TSTP Solution File: SEV215^5 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEV215^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n094.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 13:33:53 EDT 2014
% Result : Timeout 300.06s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEV215^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n094.star.cs.uiowa.edu
% % Model : x86_64 x86_64
% % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory : 32286.75MB
% % OS : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 08:30:51 CDT 2014
% % CPUTime : 300.06
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x1a838c0>, <kernel.Type object at 0x1a83998>) of role type named c_type
% Using role type
% Declaring c:Type
% FOF formula (<kernel.Constant object at 0x1eb4248>, <kernel.Type object at 0x1a83830>) of role type named iS_type
% Using role type
% Declaring iS:Type
% FOF formula (<kernel.Constant object at 0x1a837e8>, <kernel.DependentProduct object at 0x1a83950>) of role type named cR
% Using role type
% Declaring cR:(c->c)
% FOF formula (<kernel.Constant object at 0x1a83998>, <kernel.DependentProduct object at 0x1a83560>) of role type named cP
% Using role type
% Declaring cP:(iS->(iS->iS))
% FOF formula (<kernel.Constant object at 0x1a838c0>, <kernel.Constant object at 0x1a83560>) of role type named c0
% Using role type
% Declaring c0:iS
% FOF formula (<kernel.Constant object at 0x1a837e8>, <kernel.DependentProduct object at 0x1a83ab8>) of role type named cL
% Using role type
% Declaring cL:(c->c)
% FOF formula (<kernel.Constant object at 0x1a83b00>, <kernel.DependentProduct object at 0x1a83950>) of role type named cX0
% Using role type
% Declaring cX0:(c->Prop)
% FOF formula (<kernel.Constant object at 0x1a83560>, <kernel.DependentProduct object at 0x1a83440>) of role type named cX1
% Using role type
% Declaring cX1:(c->Prop)
% FOF formula (((and ((and ((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (forall (Xz:c), ((iff (cX0 Xz)) ((cX1 Xz)->False))))) (forall (Xz:c), ((cX0 Xz)->((and (((eq c) (cL Xz)) Xz)) (((eq c) (cR Xz)) Xz)))))->((ex (c->(iS->Prop))) (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg))))))) of role conjecture named cTHM_S_CHAR_T_pme
% Conjecture to prove = (((and ((and ((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (forall (Xz:c), ((iff (cX0 Xz)) ((cX1 Xz)->False))))) (forall (Xz:c), ((cX0 Xz)->((and (((eq c) (cL Xz)) Xz)) (((eq c) (cR Xz)) Xz)))))->((ex (c->(iS->Prop))) (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg))))))):Prop
% Parameter c_DUMMY:c.
% We need to prove ['(((and ((and ((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (forall (Xz:c), ((iff (cX0 Xz)) ((cX1 Xz)->False))))) (forall (Xz:c), ((cX0 Xz)->((and (((eq c) (cL Xz)) Xz)) (((eq c) (cR Xz)) Xz)))))->((ex (c->(iS->Prop))) (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))))']
% Parameter c:Type.
% Parameter iS:Type.
% Parameter cR:(c->c).
% Parameter cP:(iS->(iS->iS)).
% Parameter c0:iS.
% Parameter cL:(c->c).
% Parameter cX0:(c->Prop).
% Parameter cX1:(c->Prop).
% Trying to prove (((and ((and ((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (forall (Xz:c), ((iff (cX0 Xz)) ((cX1 Xz)->False))))) (forall (Xz:c), ((cX0 Xz)->((and (((eq c) (cL Xz)) Xz)) (((eq c) (cR Xz)) Xz)))))->((ex (c->(iS->Prop))) (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))))
% Found eq_ref00:=(eq_ref0 x0):(((eq (c->(iS->Prop))) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((eq_ref (c->(iS->Prop))) x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((eq_ref (c->(iS->Prop))) x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))))=> ((eq_ref (c->(iS->Prop))) x0)) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of ((P x0)->(P Xg))
% Found ((eq_ref0 x0) P) as proof of ((P x0)->(P Xg))
% Found (((eq_ref (c->(iS->Prop))) x0) P) as proof of ((P x0)->(P Xg))
% Found (((eq_ref (c->(iS->Prop))) x0) P) as proof of ((P x0)->(P Xg))
% Found (fun (P:((c->(iS->Prop))->Prop))=> (((eq_ref (c->(iS->Prop))) x0) P)) as proof of ((P x0)->(P Xg))
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))) (P:((c->(iS->Prop))->Prop))=> (((eq_ref (c->(iS->Prop))) x0) P)) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found x1:(P x0)
% Instantiate: x0:=Xg:(c->(iS->Prop))
% Found (fun (x1:(P x0))=> x1) as proof of (P Xg)
% Found (fun (P:((c->(iS->Prop))->Prop)) (x1:(P x0))=> x1) as proof of ((P x0)->(P Xg))
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))) (P:((c->(iS->Prop))->Prop)) (x1:(P x0))=> x1) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found eq_ref00:=(eq_ref0 x0):(((eq (c->(iS->Prop))) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((eq_ref (c->(iS->Prop))) x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((eq_ref (c->(iS->Prop))) x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))))=> ((eq_ref (c->(iS->Prop))) x0)) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of ((P x0)->(P Xg))
% Found ((eq_ref0 x0) P) as proof of ((P x0)->(P Xg))
% Found (((eq_ref (c->(iS->Prop))) x0) P) as proof of ((P x0)->(P Xg))
% Found (((eq_ref (c->(iS->Prop))) x0) P) as proof of ((P x0)->(P Xg))
% Found (fun (P:((c->(iS->Prop))->Prop))=> (((eq_ref (c->(iS->Prop))) x0) P)) as proof of ((P x0)->(P Xg))
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))) (P:((c->(iS->Prop))->Prop))=> (((eq_ref (c->(iS->Prop))) x0) P)) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found x1:(P x0)
% Instantiate: x0:=Xg:(c->(iS->Prop))
% Found (fun (x1:(P x0))=> x1) as proof of (P Xg)
% Found (fun (P:((c->(iS->Prop))->Prop)) (x1:(P x0))=> x1) as proof of ((P x0)->(P Xg))
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))) (P:((c->(iS->Prop))->Prop)) (x1:(P x0))=> x1) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found eq_ref00:=(eq_ref0 x2):(((eq (c->(iS->Prop))) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found ((eq_ref (c->(iS->Prop))) x2) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found ((eq_ref (c->(iS->Prop))) x2) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))))=> ((eq_ref (c->(iS->Prop))) x2)) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found eq_ref00:=(eq_ref0 x0):(((eq (c->(iS->Prop))) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((eq_ref (c->(iS->Prop))) x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((eq_ref (c->(iS->Prop))) x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))))=> ((eq_ref (c->(iS->Prop))) x0)) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found x3:(P x2)
% Instantiate: x2:=Xg:(c->(iS->Prop))
% Found (fun (x3:(P x2))=> x3) as proof of (P Xg)
% Found (fun (P:((c->(iS->Prop))->Prop)) (x3:(P x2))=> x3) as proof of ((P x2)->(P Xg))
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))) (P:((c->(iS->Prop))->Prop)) (x3:(P x2))=> x3) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% Found (eq_ref00 P) as proof of ((P x2)->(P Xg))
% Found ((eq_ref0 x2) P) as proof of ((P x2)->(P Xg))
% Found (((eq_ref (c->(iS->Prop))) x2) P) as proof of ((P x2)->(P Xg))
% Found (((eq_ref (c->(iS->Prop))) x2) P) as proof of ((P x2)->(P Xg))
% Found (fun (P:((c->(iS->Prop))->Prop))=> (((eq_ref (c->(iS->Prop))) x2) P)) as proof of ((P x2)->(P Xg))
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))) (P:((c->(iS->Prop))->Prop))=> (((eq_ref (c->(iS->Prop))) x2) P)) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found eq_ref00:=(eq_ref0 x2):(((eq (c->(iS->Prop))) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found ((eq_ref (c->(iS->Prop))) x2) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found ((eq_ref (c->(iS->Prop))) x2) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))))=> ((eq_ref (c->(iS->Prop))) x2)) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of ((P x0)->(P Xg))
% Found ((eq_ref0 x0) P) as proof of ((P x0)->(P Xg))
% Found (((eq_ref (c->(iS->Prop))) x0) P) as proof of ((P x0)->(P Xg))
% Found (((eq_ref (c->(iS->Prop))) x0) P) as proof of ((P x0)->(P Xg))
% Found (fun (P:((c->(iS->Prop))->Prop))=> (((eq_ref (c->(iS->Prop))) x0) P)) as proof of ((P x0)->(P Xg))
