%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV205^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n190.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 13:33:52 EDT 2014

% Result   : Timeout 300.02s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV205^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n190.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 08:28:31 CDT 2014
% % CPUTime  : 300.02 
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0xa15680>, <kernel.Type object at 0xa15170>) of role type named b_type
% Using role type
% Declaring b:Type
% FOF formula (<kernel.Constant object at 0x9f63b0>, <kernel.Type object at 0xa15ab8>) of role type named iS_type
% Using role type
% Declaring iS:Type
% FOF formula (<kernel.Constant object at 0xa15638>, <kernel.DependentProduct object at 0xa15830>) of role type named cP2
% Using role type
% Declaring cP2:(b->(b->b))
% FOF formula (<kernel.Constant object at 0xa15170>, <kernel.DependentProduct object at 0xa150e0>) of role type named cP
% Using role type
% Declaring cP:(iS->(iS->iS))
% FOF formula (<kernel.Constant object at 0xa15680>, <kernel.Constant object at 0xa150e0>) of role type named c02
% Using role type
% Declaring c02:b
% FOF formula (<kernel.Constant object at 0xa15638>, <kernel.Constant object at 0xa150e0>) of role type named c0
% Using role type
% Declaring c0:iS
% 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))) (forall (Xx:b) (Xy:b), (not (((eq b) ((cP2 Xx) Xy)) c02))))->((ex (iS->b)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg))))))) of role conjecture named cTHM_S_INIT_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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))) (forall (Xx:b) (Xy:b), (not (((eq b) ((cP2 Xx) Xy)) c02))))->((ex (iS->b)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg))))))):Prop
% 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))) (forall (Xx:b) (Xy:b), (not (((eq b) ((cP2 Xx) Xy)) c02))))->((ex (iS->b)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))))']
% Parameter b:Type.
% Parameter iS:Type.
% Parameter cP2:(b->(b->b)).
% Parameter cP:(iS->(iS->iS)).
% Parameter c02:b.
% Parameter c0:iS.
% 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))) (forall (Xx:b) (Xy:b), (not (((eq b) ((cP2 Xx) Xy)) c02))))->((ex (iS->b)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))))
% Found eq_ref00:=(eq_ref0 x0):(((eq (iS->b)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (iS->b)) x0) Xg)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) Xg)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((eq_ref (iS->b)) x0)) as proof of (((eq (iS->b)) x0) Xg)
% Found x1:(P x0)
% Instantiate: x0:=Xg:(iS->b)
% Found (fun (x1:(P x0))=> x1) as proof of (P Xg)
% Found (fun (P:((iS->b)->Prop)) (x1:(P x0))=> x1) as proof of ((P x0)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop)) (x1:(P x0))=> x1) as proof of (((eq (iS->b)) 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 (iS->b)) x0) P) as proof of ((P x0)->(P Xg))
% Found (((eq_ref (iS->b)) x0) P) as proof of ((P x0)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x0) P)) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 x0):(((eq (iS->b)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (iS->b)) x0) x')
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) x')
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((eq_ref (iS->b)) x0)) as proof of (((eq (iS->b)) x0) x')
% Found x1:(P x0)
% Instantiate: x0:=x':(iS->b)
% Found (fun (x1:(P x0))=> x1) as proof of (P x')
% Found (fun (P:((iS->b)->Prop)) (x1:(P x0))=> x1) as proof of ((P x0)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop)) (x1:(P x0))=> x1) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of ((P x0)->(P x'))
% Found ((eq_ref0 x0) P) as proof of ((P x0)->(P x'))
% Found (((eq_ref (iS->b)) x0) P) as proof of ((P x0)->(P x'))
% Found (((eq_ref (iS->b)) x0) P) as proof of ((P x0)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x0) P)) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 x2):(((eq (iS->b)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (iS->b)) x2) Xg)
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) Xg)
