fun (Xr:(a->Prop)) (Xx:a) (x4:(P Xr))=> ((((x2 Xr) Xx) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) ((((conj (P Xr)) (forall (Xx_16:a), ((iff ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))))) x4) (fun (Xx_16:a)=> (iff_refl ((or (Xr Xx_16)) (((eq a) Xx_16) Xx)))))))))))) as proof of ((forall (Xw:((a->Prop)->Prop)), (((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))->(Xw cC)))->(forall (P:((a->Prop)->Prop)), (((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))->(P cC))))
% Got proof (fun (x:(forall (Xw:((a->Prop)->Prop)), (((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))->(Xw cC)))) (P:((a->Prop)->Prop)) (x0:((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))))=> (((fun (P0:Type) (x1:((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->P0)))=> (((((and_rect (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))) P0) x1) x0)) (P cC)) (fun (x1:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x P) ((((conj (P (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) ((x1 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6))))) (fun (Xr:(a->Prop)) (Xx:a) (x4:(P Xr))=> ((((x2 Xr) Xx) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) ((((conj (P Xr)) (forall (Xx_16:a), ((iff ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))))) x4) (fun (Xx_16:a)=> (iff_refl ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))))))))))))
% Time elapsed = 278.983188s
% node=29707 cost=1411.000000 depth=31
% ::::::::::::::::::::::
% % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p
% % SZS output start Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% (fun (x:(forall (Xw:((a->Prop)->Prop)), (((and (Xw (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((Xw Xr)->(Xw (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))))))->(Xw cC)))) (P:((a->Prop)->Prop)) (x0:((and (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))))=> (((fun (P0:Type) (x1:((forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))->((forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))->P0)))=> (((((and_rect (forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z)))) P0) x1) x0)) (P cC)) (fun (x1:(forall (E:(a->Prop)), ((((ex a) (fun (Xt:a)=> (E Xt)))->False)->(P E)))) (x2:(forall (Y:(a->Prop)) (Xx:a) (Z:(a->Prop)), (((and (P Y)) (forall (Xx_16:a), ((iff (Z Xx_16)) ((or (Y Xx_16)) (((eq a) Xx_16) Xx)))))->(P Z))))=> ((x P) ((((conj (P (fun (Xx:a)=> False))) (forall (Xr:(a->Prop)) (Xx:a), ((P Xr)->(P (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx))))))) ((x1 (fun (Xx:a)=> False)) (fun (x4:((ex a) (fun (Xt:a)=> False)))=> (((fun (P0:Prop) (x5:(a->(False->P0)))=> (((((ex_ind a) (fun (Xt:a)=> False)) P0) x5) x4)) False) (fun (x5:a) (x6:False)=> x6))))) (fun (Xr:(a->Prop)) (Xx:a) (x4:(P Xr))=> ((((x2 Xr) Xx) (fun (Xt:a)=> ((or (Xr Xt)) (((eq a) Xt) Xx)))) ((((conj (P Xr)) (forall (Xx_16:a), ((iff ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))) ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))))) x4) (fun (Xx_16:a)=> (iff_refl ((or (Xr Xx_16)) (((eq a) Xx_16) Xx))))))))))))
% % SZS output end Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% EOF
%------------------------------------------------------------------------------