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))) (P:((c->(iS->Prop))->Prop))=> (((eq_ref (c->(iS->Prop))) x0) P)) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found x3:(P x0)
% Instantiate: x0:=Xg:(c->(iS->Prop))
% Found (fun (x3:(P x0))=> x3) as proof of (P Xg)
% Found (fun (P:((c->(iS->Prop))->Prop)) (x3:(P x0))=> x3) as proof of ((P x0)->(P Xg))
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))) (P:((c->(iS->Prop))->Prop)) (x3:(P x0))=> x3) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found eq_ref00:=(eq_ref0 x0):(((eq (c->(iS->Prop))) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((eq_ref (c->(iS->Prop))) x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((eq_ref (c->(iS->Prop))) x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))))=> ((eq_ref (c->(iS->Prop))) x0)) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% Found (eq_ref00 P) as proof of ((P x2)->(P Xg))
% Found ((eq_ref0 x2) P) as proof of ((P x2)->(P Xg))
% Found (((eq_ref (c->(iS->Prop))) x2) P) as proof of ((P x2)->(P Xg))
% Found (((eq_ref (c->(iS->Prop))) x2) P) as proof of ((P x2)->(P Xg))
% Found (fun (P:((c->(iS->Prop))->Prop))=> (((eq_ref (c->(iS->Prop))) x2) P)) as proof of ((P x2)->(P Xg))
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))) (P:((c->(iS->Prop))->Prop))=> (((eq_ref (c->(iS->Prop))) x2) P)) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found x3:(P x2)
% Instantiate: x2:=Xg:(c->(iS->Prop))
% Found (fun (x3:(P x2))=> x3) as proof of (P Xg)
% Found (fun (P:((c->(iS->Prop))->Prop)) (x3:(P x2))=> x3) as proof of ((P x2)->(P Xg))
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))) (P:((c->(iS->Prop))->Prop)) (x3:(P x2))=> x3) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of ((P x0)->(P Xg))
% Found ((eq_ref0 x0) P) as proof of ((P x0)->(P Xg))
% Found (((eq_ref (c->(iS->Prop))) x0) P) as proof of ((P x0)->(P Xg))
% Found (((eq_ref (c->(iS->Prop))) x0) P) as proof of ((P x0)->(P Xg))
% Found (fun (P:((c->(iS->Prop))->Prop))=> (((eq_ref (c->(iS->Prop))) x0) P)) as proof of ((P x0)->(P Xg))
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))) (P:((c->(iS->Prop))->Prop))=> (((eq_ref (c->(iS->Prop))) x0) P)) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found x3:(P x0)
% Instantiate: x0:=Xg:(c->(iS->Prop))
% Found (fun (x3:(P x0))=> x3) as proof of (P Xg)
% Found (fun (P:((c->(iS->Prop))->Prop)) (x3:(P x0))=> x3) as proof of ((P x0)->(P Xg))
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))) (P:((c->(iS->Prop))->Prop)) (x3:(P x0))=> x3) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found eq_ref00:=(eq_ref0 x4):(((eq (c->(iS->Prop))) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq (c->(iS->Prop))) x4) Xg)
% Found ((eq_ref (c->(iS->Prop))) x4) as proof of (((eq (c->(iS->Prop))) x4) Xg)
% Found ((eq_ref (c->(iS->Prop))) x4) as proof of (((eq (c->(iS->Prop))) x4) Xg)
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))))=> ((eq_ref (c->(iS->Prop))) x4)) as proof of (((eq (c->(iS->Prop))) x4) Xg)
% Found eq_ref00:=(eq_ref0 x0):(((eq (c->(iS->Prop))) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((eq_ref (c->(iS->Prop))) x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((eq_ref (c->(iS->Prop))) x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (fun (x2:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x0)) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (fun (x1:((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (x2:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x0)) as proof of ((forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))->(((eq (c->(iS->Prop))) x0) Xg))
% Found (fun (x1:((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (x2:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x0)) as proof of (((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))->((forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))->(((eq (c->(iS->Prop))) x0) Xg)))
% Found (and_rect00 (fun (x1:((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (x2:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x0))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((and_rect0 (((eq (c->(iS->Prop))) x0) Xg)) (fun (x1:((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (x2:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x0))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (((fun (P:Type) (x1:(((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))->((forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))->P)))=> (((((and_rect ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))) P) x1) x00)) (((eq (c->(iS->Prop))) x0) Xg)) (fun (x1:((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (x2:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x0))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))))=> (((fun (P:Type) (x1:(((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))->((forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))->P)))=> (((((and_rect ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))) P) x1) x00)) (((eq (c->(iS->Prop))) x0) Xg)) (fun (x1:((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (x2:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x0)))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found eq_ref00:=(eq_ref0 Xg):(((eq (c->(iS->Prop))) Xg) Xg)
% Found (eq_ref0 Xg) as proof of (((eq (c->(iS->Prop))) Xg) x0)
% Found ((eq_ref (c->(iS->Prop))) Xg) as proof of (((eq (c->(iS->Prop))) Xg) x0)
% Found ((eq_ref (c->(iS->Prop))) Xg) as proof of (((eq (c->(iS->Prop))) Xg) x0)
% Found ((eq_ref (c->(iS->Prop))) Xg) as proof of (((eq (c->(iS->Prop))) Xg) x0)
% Found (eq_sym000 ((eq_ref (c->(iS->Prop))) Xg)) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((eq_sym00 x0) ((eq_ref (c->(iS->Prop))) Xg)) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (((eq_sym0 Xg) x0) ((eq_ref (c->(iS->Prop))) Xg)) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((((eq_sym (c->(iS->Prop))) Xg) x0) ((eq_ref (c->(iS->Prop))) Xg)) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))))=> ((((eq_sym (c->(iS->Prop))) Xg) x0) ((eq_ref (c->(iS->Prop))) Xg))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found eq_ref00:=(eq_ref0 x0):(((eq (c->(iS->Prop))) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((eq_ref (c->(iS->Prop))) x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((eq_ref (c->(iS->Prop))) x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))))=> ((eq_ref (c->(iS->Prop))) x0)) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found eq_ref00:=(eq_ref0 x2):(((eq (c->(iS->Prop))) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found ((eq_ref (c->(iS->Prop))) x2) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found ((eq_ref (c->(iS->Prop))) x2) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))))=> ((eq_ref (c->(iS->Prop))) x2)) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found eq_ref00:=(eq_ref0 x0):(((eq (c->(iS->Prop))) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((eq_ref (c->(iS->Prop))) x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((eq_ref (c->(iS->Prop))) x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (fun (x2:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x0)) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (fun (x1:((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (x2:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x0)) as proof of ((forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))->(((eq (c->(iS->Prop))) x0) Xg))
% Found (fun (x1:((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (x2:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x0)) as proof of (((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))->((forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))->(((eq (c->(iS->Prop))) x0) Xg)))