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((eq_ref (iS->b)) x2)) as proof of (((eq (iS->b)) x2) Xg)
% Found eq_ref00:=(eq_ref0 x0):(((eq (iS->b)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (iS->b)) x0) Xg)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) Xg)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((eq_ref (iS->b)) x0)) as proof of (((eq (iS->b)) 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 (iS->b)) x2) P) as proof of ((P x2)->(P Xg))
% Found (((eq_ref (iS->b)) x2) P) as proof of ((P x2)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x2) P)) as proof of (((eq (iS->b)) x2) Xg)
% Found x3:(P x2)
% Instantiate: x2:=Xg:(iS->b)
% Found (fun (x3:(P x2))=> x3) as proof of (P Xg)
% Found (fun (P:((iS->b)->Prop)) (x3:(P x2))=> x3) as proof of ((P x2)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop)) (x3:(P x2))=> x3) as proof of (((eq (iS->b)) x2) Xg)
% Found eq_ref00:=(eq_ref0 x2):(((eq (iS->b)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (iS->b)) x2) x')
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) x')
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((eq_ref (iS->b)) x2)) as proof of (((eq (iS->b)) x2) x')
% Found x3:(P x0)
% Instantiate: x0:=Xg:(iS->b)
% Found (fun (x3:(P x0))=> x3) as proof of (P Xg)
% Found (fun (P:((iS->b)->Prop)) (x3:(P x0))=> x3) as proof of ((P x0)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop)) (x3:(P x0))=> x3) as proof of (((eq (iS->b)) 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 (iS->b)) x0) P) as proof of ((P x0)->(P Xg))
% Found (((eq_ref (iS->b)) x0) P) as proof of ((P x0)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x0) P)) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 x0):(((eq (iS->b)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (iS->b)) x0) x')
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) x')
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((eq_ref (iS->b)) x0)) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% Found (eq_ref00 P) as proof of ((P x2)->(P x'))
% Found ((eq_ref0 x2) P) as proof of ((P x2)->(P x'))
% Found (((eq_ref (iS->b)) x2) P) as proof of ((P x2)->(P x'))
% Found (((eq_ref (iS->b)) x2) P) as proof of ((P x2)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x2) P)) as proof of (((eq (iS->b)) x2) x')
% Found x3:(P x2)
% Instantiate: x2:=x':(iS->b)
% Found (fun (x3:(P x2))=> x3) as proof of (P x')
% Found (fun (P:((iS->b)->Prop)) (x3:(P x2))=> x3) as proof of ((P x2)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop)) (x3:(P x2))=> x3) as proof of (((eq (iS->b)) x2) x')
% Found eq_ref00:=(eq_ref0 (x0 c0)):(((eq b) (x0 c0)) (x0 c0))
% Found (eq_ref0 (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of ((P x0)->(P x'))
% Found ((eq_ref0 x0) P) as proof of ((P x0)->(P x'))
% Found (((eq_ref (iS->b)) x0) P) as proof of ((P x0)->(P x'))
% Found (((eq_ref (iS->b)) x0) P) as proof of ((P x0)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x0) P)) as proof of (((eq (iS->b)) x0) x')
% Found x3:(P x0)
% Instantiate: x0:=x':(iS->b)
% Found (fun (x3:(P x0))=> x3) as proof of (P x')
% Found (fun (P:((iS->b)->Prop)) (x3:(P x0))=> x3) as proof of ((P x0)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop)) (x3:(P x0))=> x3) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 x0):(((eq (iS->b)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (iS->b)) x0) Xg)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) Xg)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x2:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x0)) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x1:(((eq b) (Xg c0)) c02)) (x2:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x0)) as proof of ((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->(((eq (iS->b)) x0) Xg))
% Found (fun (x1:(((eq b) (Xg c0)) c02)) (x2:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x0)) as proof of ((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->(((eq (iS->b)) x0) Xg)))
% Found (and_rect00 (fun (x1:(((eq b) (Xg c0)) c02)) (x2:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) Xg)