% Found (and_rect00 (fun (x1:((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (x2:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x0))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((and_rect0 (((eq (c->(iS->Prop))) x0) Xg)) (fun (x1:((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (x2:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x0))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (((fun (P:Type) (x1:(((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))->((forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))->P)))=> (((((and_rect ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))) P) x1) x00)) (((eq (c->(iS->Prop))) x0) Xg)) (fun (x1:((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (x2:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x0))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))))=> (((fun (P:Type) (x1:(((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))->((forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))->P)))=> (((((and_rect ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))) P) x1) x00)) (((eq (c->(iS->Prop))) x0) Xg)) (fun (x1:((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (x2:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x0)))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found eq_ref000:=(eq_ref00 P):((P x4)->(P x4))
% Found (eq_ref00 P) as proof of ((P x4)->(P Xg))
% Found ((eq_ref0 x4) P) as proof of ((P x4)->(P Xg))
% Found (((eq_ref (c->(iS->Prop))) x4) P) as proof of ((P x4)->(P Xg))
% Found (((eq_ref (c->(iS->Prop))) x4) P) as proof of ((P x4)->(P Xg))
% Found (fun (P:((c->(iS->Prop))->Prop))=> (((eq_ref (c->(iS->Prop))) x4) P)) as proof of ((P x4)->(P Xg))
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))) (P:((c->(iS->Prop))->Prop))=> (((eq_ref (c->(iS->Prop))) x4) P)) as proof of (((eq (c->(iS->Prop))) x4) Xg)
% Found x5:(P x4)
% Instantiate: x4:=Xg:(c->(iS->Prop))
% Found (fun (x5:(P x4))=> x5) as proof of (P Xg)
% Found (fun (P:((c->(iS->Prop))->Prop)) (x5:(P x4))=> x5) as proof of ((P x4)->(P Xg))
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))) (P:((c->(iS->Prop))->Prop)) (x5:(P x4))=> x5) as proof of (((eq (c->(iS->Prop))) x4) Xg)
% Found eq_ref00:=(eq_ref0 x4):(((eq (c->(iS->Prop))) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq (c->(iS->Prop))) x4) Xg)
% Found ((eq_ref (c->(iS->Prop))) x4) as proof of (((eq (c->(iS->Prop))) x4) Xg)
% Found ((eq_ref (c->(iS->Prop))) x4) as proof of (((eq (c->(iS->Prop))) x4) Xg)
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))))=> ((eq_ref (c->(iS->Prop))) x4)) as proof of (((eq (c->(iS->Prop))) x4) Xg)
% Found eq_ref00:=(eq_ref0 (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or 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((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))):(((eq ((c->(iS->Prop))->Prop)) (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf 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Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun 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(((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg))))))
% Found (eq_ref0 (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))) as proof of (((eq ((c->(iS->Prop))->Prop)) (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))) b)
% Found ((eq_ref ((c->(iS->Prop))->Prop)) (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))) as proof of (((eq ((c->(iS->Prop))->Prop)) (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))) b)
% Found ((eq_ref ((c->(iS->Prop))->Prop)) (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))) as proof of (((eq ((c->(iS->Prop))->Prop)) (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))) b)
% Found ((eq_ref ((c->(iS->Prop))->Prop)) (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))) as proof of (((eq ((c->(iS->Prop))->Prop)) (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))) b)
% Found eta_expansion000:=(eta_expansion00 Xg):(((eq (c->(iS->Prop))) Xg) (fun (x:c)=> (Xg x)))
% Found (eta_expansion00 Xg) as proof of (((eq (c->(iS->Prop))) Xg) x0)
% Found ((eta_expansion0 (iS->Prop)) Xg) as proof of (((eq (c->(iS->Prop))) Xg) x0)
% Found (((eta_expansion c) (iS->Prop)) Xg) as proof of (((eq (c->(iS->Prop))) Xg) x0)
% Found (((eta_expansion c) (iS->Prop)) Xg) as proof of (((eq (c->(iS->Prop))) Xg) x0)
% Found (((eta_expansion c) (iS->Prop)) Xg) as proof of (((eq (c->(iS->Prop))) Xg) x0)
% Found (eq_sym000 (((eta_expansion c) (iS->Prop)) Xg)) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((eq_sym00 x0) (((eta_expansion c) (iS->Prop)) Xg)) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (((eq_sym0 Xg) x0) (((eta_expansion c) (iS->Prop)) Xg)) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((((eq_sym (c->(iS->Prop))) Xg) x0) (((eta_expansion c) (iS->Prop)) Xg)) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))))=> ((((eq_sym (c->(iS->Prop))) Xg) x0) (((eta_expansion c) (iS->Prop)) Xg))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found eq_ref00:=(eq_ref0 x0):(((eq (c->(iS->Prop))) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((eq_ref (c->(iS->Prop))) x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((eq_ref (c->(iS->Prop))) x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (fun (x2:(forall (Xz:c), ((cX0 Xz)->((and (((eq c) (cL Xz)) Xz)) (((eq c) (cR Xz)) Xz)))))=> ((eq_ref (c->(iS->Prop))) x0)) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (fun (x1:((and ((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (forall (Xz:c), ((iff (cX0 Xz)) ((cX1 Xz)->False))))) (x2:(forall (Xz:c), ((cX0 Xz)->((and (((eq c) (cL Xz)) Xz)) (((eq c) (cR Xz)) Xz)))))=> ((eq_ref (c->(iS->Prop))) x0)) as proof of ((forall (Xz:c), ((cX0 Xz)->((and (((eq c) (cL Xz)) Xz)) (((eq c) (cR Xz)) Xz))))->(((eq (c->(iS->Prop))) x0) Xg))
% Found (fun (x1:((and ((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (forall (Xz:c), ((iff (cX0 Xz)) ((cX1 Xz)->False))))) (x2:(forall (Xz:c), ((cX0 Xz)->((and (((eq c) (cL Xz)) Xz)) (((eq c) (cR Xz)) Xz)))))=> ((eq_ref (c->(iS->Prop))) x0)) as proof of (((and ((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (forall (Xz:c), ((iff (cX0 Xz)) ((cX1 Xz)->False))))->((forall (Xz:c), ((cX0 Xz)->((and (((eq c) (cL Xz)) Xz)) (((eq c) (cR Xz)) Xz))))->(((eq (c->(iS->Prop))) x0) Xg)))
% Found (and_rect00 (fun (x1:((and ((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (forall (Xz:c), ((iff (cX0 Xz)) ((cX1 Xz)->False))))) (x2:(forall (Xz:c), ((cX0 Xz)->((and (((eq c) (cL Xz)) Xz)) (((eq c) (cR Xz)) Xz)))))=> ((eq_ref (c->(iS->Prop))) x0))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((and_rect0 (((eq (c->(iS->Prop))) x0) Xg)) (fun (x1:((and ((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (forall (Xz:c), ((iff (cX0 Xz)) ((cX1 Xz)->False))))) (x2:(forall (Xz:c), ((cX0 Xz)->((and (((eq c) (cL Xz)) Xz)) (((eq c) (cR Xz)) Xz)))))=> ((eq_ref (c->(iS->Prop))) x0))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (((fun (P:Type) (x1:(((and ((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (forall (Xz:c), ((iff (cX0 Xz)) ((cX1 Xz)->False))))->((forall (Xz:c), ((cX0 Xz)->((and (((eq c) (cL Xz)) Xz)) (((eq c) (cR Xz)) Xz))))->P)))=> (((((and_rect ((and ((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (forall (Xz:c), ((iff (cX0 Xz)) ((cX1 Xz)->False))))) (forall (Xz:c), ((cX0 Xz)->((and (((eq c) (cL Xz)) Xz)) (((eq c) (cR Xz)) Xz))))) P) x1) x)) (((eq (c->(iS->Prop))) x0) Xg)) (fun (x1:((and ((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (forall (Xz:c), ((iff (cX0 Xz)) ((cX1 Xz)->False))))) (x2:(forall (Xz:c), ((cX0 Xz)->((and (((eq c) (cL Xz)) Xz)) (((eq c) (cR Xz)) Xz)))))=> ((eq_ref (c->(iS->Prop))) x0))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))))=> (((fun (P:Type) (x1:(((and ((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (forall (Xz:c), ((iff (cX0 Xz)) ((cX1 Xz)->False))))->((forall (Xz:c), ((cX0 Xz)->((and (((eq c) (cL Xz)) Xz)) (((eq c) (cR Xz)) Xz))))->P)))=> (((((and_rect ((and ((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (forall (Xz:c), ((iff (cX0 Xz)) ((cX1 Xz)->False))))) (forall (Xz:c), ((cX0 Xz)->((and (((eq c) (cL Xz)) Xz)) (((eq c) (cR Xz)) Xz))))) P) x1) x)) (((eq (c->(iS->Prop))) x0) Xg)) (fun (x1:((and ((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (forall (Xz:c), ((iff (cX0 Xz)) ((cX1 Xz)->False))))) (x2:(forall (Xz:c), ((cX0 Xz)->((and (((eq c) (cL Xz)) Xz)) (((eq c) (cR Xz)) Xz)))))=> ((eq_ref (c->(iS->Prop))) x0)))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of ((P x0)->(P Xg))
% Found ((eq_ref0 x0) P) as proof of ((P x0)->(P Xg))
% Found (((eq_ref (c->(iS->Prop))) x0) P) as proof of ((P x0)->(P Xg))
% Found (((eq_ref (c->(iS->Prop))) x0) P) as proof of ((P x0)->(P Xg))
% Found (fun (P:((c->(iS->Prop))->Prop))=> (((eq_ref (c->(iS->Prop))) x0) P)) as proof of ((P x0)->(P Xg))
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))) (P:((c->(iS->Prop))->Prop))=> (((eq_ref (c->(iS->Prop))) x0) P)) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found x5:(P x0)
% Instantiate: x0:=Xg:(c->(iS->Prop))
% Found (fun (x5:(P x0))=> x5) as proof of (P Xg)
% Found (fun (P:((c->(iS->Prop))->Prop)) (x5:(P x0))=> x5) as proof of ((P x0)->(P Xg))
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))) (P:((c->(iS->Prop))->Prop)) (x5:(P x0))=> x5) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% Found (eq_ref00 P) as proof of ((P x2)->(P Xg))
% Found ((eq_ref0 x2) P) as proof of ((P x2)->(P Xg))
% Found (((eq_ref (c->(iS->Prop))) x2) P) as proof of ((P x2)->(P Xg))
% Found (((eq_ref (c->(iS->Prop))) x2) P) as proof of ((P x2)->(P Xg))
% Found (fun (P:((c->(iS->Prop))->Prop))=> (((eq_ref (c->(iS->Prop))) x2) P)) as proof of ((P x2)->(P Xg))
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))) (P:((c->(iS->Prop))->Prop))=> (((eq_ref (c->(iS->Prop))) x2) P)) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found x5:(P x2)
% Instantiate: x2:=Xg:(c->(iS->Prop))
% Found (fun (x5:(P x2))=> x5) as proof of (P Xg)
% Found (fun (P:((c->(iS->Prop))->Prop)) (x5:(P x2))=> x5) as proof of ((P x2)->(P Xg))
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))) (P:((c->(iS->Prop))->Prop)) (x5:(P x2))=> x5) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found eq_ref00:=(eq_ref0 x0):(((eq (c->(iS->Prop))) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((eq_ref (c->(iS->Prop))) x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((eq_ref (c->(iS->Prop))) x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))))=> ((eq_ref (c->(iS->Prop))) x0)) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found eq_ref00:=(eq_ref0 x2):(((eq (c->(iS->Prop))) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found ((eq_ref (c->(iS->Prop))) x2) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found ((eq_ref (c->(iS->Prop))) x2) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))))=> ((eq_ref (c->(iS->Prop))) x2)) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found eq_ref000:=(eq_ref00 P):((P x4)->(P x4))
% Found (eq_ref00 P) as proof of ((P x4)->(P Xg))
% Found ((eq_ref0 x4) P) as proof of ((P x4)->(P Xg))
% Found (((eq_ref (c->(iS->Prop))) x4) P) as proof of ((P x4)->(P Xg))
% Found (((eq_ref (c->(iS->Prop))) x4) P) as proof of ((P x4)->(P Xg))
% Found (fun (P:((c->(iS->Prop))->Prop))=> (((eq_ref (c->(iS->Prop))) x4) P)) as proof of ((P x4)->(P Xg))
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))) (P:((c->(iS->Prop))->Prop))=> (((eq_ref (c->(iS->Prop))) x4) P)) as proof of (((eq (c->(iS->Prop))) x4) Xg)
% Found x5:(P x4)
% Instantiate: x4:=Xg:(c->(iS->Prop))
% Found (fun (x5:(P x4))=> x5) as proof of (P Xg)
% Found (fun (P:((c->(iS->Prop))->Prop)) (x5:(P x4))=> x5) as proof of ((P x4)->(P Xg))
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))) (P:((c->(iS->Prop))->Prop)) (x5:(P x4))=> x5) as proof of (((eq (c->(iS->Prop))) x4) Xg)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq (iS->Prop)) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq (iS->Prop)) (x0 x1)) (Xg x1))
% Found ((eq_ref (iS->Prop)) (x0 x1)) as proof of (((eq (iS->Prop)) (x0 x1)) (Xg x1))
% Found ((eq_ref (iS->Prop)) (x0 x1)) as proof of (((eq (iS->Prop)) (x0 x1)) (Xg x1))
% Found (fun (x1:c)=> ((eq_ref (iS->Prop)) (x0 x1))) as proof of (((eq (iS->Prop)) (x0 x1)) (Xg x1))
% Found (fun (x1:c)=> ((eq_ref (iS->Prop)) (x0 x1))) as proof of (forall (x:c), (((eq (iS->Prop)) (x0 x)) (Xg x)))
% Found (functional_extensionality_dep0000 (fun (x1:c)=> ((eq_ref (iS->Prop)) (x0 x1)))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((functional_extensionality_dep000 Xg) (fun (x1:c)=> ((eq_ref (iS->Prop)) (x0 x1)))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (((functional_extensionality_dep00 x0) Xg) (fun (x1:c)=> ((eq_ref (iS->Prop)) (x0 x1)))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((((functional_extensionality_dep0 (fun (x3:c)=> (iS->Prop))) x0) Xg) (fun (x1:c)=> ((eq_ref (iS->Prop)) (x0 x1)))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (((((functional_extensionality_dep c) (fun (x3:c)=> (iS->Prop))) x0) Xg) (fun (x1:c)=> ((eq_ref (iS->Prop)) (x0 x1)))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))))=> (((((functional_extensionality_dep c) (fun (x3:c)=> (iS->Prop))) x0) Xg) (fun (x1:c)=> ((eq_ref (iS->Prop)) (x0 x1))))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq (iS->Prop)) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq (iS->Prop)) (x0 x1)) (Xg x1))
% Found ((eq_ref (iS->Prop)) (x0 x1)) as proof of (((eq (iS->Prop)) (x0 x1)) (Xg x1))
% Found ((eq_ref (iS->Prop)) (x0 x1)) as proof of (((eq (iS->Prop)) (x0 x1)) (Xg x1))
% Found (fun (x1:c)=> ((eq_ref (iS->Prop)) (x0 x1))) as proof of (((eq (iS->Prop)) (x0 x1)) (Xg x1))
% Found (fun (x1:c)=> ((eq_ref (iS->Prop)) (x0 x1))) as proof of (forall (x:c), (((eq (iS->Prop)) (x0 x)) (Xg x)))
% Found (functional_extensionality0000 (fun (x1:c)=> ((eq_ref (iS->Prop)) (x0 x1)))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((functional_extensionality000 Xg) (fun (x1:c)=> ((eq_ref (iS->Prop)) (x0 x1)))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (((functional_extensionality00 x0) Xg) (fun (x1:c)=> ((eq_ref (iS->Prop)) (x0 x1)))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((((functional_extensionality0 (iS->Prop)) x0) Xg) (fun (x1:c)=> ((eq_ref (iS->Prop)) (x0 x1)))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (((((functional_extensionality c) (iS->Prop)) x0) Xg) (fun (x1:c)=> ((eq_ref (iS->Prop)) (x0 x1)))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))))=> (((((functional_extensionality c) (iS->Prop)) x0) Xg) (fun (x1:c)=> ((eq_ref (iS->Prop)) (x0 x1))))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found eq_ref00:=(eq_ref0 (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or 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((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))):(((eq ((c->(iS->Prop))->Prop)) (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))) (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg))))))
% Found (eq_ref0 (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))) as proof of (((eq ((c->(iS->Prop))->Prop)) (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))) b)
% Found ((eq_ref ((c->(iS->Prop))->Prop)) (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))) as proof of (((eq ((c->(iS->Prop))->Prop)) (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))) b)
% Found ((eq_ref ((c->(iS->Prop))->Prop)) (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))) as proof of (((eq ((c->(iS->Prop))->Prop)) (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))) b)
% Found ((eq_ref ((c->(iS->Prop))->Prop)) (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))) as proof of (((eq ((c->(iS->Prop))->Prop)) (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))) b)
% Found eq_ref00:=(eq_ref0 x0):(((eq (c->(iS->Prop))) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((eq_ref (c->(iS->Prop))) x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((eq_ref (c->(iS->Prop))) x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (fun (x2:(forall (Xz:c), ((cX0 Xz)->((and (((eq c) (cL Xz)) Xz)) (((eq c) (cR Xz)) Xz)))))=> ((eq_ref (c->(iS->Prop))) x0)) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (fun (x1:((and ((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (forall (Xz:c), ((iff (cX0 Xz)) (not (cX1 Xz)))))) (x2:(forall (Xz:c), ((cX0 Xz)->((and (((eq c) (cL Xz)) Xz)) (((eq c) (cR Xz)) Xz)))))=> ((eq_ref (c->(iS->Prop))) x0)) as proof of ((forall (Xz:c), ((cX0 Xz)->((and (((eq c) (cL Xz)) Xz)) (((eq c) (cR Xz)) Xz))))->(((eq (c->(iS->Prop))) x0) Xg))