% Found ((and_rect0 (((eq (iS->b)) x0) Xg)) (fun (x1:(((eq b) (Xg c0)) c02)) (x2:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) Xg)
% Found (((fun (P:Type) (x1:((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->P)))=> (((((and_rect (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))) P) x1) x00)) (((eq (iS->b)) x0) Xg)) (fun (x1:(((eq b) (Xg c0)) c02)) (x2:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((fun (P:Type) (x1:((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->P)))=> (((((and_rect (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))) P) x1) x00)) (((eq (iS->b)) x0) Xg)) (fun (x1:(((eq b) (Xg c0)) c02)) (x2:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x0)))) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 x4):(((eq (iS->b)) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq (iS->b)) x4) Xg)
% Found ((eq_ref (iS->b)) x4) as proof of (((eq (iS->b)) x4) Xg)
% Found ((eq_ref (iS->b)) x4) as proof of (((eq (iS->b)) x4) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((eq_ref (iS->b)) x4)) as proof of (((eq (iS->b)) x4) Xg)
% Found eq_ref00:=(eq_ref0 (x0 c0)):(((eq b) (x0 c0)) (x0 c0))
% Found (eq_ref0 (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found eq_ref00:=(eq_ref0 Xg):(((eq (iS->b)) Xg) Xg)
% Found (eq_ref0 Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found ((eq_ref (iS->b)) Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found ((eq_ref (iS->b)) Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found ((eq_ref (iS->b)) Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found (eq_sym000 ((eq_ref (iS->b)) Xg)) as proof of (((eq (iS->b)) x0) Xg)
% Found ((eq_sym00 x0) ((eq_ref (iS->b)) Xg)) as proof of (((eq (iS->b)) x0) Xg)
% Found (((eq_sym0 Xg) x0) ((eq_ref (iS->b)) Xg)) as proof of (((eq (iS->b)) x0) Xg)
% Found ((((eq_sym (iS->b)) Xg) x0) ((eq_ref (iS->b)) Xg)) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((((eq_sym (iS->b)) Xg) x0) ((eq_ref (iS->b)) Xg))) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 x0):(((eq (iS->b)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (iS->b)) x0) Xg)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) Xg)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((eq_ref (iS->b)) x0)) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 x2):(((eq (iS->b)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (iS->b)) x2) Xg)
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) Xg)
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((eq_ref (iS->b)) x2)) as proof of (((eq (iS->b)) x2) Xg)
% Found x5:(P x4)
% Instantiate: x4:=Xg:(iS->b)
% Found (fun (x5:(P x4))=> x5) as proof of (P Xg)
% Found (fun (P:((iS->b)->Prop)) (x5:(P x4))=> x5) as proof of ((P x4)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop)) (x5:(P x4))=> x5) as proof of (((eq (iS->b)) x4) 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 (iS->b)) x4) P) as proof of ((P x4)->(P Xg))
% Found (((eq_ref (iS->b)) x4) P) as proof of ((P x4)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x4) P)) as proof of ((P x4)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x4) P)) as proof of (((eq (iS->b)) x4) Xg)
% Found eq_ref00:=(eq_ref0 x0):(((eq (iS->b)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (iS->b)) x0) x')
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) x')
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x2:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x0)) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x1:(((eq b) (x' c0)) c02)) (x2:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x0)) as proof of ((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->(((eq (iS->b)) x0) x'))
% Found (fun (x1:(((eq b) (x' c0)) c02)) (x2:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x0)) as proof of ((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->(((eq (iS->b)) x0) x')))
% Found (and_rect00 (fun (x1:(((eq b) (x' c0)) c02)) (x2:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) x')
% Found ((and_rect0 (((eq (iS->b)) x0) x')) (fun (x1:(((eq b) (x' c0)) c02)) (x2:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) x')