% Found (fun (x1:((and ((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (forall (Xz:c), ((iff (cX0 Xz)) (not (cX1 Xz)))))) (x2:(forall (Xz:c), ((cX0 Xz)->((and (((eq c) (cL Xz)) Xz)) (((eq c) (cR Xz)) Xz)))))=> ((eq_ref (c->(iS->Prop))) x0)) as proof of (((and ((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (forall (Xz:c), ((iff (cX0 Xz)) (not (cX1 Xz)))))->((forall (Xz:c), ((cX0 Xz)->((and (((eq c) (cL Xz)) Xz)) (((eq c) (cR Xz)) Xz))))->(((eq (c->(iS->Prop))) x0) Xg)))
% Found (and_rect00 (fun (x1:((and ((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (forall (Xz:c), ((iff (cX0 Xz)) (not (cX1 Xz)))))) (x2:(forall (Xz:c), ((cX0 Xz)->((and (((eq c) (cL Xz)) Xz)) (((eq c) (cR Xz)) Xz)))))=> ((eq_ref (c->(iS->Prop))) x0))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((and_rect0 (((eq (c->(iS->Prop))) x0) Xg)) (fun (x1:((and ((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (forall (Xz:c), ((iff (cX0 Xz)) (not (cX1 Xz)))))) (x2:(forall (Xz:c), ((cX0 Xz)->((and (((eq c) (cL Xz)) Xz)) (((eq c) (cR Xz)) Xz)))))=> ((eq_ref (c->(iS->Prop))) x0))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (((fun (P:Type) (x1:(((and ((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (forall (Xz:c), ((iff (cX0 Xz)) (not (cX1 Xz)))))->((forall (Xz:c), ((cX0 Xz)->((and (((eq c) (cL Xz)) Xz)) (((eq c) (cR Xz)) Xz))))->P)))=> (((((and_rect ((and ((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (forall (Xz:c), ((iff (cX0 Xz)) (not (cX1 Xz)))))) (forall (Xz:c), ((cX0 Xz)->((and (((eq c) (cL Xz)) Xz)) (((eq c) (cR Xz)) Xz))))) P) x1) x)) (((eq (c->(iS->Prop))) x0) Xg)) (fun (x1:((and ((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (forall (Xz:c), ((iff (cX0 Xz)) (not (cX1 Xz)))))) (x2:(forall (Xz:c), ((cX0 Xz)->((and (((eq c) (cL Xz)) Xz)) (((eq c) (cR Xz)) Xz)))))=> ((eq_ref (c->(iS->Prop))) x0))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))))=> (((fun (P:Type) (x1:(((and ((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (forall (Xz:c), ((iff (cX0 Xz)) (not (cX1 Xz)))))->((forall (Xz:c), ((cX0 Xz)->((and (((eq c) (cL Xz)) Xz)) (((eq c) (cR Xz)) Xz))))->P)))=> (((((and_rect ((and ((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (forall (Xz:c), ((iff (cX0 Xz)) (not (cX1 Xz)))))) (forall (Xz:c), ((cX0 Xz)->((and (((eq c) (cL Xz)) Xz)) (((eq c) (cR Xz)) Xz))))) P) x1) x)) (((eq (c->(iS->Prop))) x0) Xg)) (fun (x1:((and ((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (forall (Xz:c), ((iff (cX0 Xz)) (not (cX1 Xz)))))) (x2:(forall (Xz:c), ((cX0 Xz)->((and (((eq c) (cL Xz)) Xz)) (((eq c) (cR Xz)) Xz)))))=> ((eq_ref (c->(iS->Prop))) x0)))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found eq_ref00:=(eq_ref0 x2):(((eq (c->(iS->Prop))) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found ((eq_ref (c->(iS->Prop))) x2) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found ((eq_ref (c->(iS->Prop))) x2) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found (fun (x4:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x2)) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found (fun (x3:((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (x4:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x2)) as proof of ((forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))->(((eq (c->(iS->Prop))) x2) Xg))
% Found (fun (x3:((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (x4:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x2)) as proof of (((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))->((forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))->(((eq (c->(iS->Prop))) x2) Xg)))
% Found (and_rect10 (fun (x3:((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (x4:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x2))) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found ((and_rect1 (((eq (c->(iS->Prop))) x2) Xg)) (fun (x3:((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (x4:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x2))) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found (((fun (P:Type) (x3:(((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))->((forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))->P)))=> (((((and_rect ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))) P) x3) x00)) (((eq (c->(iS->Prop))) x2) Xg)) (fun (x3:((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (x4:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x2))) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))))=> (((fun (P:Type) (x3:(((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))->((forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))->P)))=> (((((and_rect ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))) P) x3) x00)) (((eq (c->(iS->Prop))) x2) Xg)) (fun (x3:((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (x4:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x2)))) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found x5:(P x0)
% Instantiate: x0:=Xg:(c->(iS->Prop))
% Found (fun (x5:(P x0))=> x5) as proof of (P Xg)
% Found (fun (P:((c->(iS->Prop))->Prop)) (x5:(P x0))=> x5) as proof of ((P x0)->(P Xg))
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))) (P:((c->(iS->Prop))->Prop)) (x5:(P x0))=> x5) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of ((P x0)->(P Xg))
% Found ((eq_ref0 x0) P) as proof of ((P x0)->(P Xg))
% Found (((eq_ref (c->(iS->Prop))) x0) P) as proof of ((P x0)->(P Xg))
% Found (((eq_ref (c->(iS->Prop))) x0) P) as proof of ((P x0)->(P Xg))
% Found (fun (P:((c->(iS->Prop))->Prop))=> (((eq_ref (c->(iS->Prop))) x0) P)) as proof of ((P x0)->(P Xg))
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))) (P:((c->(iS->Prop))->Prop))=> (((eq_ref (c->(iS->Prop))) x0) P)) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found x5:(P x2)
% Instantiate: x2:=Xg:(c->(iS->Prop))
% Found (fun (x5:(P x2))=> x5) as proof of (P Xg)
% Found (fun (P:((c->(iS->Prop))->Prop)) (x5:(P x2))=> x5) as proof of ((P x2)->(P Xg))
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))) (P:((c->(iS->Prop))->Prop)) (x5:(P x2))=> x5) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% Found (eq_ref00 P) as proof of ((P x2)->(P Xg))
% Found ((eq_ref0 x2) P) as proof of ((P x2)->(P Xg))
% Found (((eq_ref (c->(iS->Prop))) x2) P) as proof of ((P x2)->(P Xg))
% Found (((eq_ref (c->(iS->Prop))) x2) P) as proof of ((P x2)->(P Xg))
% Found (fun (P:((c->(iS->Prop))->Prop))=> (((eq_ref (c->(iS->Prop))) x2) P)) as proof of ((P x2)->(P Xg))
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))) (P:((c->(iS->Prop))->Prop))=> (((eq_ref (c->(iS->Prop))) x2) P)) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found eta_expansion000:=(eta_expansion00 Xg):(((eq (c->(iS->Prop))) Xg) (fun (x:c)=> (Xg x)))
% Found (eta_expansion00 Xg) as proof of (((eq (c->(iS->Prop))) Xg) x2)
% Found ((eta_expansion0 (iS->Prop)) Xg) as proof of (((eq (c->(iS->Prop))) Xg) x2)
% Found (((eta_expansion c) (iS->Prop)) Xg) as proof of (((eq (c->(iS->Prop))) Xg) x2)
% Found (((eta_expansion c) (iS->Prop)) Xg) as proof of (((eq (c->(iS->Prop))) Xg) x2)
% Found (((eta_expansion c) (iS->Prop)) Xg) as proof of (((eq (c->(iS->Prop))) Xg) x2)
% Found (eq_sym000 (((eta_expansion c) (iS->Prop)) Xg)) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found ((eq_sym00 x2) (((eta_expansion c) (iS->Prop)) Xg)) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found (((eq_sym0 Xg) x2) (((eta_expansion c) (iS->Prop)) Xg)) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found ((((eq_sym (c->(iS->Prop))) Xg) x2) (((eta_expansion c) (iS->Prop)) Xg)) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))))=> ((((eq_sym (c->(iS->Prop))) Xg) x2) (((eta_expansion c) (iS->Prop)) Xg))) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq (iS->Prop)) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq (iS->Prop)) (x0 x1)) (Xg x1))
% Found ((eq_ref (iS->Prop)) (x0 x1)) as proof of (((eq (iS->Prop)) (x0 x1)) (Xg x1))
% Found ((eq_ref (iS->Prop)) (x0 x1)) as proof of (((eq (iS->Prop)) (x0 x1)) (Xg x1))
% Found (fun (x1:c)=> ((eq_ref (iS->Prop)) (x0 x1))) as proof of (((eq (iS->Prop)) (x0 x1)) (Xg x1))
% Found (fun (x1:c)=> ((eq_ref (iS->Prop)) (x0 x1))) as proof of (forall (x:c), (((eq (iS->Prop)) (x0 x)) (Xg x)))
% Found (functional_extensionality0000 (fun (x1:c)=> ((eq_ref (iS->Prop)) (x0 x1)))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((functional_extensionality000 Xg) (fun (x1:c)=> ((eq_ref (iS->Prop)) (x0 x1)))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (((functional_extensionality00 x0) Xg) (fun (x1:c)=> ((eq_ref (iS->Prop)) (x0 x1)))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((((functional_extensionality0 (iS->Prop)) x0) Xg) (fun (x1:c)=> ((eq_ref (iS->Prop)) (x0 x1)))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (((((functional_extensionality c) (iS->Prop)) x0) Xg) (fun (x1:c)=> ((eq_ref (iS->Prop)) (x0 x1)))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))))=> (((((functional_extensionality c) (iS->Prop)) x0) Xg) (fun (x1:c)=> ((eq_ref (iS->Prop)) (x0 x1))))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq (iS->Prop)) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq (iS->Prop)) (x0 x1)) (Xg x1))
% Found ((eq_ref (iS->Prop)) (x0 x1)) as proof of (((eq (iS->Prop)) (x0 x1)) (Xg x1))
% Found ((eq_ref (iS->Prop)) (x0 x1)) as proof of (((eq (iS->Prop)) (x0 x1)) (Xg x1))
% Found (fun (x1:c)=> ((eq_ref (iS->Prop)) (x0 x1))) as proof of (((eq (iS->Prop)) (x0 x1)) (Xg x1))
% Found (fun (x1:c)=> ((eq_ref (iS->Prop)) (x0 x1))) as proof of (forall (x:c), (((eq (iS->Prop)) (x0 x)) (Xg x)))
% Found (functional_extensionality_dep0000 (fun (x1:c)=> ((eq_ref (iS->Prop)) (x0 x1)))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((functional_extensionality_dep000 Xg) (fun (x1:c)=> ((eq_ref (iS->Prop)) (x0 x1)))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (((functional_extensionality_dep00 x0) Xg) (fun (x1:c)=> ((eq_ref (iS->Prop)) (x0 x1)))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((((functional_extensionality_dep0 (fun (x3:c)=> (iS->Prop))) x0) Xg) (fun (x1:c)=> ((eq_ref (iS->Prop)) (x0 x1)))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (((((functional_extensionality_dep c) (fun (x3:c)=> (iS->Prop))) x0) Xg) (fun (x1:c)=> ((eq_ref (iS->Prop)) (x0 x1)))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))))=> (((((functional_extensionality_dep c) (fun (x3:c)=> (iS->Prop))) x0) Xg) (fun (x1:c)=> ((eq_ref (iS->Prop)) (x0 x1))))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found eq_ref00:=(eq_ref0 (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or 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((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))):(((eq ((c->(iS->Prop))->Prop)) (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))) (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg))))))
% Found (eq_ref0 (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))) as proof of (((eq ((c->(iS->Prop))->Prop)) (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))) b)
% Found ((eq_ref ((c->(iS->Prop))->Prop)) (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))) as proof of (((eq ((c->(iS->Prop))->Prop)) (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))) b)
% Found ((eq_ref ((c->(iS->Prop))->Prop)) (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))) as proof of (((eq ((c->(iS->Prop))->Prop)) (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))) b)
% Found ((eq_ref ((c->(iS->Prop))->Prop)) (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))) as proof of (((eq ((c->(iS->Prop))->Prop)) (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))) b)
% Found eq_ref00:=(eq_ref0 x6):(((eq (c->(iS->Prop))) x6) x6)
% Found (eq_ref0 x6) as proof of (((eq (c->(iS->Prop))) x6) Xg)
% Found ((eq_ref (c->(iS->Prop))) x6) as proof of (((eq (c->(iS->Prop))) x6) Xg)
% Found ((eq_ref (c->(iS->Prop))) x6) as proof of (((eq (c->(iS->Prop))) x6) Xg)
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))))=> ((eq_ref (c->(iS->Prop))) x6)) as proof of (((eq (c->(iS->Prop))) x6) Xg)
% Found eq_ref00:=(eq_ref0 x0):(((eq (c->(iS->Prop))) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((eq_ref (c->(iS->Prop))) x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((eq_ref (c->(iS->Prop))) x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (fun (x4:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x0)) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (fun (x3:((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (x4:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x0)) as proof of ((forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))->(((eq (c->(iS->Prop))) x0) Xg))
% Found (fun (x3:((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (x4:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x0)) as proof of (((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))->((forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))->(((eq (c->(iS->Prop))) x0) Xg)))
% Found (and_rect10 (fun (x3:((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (x4:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x0))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((and_rect1 (((eq (c->(iS->Prop))) x0) Xg)) (fun (x3:((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (x4:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x0))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (((fun (P:Type) (x3:(((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))->((forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))->P)))=> (((((and_rect ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))) P) x3) x00)) (((eq (c->(iS->Prop))) x0) Xg)) (fun (x3:((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (x4:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x0))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))))=> (((fun (P:Type) (x3:(((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))->((forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))->P)))=> (((((and_rect ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))) P) x3) x00)) (((eq (c->(iS->Prop))) x0) Xg)) (fun (x3:((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (x4:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x0)))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found eq_ref00:=(eq_ref0 x2):(((eq (c->(iS->Prop))) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found ((eq_ref (c->(iS->Prop))) x2) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found ((eq_ref (c->(iS->Prop))) x2) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found (fun (x4:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x2)) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found (fun (x3:((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (x4:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x2)) as proof of ((forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))->(((eq (c->(iS->Prop))) x2) Xg))
% Found (fun (x3:((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (x4:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x2)) as proof of (((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))->((forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))->(((eq (c->(iS->Prop))) x2) Xg)))
% Found (and_rect10 (fun (x3:((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (x4:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x2))) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found ((and_rect1 (((eq (c->(iS->Prop))) x2) Xg)) (fun (x3:((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (x4:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x2))) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found (((fun (P:Type) (x3:(((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))->((forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))->P)))=> (((((and_rect ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))) P) x3) x00)) (((eq (c->(iS->Prop))) x2) Xg)) (fun (x3:((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (x4:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x2))) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))))=> (((fun (P:Type) (x3:(((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))->((forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))->P)))=> (((((and_rect ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))) P) x3) x00)) (((eq (c->(iS->Prop))) x2) Xg)) (fun (x3:((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (x4:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x2)))) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found eta_expansion000:=(eta_expansion00 Xg):(((eq (c->(iS->Prop))) Xg) (fun (x:c)=> (Xg x)))