% Found (((fun (P:Type) (x1:((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->P)))=> (((((and_rect (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))) P) x1) x00)) (((eq (iS->b)) x0) x')) (fun (x1:(((eq b) (x' c0)) c02)) (x2:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((fun (P:Type) (x1:((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->P)))=> (((((and_rect (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))) P) x1) x00)) (((eq (iS->b)) x0) x')) (fun (x1:(((eq b) (x' c0)) c02)) (x2:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x0)))) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 x4):(((eq (iS->b)) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq (iS->b)) x4) x')
% Found ((eq_ref (iS->b)) x4) as proof of (((eq (iS->b)) x4) x')
% Found ((eq_ref (iS->b)) x4) as proof of (((eq (iS->b)) x4) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((eq_ref (iS->b)) x4)) as proof of (((eq (iS->b)) x4) x')
% Found eq_ref00:=(eq_ref0 (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))):(((eq ((iS->b)->Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg))))))
% Found (eq_ref0 (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) as proof of (((eq ((iS->b)->Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) b0)
% Found ((eq_ref ((iS->b)->Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) as proof of (((eq ((iS->b)->Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) b0)
% Found ((eq_ref ((iS->b)->Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) as proof of (((eq ((iS->b)->Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) b0)
% Found ((eq_ref ((iS->b)->Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) as proof of (((eq ((iS->b)->Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) b0)
% Found eq_ref00:=(eq_ref0 x'):(((eq (iS->b)) x') x')
% Found (eq_ref0 x') as proof of (((eq (iS->b)) x') x0)
% Found ((eq_ref (iS->b)) x') as proof of (((eq (iS->b)) x') x0)
% Found ((eq_ref (iS->b)) x') as proof of (((eq (iS->b)) x') x0)
% Found ((eq_ref (iS->b)) x') as proof of (((eq (iS->b)) x') x0)
% Found (eq_sym000 ((eq_ref (iS->b)) x')) as proof of (((eq (iS->b)) x0) x')
% Found ((eq_sym00 x0) ((eq_ref (iS->b)) x')) as proof of (((eq (iS->b)) x0) x')
% Found (((eq_sym0 x') x0) ((eq_ref (iS->b)) x')) as proof of (((eq (iS->b)) x0) x')
% Found ((((eq_sym (iS->b)) x') x0) ((eq_ref (iS->b)) x')) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((((eq_sym (iS->b)) x') x0) ((eq_ref (iS->b)) x'))) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 (x2 c0)):(((eq b) (x2 c0)) (x2 c0))
% Found (eq_ref0 (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found eq_ref00:=(eq_ref0 x0):(((eq (iS->b)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (iS->b)) x0) Xg)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) Xg)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x2:(forall (Xx:b) (Xy:b), (not (((eq b) ((cP2 Xx) Xy)) c02))))=> ((eq_ref (iS->b)) x0)) as proof of (((eq (iS->b)) 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))) (x2:(forall (Xx:b) (Xy:b), (not (((eq b) ((cP2 Xx) Xy)) c02))))=> ((eq_ref (iS->b)) x0)) as proof of ((forall (Xx:b) (Xy:b), (not (((eq b) ((cP2 Xx) Xy)) c02)))->(((eq (iS->b)) 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))) (x2:(forall (Xx:b) (Xy:b), (not (((eq b) ((cP2 Xx) Xy)) c02))))=> ((eq_ref (iS->b)) 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))->((forall (Xx:b) (Xy:b), (not (((eq b) ((cP2 Xx) Xy)) c02)))->(((eq (iS->b)) 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))) (x2:(forall (Xx:b) (Xy:b), (not (((eq b) ((cP2 Xx) Xy)) c02))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) Xg)
% Found ((and_rect0 (((eq (iS->b)) 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))) (x2:(forall (Xx:b) (Xy:b), (not (((eq b) ((cP2 Xx) Xy)) c02))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))->((forall (Xx:b) (Xy:b), (not (((eq b) ((cP2 Xx) Xy)) c02)))->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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))) (forall (Xx:b) (Xy:b), (not (((eq b) ((cP2 Xx) Xy)) c02)))) P) x1) x)) (((eq (iS->b)) 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))) (x2:(forall (Xx:b) (Xy:b), (not (((eq b) ((cP2 Xx) Xy)) c02))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))->((forall (Xx:b) (Xy:b), (not (((eq b) ((cP2 Xx) Xy)) c02)))->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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))) (forall (Xx:b) (Xy:b), (not (((eq b) ((cP2 Xx) Xy)) c02)))) P) x1) x)) (((eq (iS->b)) 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))) (x2:(forall (Xx:b) (Xy:b), (not (((eq b) ((cP2 Xx) Xy)) c02))))=> ((eq_ref (iS->b)) x0)))) as proof of (((eq (iS->b)) x0) Xg)
% Found x5:(P x0)
% Instantiate: x0:=Xg:(iS->b)
% Found (fun (x5:(P x0))=> x5) as proof of (P Xg)
% Found (fun (P:((iS->b)->Prop)) (x5:(P x0))=> x5) as proof of ((P x0)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop)) (x5:(P x0))=> x5) as proof of (((eq (iS->b)) 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 (iS->b)) x0) P) as proof of ((P x0)->(P Xg))
% Found (((eq_ref (iS->b)) x0) P) as proof of ((P x0)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x0) P)) as proof of (((eq (iS->b)) 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 (iS->b)) x2) P) as proof of ((P x2)->(P Xg))
% Found (((eq_ref (iS->b)) x2) P) as proof of ((P x2)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x2) P)) as proof of (((eq (iS->b)) x2) Xg)
% Found x5:(P x2)
% Instantiate: x2:=Xg:(iS->b)
% Found (fun (x5:(P x2))=> x5) as proof of (P Xg)
% Found (fun (P:((iS->b)->Prop)) (x5:(P x2))=> x5) as proof of ((P x2)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop)) (x5:(P x2))=> x5) as proof of (((eq (iS->b)) x2) Xg)
% Found eq_ref00:=(eq_ref0 x0):(((eq (iS->b)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (iS->b)) x0) x')
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) x')
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((eq_ref (iS->b)) x0)) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 x2):(((eq (iS->b)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (iS->b)) x2) x')
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) x')
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((eq_ref (iS->b)) x2)) as proof of (((eq (iS->b)) x2) x')
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq b) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq b) (x0 x1)) (Xg x1))
% Found ((eq_ref b) (x0 x1)) as proof of (((eq b) (x0 x1)) (Xg x1))
% Found ((eq_ref b) (x0 x1)) as proof of (((eq b) (x0 x1)) (Xg x1))
% Found (fun (x1:iS)=> ((eq_ref b) (x0 x1))) as proof of (((eq b) (x0 x1)) (Xg x1))
% Found (fun (x1:iS)=> ((eq_ref b) (x0 x1))) as proof of (forall (x:iS), (((eq b) (x0 x)) (Xg x)))
% Found (functional_extensionality0000 (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of (((eq (iS->b)) x0) Xg)
% Found ((functional_extensionality000 Xg) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of (((eq (iS->b)) x0) Xg)
% Found (((functional_extensionality00 x0) Xg) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of (((eq (iS->b)) x0) Xg)
% Found ((((functional_extensionality0 b) x0) Xg) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of (((eq (iS->b)) x0) Xg)
% Found (((((functional_extensionality iS) b) x0) Xg) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((((functional_extensionality iS) b) x0) Xg) (fun (x1:iS)=> ((eq_ref b) (x0 x1))))) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq b) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq b) (x0 x1)) (Xg x1))
% Found ((eq_ref b) (x0 x1)) as proof of (((eq b) (x0 x1)) (Xg x1))
% Found ((eq_ref b) (x0 x1)) as proof of (((eq b) (x0 x1)) (Xg x1))
% Found (fun (x1:iS)=> ((eq_ref b) (x0 x1))) as proof of (((eq b) (x0 x1)) (Xg x1))
% Found (fun (x1:iS)=> ((eq_ref b) (x0 x1))) as proof of (forall (x:iS), (((eq b) (x0 x)) (Xg x)))
% Found (functional_extensionality_dep0000 (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of (((eq (iS->b)) x0) Xg)
% Found ((functional_extensionality_dep000 Xg) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of (((eq (iS->b)) x0) Xg)
% Found (((functional_extensionality_dep00 x0) Xg) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of (((eq (iS->b)) x0) Xg)
% Found ((((functional_extensionality_dep0 (fun (x3:iS)=> b)) x0) Xg) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of (((eq (iS->b)) x0) Xg)