% Found (eta_expansion00 Xg) as proof of (((eq (c->(iS->Prop))) Xg) x0)
% Found ((eta_expansion0 (iS->Prop)) Xg) as proof of (((eq (c->(iS->Prop))) Xg) x0)
% Found (((eta_expansion c) (iS->Prop)) Xg) as proof of (((eq (c->(iS->Prop))) Xg) x0)
% Found (((eta_expansion c) (iS->Prop)) Xg) as proof of (((eq (c->(iS->Prop))) Xg) x0)
% Found (((eta_expansion c) (iS->Prop)) Xg) as proof of (((eq (c->(iS->Prop))) Xg) x0)
% Found (eq_sym000 (((eta_expansion c) (iS->Prop)) Xg)) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((eq_sym00 x0) (((eta_expansion c) (iS->Prop)) Xg)) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (((eq_sym0 Xg) x0) (((eta_expansion c) (iS->Prop)) Xg)) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((((eq_sym (c->(iS->Prop))) Xg) x0) (((eta_expansion c) (iS->Prop)) Xg)) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))))=> ((((eq_sym (c->(iS->Prop))) Xg) x0) (((eta_expansion c) (iS->Prop)) Xg))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found eq_ref00:=(eq_ref0 ((x0 x1) y)):(((eq Prop) ((x0 x1) y)) ((x0 x1) y))
% Found (eq_ref0 ((x0 x1) y)) as proof of (((eq Prop) ((x0 x1) y)) ((Xg x1) y))
% Found ((eq_ref Prop) ((x0 x1) y)) as proof of (((eq Prop) ((x0 x1) y)) ((Xg x1) y))
% Found ((eq_ref Prop) ((x0 x1) y)) as proof of (((eq Prop) ((x0 x1) y)) ((Xg x1) y))
% Found (fun (y:iS)=> ((eq_ref Prop) ((x0 x1) y))) as proof of (((eq Prop) ((x0 x1) y)) ((Xg x1) y))
% Found (fun (x1:c) (y:iS)=> ((eq_ref Prop) ((x0 x1) y))) as proof of (forall (y:iS), (((eq Prop) ((x0 x1) y)) ((Xg x1) y)))
% Found (fun (x1:c) (y:iS)=> ((eq_ref Prop) ((x0 x1) y))) as proof of (forall (x:c) (y:iS), (((eq Prop) ((x0 x) y)) ((Xg x) y)))
% Found (functional_extensionality_double00000 (fun (x1:c) (y:iS)=> ((eq_ref Prop) ((x0 x1) y)))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((functional_extensionality_double0000 Xg) (fun (x1:c) (y:iS)=> ((eq_ref Prop) ((x0 x1) y)))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (((functional_extensionality_double000 x0) Xg) (fun (x1:c) (y:iS)=> ((eq_ref Prop) ((x0 x1) y)))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((((functional_extensionality_double00 Prop) x0) Xg) (fun (x1:c) (y:iS)=> ((eq_ref Prop) ((x0 x1) y)))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (((((functional_extensionality_double0 iS) Prop) x0) Xg) (fun (x1:c) (y:iS)=> ((eq_ref Prop) ((x0 x1) y)))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((((((functional_extensionality_double c) iS) Prop) x0) Xg) (fun (x1:c) (y:iS)=> ((eq_ref Prop) ((x0 x1) y)))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))))=> ((((((functional_extensionality_double c) iS) Prop) x0) Xg) (fun (x1:c) (y:iS)=> ((eq_ref Prop) ((x0 x1) y))))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found eta_expansion000:=(eta_expansion00 (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) 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(iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) 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Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) 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(Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))) b)
% Found (((eta_expansion (c->(iS->Prop))) Prop) (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))) as proof of (((eq ((c->(iS->Prop))->Prop)) (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))) b)
% Found (((eta_expansion (c->(iS->Prop))) Prop) (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))) as proof of (((eq ((c->(iS->Prop))->Prop)) (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))) b)
% Found (((eta_expansion (c->(iS->Prop))) Prop) (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))) as proof of (((eq ((c->(iS->Prop))->Prop)) (fun (Xf:(c->(iS->Prop)))=> ((and ((and ((and (forall (Xb:c), ((and ((and ((Xf Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xf Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xf Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xf Xb) Xx)) ((Xf Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xf Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xf Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xf Xb) ((cP Xx) Xy))))))) (Xf (cR Xb))))))) (forall (Xg:(c->(iS->Prop))), (((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))->(((eq (c->(iS->Prop))) Xf) Xg)))))) b)
% Found eq_ref00:=(eq_ref0 Xg):(((eq (c->(iS->Prop))) Xg) Xg)
% Found (eq_ref0 Xg) as proof of (((eq (c->(iS->Prop))) Xg) x2)
% Found ((eq_ref (c->(iS->Prop))) Xg) as proof of (((eq (c->(iS->Prop))) Xg) x2)
% Found ((eq_ref (c->(iS->Prop))) Xg) as proof of (((eq (c->(iS->Prop))) Xg) x2)
% Found ((eq_ref (c->(iS->Prop))) Xg) as proof of (((eq (c->(iS->Prop))) Xg) x2)
% Found (eq_sym000 ((eq_ref (c->(iS->Prop))) Xg)) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found ((eq_sym00 x2) ((eq_ref (c->(iS->Prop))) Xg)) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found (((eq_sym0 Xg) x2) ((eq_ref (c->(iS->Prop))) Xg)) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found ((((eq_sym (c->(iS->Prop))) Xg) x2) ((eq_ref (c->(iS->Prop))) Xg)) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))))=> ((((eq_sym (c->(iS->Prop))) Xg) x2) ((eq_ref (c->(iS->Prop))) Xg))) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found eq_ref00:=(eq_ref0 x0):(((eq (c->(iS->Prop))) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((eq_ref (c->(iS->Prop))) x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((eq_ref (c->(iS->Prop))) x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))))=> ((eq_ref (c->(iS->Prop))) x0)) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found eq_ref00:=(eq_ref0 x2):(((eq (c->(iS->Prop))) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found ((eq_ref (c->(iS->Prop))) x2) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found ((eq_ref (c->(iS->Prop))) x2) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))))=> ((eq_ref (c->(iS->Prop))) x2)) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found eq_ref00:=(eq_ref0 x4):(((eq (c->(iS->Prop))) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq (c->(iS->Prop))) x4) Xg)
% Found ((eq_ref (c->(iS->Prop))) x4) as proof of (((eq (c->(iS->Prop))) x4) Xg)
% Found ((eq_ref (c->(iS->Prop))) x4) as proof of (((eq (c->(iS->Prop))) x4) Xg)
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))))=> ((eq_ref (c->(iS->Prop))) x4)) as proof of (((eq (c->(iS->Prop))) x4) Xg)
% Found eq_ref000:=(eq_ref00 P):((P x6)->(P x6))
% Found (eq_ref00 P) as proof of ((P x6)->(P Xg))
% Found ((eq_ref0 x6) P) as proof of ((P x6)->(P Xg))
% Found (((eq_ref (c->(iS->Prop))) x6) P) as proof of ((P x6)->(P Xg))
% Found (((eq_ref (c->(iS->Prop))) x6) P) as proof of ((P x6)->(P Xg))
% Found (fun (P:((c->(iS->Prop))->Prop))=> (((eq_ref (c->(iS->Prop))) x6) P)) as proof of ((P x6)->(P Xg))
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))) (P:((c->(iS->Prop))->Prop))=> (((eq_ref (c->(iS->Prop))) x6) P)) as proof of (((eq (c->(iS->Prop))) x6) Xg)
% Found x7:(P x6)
% Instantiate: x6:=Xg:(c->(iS->Prop))
% Found (fun (x7:(P x6))=> x7) as proof of (P Xg)
% Found (fun (P:((c->(iS->Prop))->Prop)) (x7:(P x6))=> x7) as proof of ((P x6)->(P Xg))
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))) (P:((c->(iS->Prop))->Prop)) (x7:(P x6))=> x7) as proof of (((eq (c->(iS->Prop))) x6) Xg)
% Found eq_ref00:=(eq_ref0 x0):(((eq (c->(iS->Prop))) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((eq_ref (c->(iS->Prop))) x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((eq_ref (c->(iS->Prop))) x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (fun (x4:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x0)) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (fun (x3:((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (x4:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x0)) as proof of ((forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))->(((eq (c->(iS->Prop))) x0) Xg))