% Found (((((functional_extensionality_dep iS) (fun (x3:iS)=> b)) x0) Xg) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((((functional_extensionality_dep iS) (fun (x3:iS)=> b)) x0) Xg) (fun (x1:iS)=> ((eq_ref b) (x0 x1))))) as proof of (((eq (iS->b)) x0) Xg)
% Found x5:(P x4)
% Instantiate: x4:=x':(iS->b)
% Found (fun (x5:(P x4))=> x5) as proof of (P x')
% Found (fun (P:((iS->b)->Prop)) (x5:(P x4))=> x5) as proof of ((P x4)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop)) (x5:(P x4))=> x5) as proof of (((eq (iS->b)) x4) x')
% Found eq_ref000:=(eq_ref00 P):((P x4)->(P x4))
% Found (eq_ref00 P) as proof of ((P x4)->(P x'))
% Found ((eq_ref0 x4) P) as proof of ((P x4)->(P x'))
% Found (((eq_ref (iS->b)) x4) P) as proof of ((P x4)->(P x'))
% Found (((eq_ref (iS->b)) x4) P) as proof of ((P x4)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x4) P)) as proof of ((P x4)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x4) P)) as proof of (((eq (iS->b)) x4) x')
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((unique (iS->b)) (fun (x1:(iS->b))=> ((and (((eq b) (x1 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x1 ((cP Xx) Xy))) ((cP2 (x1 Xx)) (x1 Xy)))))))):(((eq ((iS->b)->Prop)) ((unique (iS->b)) (fun (x1:(iS->b))=> ((and (((eq b) (x1 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x1 ((cP Xx) Xy))) ((cP2 (x1 Xx)) (x1 Xy)))))))) (fun (x:(iS->b))=> (((unique (iS->b)) (fun (x1:(iS->b))=> ((and (((eq b) (x1 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x1 ((cP Xx) Xy))) ((cP2 (x1 Xx)) (x1 Xy))))))) x)))
% Found (eta_expansion_dep00 ((unique (iS->b)) (fun (x1:(iS->b))=> ((and (((eq b) (x1 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x1 ((cP Xx) Xy))) ((cP2 (x1 Xx)) (x1 Xy)))))))) as proof of (((eq ((iS->b)->Prop)) ((unique (iS->b)) (fun (x1:(iS->b))=> ((and (((eq b) (x1 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x1 ((cP Xx) Xy))) ((cP2 (x1 Xx)) (x1 Xy)))))))) b0)
% Found ((eta_expansion_dep0 (fun (x1:(iS->b))=> Prop)) ((unique (iS->b)) (fun (x1:(iS->b))=> ((and (((eq b) (x1 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x1 ((cP Xx) Xy))) ((cP2 (x1 Xx)) (x1 Xy)))))))) as proof of (((eq ((iS->b)->Prop)) ((unique (iS->b)) (fun (x1:(iS->b))=> ((and (((eq b) (x1 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x1 ((cP Xx) Xy))) ((cP2 (x1 Xx)) (x1 Xy)))))))) b0)
% Found (((eta_expansion_dep (iS->b)) (fun (x1:(iS->b))=> Prop)) ((unique (iS->b)) (fun (x1:(iS->b))=> ((and (((eq b) (x1 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x1 ((cP Xx) Xy))) ((cP2 (x1 Xx)) (x1 Xy)))))))) as proof of (((eq ((iS->b)->Prop)) ((unique (iS->b)) (fun (x1:(iS->b))=> ((and (((eq b) (x1 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x1 ((cP Xx) Xy))) ((cP2 (x1 Xx)) (x1 Xy)))))))) b0)
% Found (((eta_expansion_dep (iS->b)) (fun (x1:(iS->b))=> Prop)) ((unique (iS->b)) (fun (x1:(iS->b))=> ((and (((eq b) (x1 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x1 ((cP Xx) Xy))) ((cP2 (x1 Xx)) (x1 Xy)))))))) as proof of (((eq ((iS->b)->Prop)) ((unique (iS->b)) (fun (x1:(iS->b))=> ((and (((eq b) (x1 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x1 ((cP Xx) Xy))) ((cP2 (x1 Xx)) (x1 Xy)))))))) b0)
% Found (((eta_expansion_dep (iS->b)) (fun (x1:(iS->b))=> Prop)) ((unique (iS->b)) (fun (x1:(iS->b))=> ((and (((eq b) (x1 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x1 ((cP Xx) Xy))) ((cP2 (x1 Xx)) (x1 Xy)))))))) as proof of (((eq ((iS->b)->Prop)) ((unique (iS->b)) (fun (x1:(iS->b))=> ((and (((eq b) (x1 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x1 ((cP Xx) Xy))) ((cP2 (x1 Xx)) (x1 Xy)))))))) b0)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and ((and (((eq b) (x0 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x0 ((cP Xx) Xy))) ((cP2 (x0 Xx)) (x0 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x0) Xg)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((and (((eq b) (x0 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x0 ((cP Xx) Xy))) ((cP2 (x0 Xx)) (x0 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x0) Xg)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((and (((eq b) (x0 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x0 ((cP Xx) Xy))) ((cP2 (x0 Xx)) (x0 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x0) Xg)))))