% Found (fun (x3:((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (x4:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x0)) as proof of (((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))->((forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))->(((eq (c->(iS->Prop))) x0) Xg)))
% Found (and_rect10 (fun (x3:((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (x4:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x0))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((and_rect1 (((eq (c->(iS->Prop))) x0) Xg)) (fun (x3:((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (x4:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x0))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (((fun (P:Type) (x3:(((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))->((forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))->P)))=> (((((and_rect ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))) P) x3) x00)) (((eq (c->(iS->Prop))) x0) Xg)) (fun (x3:((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (x4:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x0))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))))=> (((fun (P:Type) (x3:(((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))->((forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))->P)))=> (((((and_rect ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))) P) x3) x00)) (((eq (c->(iS->Prop))) x0) Xg)) (fun (x3:((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (x4:(forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb))))))=> ((eq_ref (c->(iS->Prop))) x0)))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found eq_ref00:=(eq_ref0 x0):(((eq (c->(iS->Prop))) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((eq_ref (c->(iS->Prop))) x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((eq_ref (c->(iS->Prop))) x0) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (fun (x4:(forall (Xz:c), ((iff (cX0 Xz)) ((cX1 Xz)->False))))=> ((eq_ref (c->(iS->Prop))) x0)) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (fun (x3:((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (x4:(forall (Xz:c), ((iff (cX0 Xz)) ((cX1 Xz)->False))))=> ((eq_ref (c->(iS->Prop))) x0)) as proof of ((forall (Xz:c), ((iff (cX0 Xz)) ((cX1 Xz)->False)))->(((eq (c->(iS->Prop))) x0) Xg))
% Found (fun (x3:((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (x4:(forall (Xz:c), ((iff (cX0 Xz)) ((cX1 Xz)->False))))=> ((eq_ref (c->(iS->Prop))) x0)) as proof of (((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))->((forall (Xz:c), ((iff (cX0 Xz)) ((cX1 Xz)->False)))->(((eq (c->(iS->Prop))) x0) Xg)))
% Found (and_rect10 (fun (x3:((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (x4:(forall (Xz:c), ((iff (cX0 Xz)) ((cX1 Xz)->False))))=> ((eq_ref (c->(iS->Prop))) x0))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found ((and_rect1 (((eq (c->(iS->Prop))) x0) Xg)) (fun (x3:((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (x4:(forall (Xz:c), ((iff (cX0 Xz)) ((cX1 Xz)->False))))=> ((eq_ref (c->(iS->Prop))) x0))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (((fun (P:Type) (x3:(((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))->((forall (Xz:c), ((iff (cX0 Xz)) ((cX1 Xz)->False)))->P)))=> (((((and_rect ((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (forall (Xz:c), ((iff (cX0 Xz)) ((cX1 Xz)->False)))) P) x3) x1)) (((eq (c->(iS->Prop))) x0) Xg)) (fun (x3:((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (x4:(forall (Xz:c), ((iff (cX0 Xz)) ((cX1 Xz)->False))))=> ((eq_ref (c->(iS->Prop))) x0))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))))=> (((fun (P:Type) (x3:(((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))->((forall (Xz:c), ((iff (cX0 Xz)) ((cX1 Xz)->False)))->P)))=> (((((and_rect ((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (forall (Xz:c), ((iff (cX0 Xz)) ((cX1 Xz)->False)))) P) x3) x1)) (((eq (c->(iS->Prop))) x0) Xg)) (fun (x3:((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (x4:(forall (Xz:c), ((iff (cX0 Xz)) ((cX1 Xz)->False))))=> ((eq_ref (c->(iS->Prop))) x0)))) as proof of (((eq (c->(iS->Prop))) x0) Xg)
% Found eq_ref00:=(eq_ref0 x6):(((eq (c->(iS->Prop))) x6) x6)
% Found (eq_ref0 x6) as proof of (((eq (c->(iS->Prop))) x6) Xg)
% Found ((eq_ref (c->(iS->Prop))) x6) as proof of (((eq (c->(iS->Prop))) x6) Xg)
% Found ((eq_ref (c->(iS->Prop))) x6) as proof of (((eq (c->(iS->Prop))) x6) Xg)
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (iS->Prop)) (fun (Xx:iS)=> ((or (((eq iS) Xx) c0)) ((ex iS) (fun (Xy:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cL Xb)))) (((eq (iS->Prop)) (fun (Xy:iS)=> ((or (((eq iS) Xy) c0)) ((ex iS) (fun (Xx:iS)=> ((Xg Xb) ((cP Xx) Xy))))))) (Xg (cR Xb)))))))=> ((eq_ref (c->(iS->Prop))) x6)) as proof of (((eq (c->(iS->Prop))) x6) Xg)
% Found eq_ref00:=(eq_ref0 x2):(((eq (c->(iS->Prop))) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found ((eq_ref (c->(iS->Prop))) x2) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found ((eq_ref (c->(iS->Prop))) x2) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found (fun (x4:(forall (Xz:c), ((iff (cX0 Xz)) ((cX1 Xz)->False))))=> ((eq_ref (c->(iS->Prop))) x2)) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found (fun (x3:((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (x4:(forall (Xz:c), ((iff (cX0 Xz)) ((cX1 Xz)->False))))=> ((eq_ref (c->(iS->Prop))) x2)) as proof of ((forall (Xz:c), ((iff (cX0 Xz)) ((cX1 Xz)->False)))->(((eq (c->(iS->Prop))) x2) Xg))
% Found (fun (x3:((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (x4:(forall (Xz:c), ((iff (cX0 Xz)) ((cX1 Xz)->False))))=> ((eq_ref (c->(iS->Prop))) x2)) as proof of (((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))->((forall (Xz:c), ((iff (cX0 Xz)) ((cX1 Xz)->False)))->(((eq (c->(iS->Prop))) x2) Xg)))
% Found (and_rect10 (fun (x3:((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (x4:(forall (Xz:c), ((iff (cX0 Xz)) ((cX1 Xz)->False))))=> ((eq_ref (c->(iS->Prop))) x2))) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found ((and_rect1 (((eq (c->(iS->Prop))) x2) Xg)) (fun (x3:((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (x4:(forall (Xz:c), ((iff (cX0 Xz)) ((cX1 Xz)->False))))=> ((eq_ref (c->(iS->Prop))) x2))) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found (((fun (P:Type) (x3:(((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))->((forall (Xz:c), ((iff (cX0 Xz)) ((cX1 Xz)->False)))->P)))=> (((((and_rect ((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (forall (Xz:c), ((iff (cX0 Xz)) ((cX1 Xz)->False)))) P) x3) x0)) (((eq (c->(iS->Prop))) x2) Xg)) (fun (x3:((and ((and (forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (forall (Xx:iS) (Xy:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx) Xu)) ((cP Xy) Xv))->((and (((eq iS) Xx) Xy)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx:iS) (Xy:iS), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(forall (Xx:iS), (X Xx)))))) (x4:(forall (Xz:c), ((iff (cX0 Xz)) ((cX1 Xz)->False))))=> ((eq_ref (c->(iS->Prop))) x2))) as proof of (((eq (c->(iS->Prop))) x2) Xg)
% Found (fun (x00:((and ((and (forall (Xb:c), ((and ((and ((Xg Xb) c0)) (forall (Xx:iS) (Xy:iS), (((and ((Xg Xb) Xy)) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xy))))->((Xg Xb) Xx))))) (forall (Xx:iS) (Xy:iS) (Xz:iS), (((and ((and ((Xg Xb) Xx)) ((Xg Xb) Xy))) (forall (R0:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb0:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb0) Xc))) ((and (((eq iS) Xb0) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb0) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R0 Xx1) Xy1) Xz1))) (((R0 Xx2) Xy2) Xz2)))))))))))))))->(((R0 Xa) Xb0) Xc))))->(((R0 Xx) Xy) Xz))))->((Xg Xb) Xz)))))) (forall (Xc:c), ((iff (cX0 Xc)) (((eq (iS->Prop)) (Xg Xc)) (fun (Xy:iS)=> (((eq iS) c0) Xy))))))) (forall (Xb:c), ((and (((eq (i
% EOF
%------------------------------------------------------------------------------