% Found (fun (x0:(iS->b))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and ((and (((eq b) (x0 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x0 ((cP Xx) Xy))) ((cP2 (x0 Xx)) (x0 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x0) Xg)))))
% Found (fun (x0:(iS->b))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(iS->b)), (((eq Prop) (f x)) ((and ((and (((eq b) (x c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x ((cP Xx) Xy))) ((cP2 (x Xx)) (x Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x) Xg))))))
% Found eq_ref00:=(eq_ref0 (x0 c0)):(((eq b) (x0 c0)) (x0 c0))
% Found (eq_ref0 (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and ((and (((eq b) (x0 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x0 ((cP Xx) Xy))) ((cP2 (x0 Xx)) (x0 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x0) Xg)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((and (((eq b) (x0 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x0 ((cP Xx) Xy))) ((cP2 (x0 Xx)) (x0 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x0) Xg)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((and (((eq b) (x0 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x0 ((cP Xx) Xy))) ((cP2 (x0 Xx)) (x0 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x0) Xg)))))
% Found (fun (x0:(iS->b))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and ((and (((eq b) (x0 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x0 ((cP Xx) Xy))) ((cP2 (x0 Xx)) (x0 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x0) Xg)))))
% Found (fun (x0:(iS->b))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(iS->b)), (((eq Prop) (f x)) ((and ((and (((eq b) (x c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x ((cP Xx) Xy))) ((cP2 (x Xx)) (x Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x) Xg))))))
% Found eq_ref00:=(eq_ref0 (x2 c0)):(((eq b) (x2 c0)) (x2 c0))
% Found (eq_ref0 (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found eq_ref00:=(eq_ref0 x0):(((eq (iS->b)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (iS->b)) x0) x')
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) x')
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x2:(forall (Xx:b) (Xy:b), (not (((eq b) ((cP2 Xx) Xy)) c02))))=> ((eq_ref (iS->b)) x0)) as proof of (((eq (iS->b)) x0) x')
% 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))) (x2:(forall (Xx:b) (Xy:b), (not (((eq b) ((cP2 Xx) Xy)) c02))))=> ((eq_ref (iS->b)) x0)) as proof of ((forall (Xx:b) (Xy:b), (not (((eq b) ((cP2 Xx) Xy)) c02)))->(((eq (iS->b)) x0) x'))
% 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))) (x2:(forall (Xx:b) (Xy:b), (not (((eq b) ((cP2 Xx) Xy)) c02))))=> ((eq_ref (iS->b)) 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))->((forall (Xx:b) (Xy:b), (not (((eq b) ((cP2 Xx) Xy)) c02)))->(((eq (iS->b)) x0) x')))
% 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))) (x2:(forall (Xx:b) (Xy:b), (not (((eq b) ((cP2 Xx) Xy)) c02))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) x')
% Found ((and_rect0 (((eq (iS->b)) x0) x')) (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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))) (x2:(forall (Xx:b) (Xy:b), (not (((eq b) ((cP2 Xx) Xy)) c02))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) x')
% 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))->((forall (Xx:b) (Xy:b), (not (((eq b) ((cP2 Xx) Xy)) c02)))->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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))) (forall (Xx:b) (Xy:b), (not (((eq b) ((cP2 Xx) Xy)) c02)))) P) x1) x)) (((eq (iS->b)) x0) x')) (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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))) (x2:(forall (Xx:b) (Xy:b), (not (((eq b) ((cP2 Xx) Xy)) c02))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))->((forall (Xx:b) (Xy:b), (not (((eq b) ((cP2 Xx) Xy)) c02)))->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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))) (forall (Xx:b) (Xy:b), (not (((eq b) ((cP2 Xx) Xy)) c02)))) P) x1) x)) (((eq (iS->b)) x0) x')) (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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))) (x2:(forall (Xx:b) (Xy:b), (not (((eq b) ((cP2 Xx) Xy)) c02))))=> ((eq_ref (iS->b)) x0)))) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 x2):(((eq (iS->b)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (iS->b)) x2) Xg)
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) Xg)
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) Xg)
% Found (fun (x4:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x2)) as proof of (((eq (iS->b)) x2) Xg)
% Found (fun (x3:(((eq b) (Xg c0)) c02)) (x4:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x2)) as proof of ((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->(((eq (iS->b)) x2) Xg))
% Found (fun (x3:(((eq b) (Xg c0)) c02)) (x4:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x2)) as proof of ((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->(((eq (iS->b)) x2) Xg)))
% Found (and_rect10 (fun (x3:(((eq b) (Xg c0)) c02)) (x4:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) Xg)
% Found ((and_rect1 (((eq (iS->b)) x2) Xg)) (fun (x3:(((eq b) (Xg c0)) c02)) (x4:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) Xg)
% Found (((fun (P:Type) (x3:((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->P)))=> (((((and_rect (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))) P) x3) x00)) (((eq (iS->b)) x2) Xg)) (fun (x3:(((eq b) (Xg c0)) c02)) (x4:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((fun (P:Type) (x3:((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->P)))=> (((((and_rect (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))) P) x3) x00)) (((eq (iS->b)) x2) Xg)) (fun (x3:(((eq b) (Xg c0)) c02)) (x4:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x2)))) as proof of (((eq (iS->b)) x2) Xg)
% Found x5:(P x0)
% Instantiate: x0:=x':(iS->b)
% Found (fun (x5:(P x0))=> x5) as proof of (P x')
% Found (fun (P:((iS->b)->Prop)) (x5:(P x0))=> x5) as proof of ((P x0)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop)) (x5:(P x0))=> x5) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of ((P x0)->(P x'))
% Found ((eq_ref0 x0) P) as proof of ((P x0)->(P x'))
% Found (((eq_ref (iS->b)) x0) P) as proof of ((P x0)->(P x'))
% Found (((eq_ref (iS->b)) x0) P) as proof of ((P x0)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x0) P)) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% Found (eq_ref00 P) as proof of ((P x2)->(P x'))
% Found ((eq_ref0 x2) P) as proof of ((P x2)->(P x'))
% Found (((eq_ref (iS->b)) x2) P) as proof of ((P x2)->(P x'))
% Found (((eq_ref (iS->b)) x2) P) as proof of ((P x2)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x2) P)) as proof of (((eq (iS->b)) x2) x')
% Found x5:(P x2)
% Instantiate: x2:=x':(iS->b)
% Found (fun (x5:(P x2))=> x5) as proof of (P x')
% Found (fun (P:((iS->b)->Prop)) (x5:(P x2))=> x5) as proof of ((P x2)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop)) (x5:(P x2))=> x5) as proof of (((eq (iS->b)) x2) x')
% Found eq_ref00:=(eq_ref0 Xg):(((eq (iS->b)) Xg) Xg)
% Found (eq_ref0 Xg) as proof of (((eq (iS->b)) Xg) x2)
% Found ((eq_ref (iS->b)) Xg) as proof of (((eq (iS->b)) Xg) x2)
% Found ((eq_ref (iS->b)) Xg) as proof of (((eq (iS->b)) Xg) x2)
% Found ((eq_ref (iS->b)) Xg) as proof of (((eq (iS->b)) Xg) x2)
% Found (eq_sym000 ((eq_ref (iS->b)) Xg)) as proof of (((eq (iS->b)) x2) Xg)
% Found ((eq_sym00 x2) ((eq_ref (iS->b)) Xg)) as proof of (((eq (iS->b)) x2) Xg)
% Found (((eq_sym0 Xg) x2) ((eq_ref (iS->b)) Xg)) as proof of (((eq (iS->b)) x2) Xg)
% Found ((((eq_sym (iS->b)) Xg) x2) ((eq_ref (iS->b)) Xg)) as proof of (((eq (iS->b)) x2) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((((eq_sym (iS->b)) Xg) x2) ((eq_ref (iS->b)) Xg))) as proof of (((eq (iS->b)) x2) Xg)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq b) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq b) (x0 x1)) (x' x1))
% Found ((eq_ref b) (x0 x1)) as proof of (((eq b) (x0 x1)) (x' x1))
% Found ((eq_ref b) (x0 x1)) as proof of (((eq b) (x0 x1)) (x' x1))
% Found (fun (x1:iS)=> ((eq_ref b) (x0 x1))) as proof of (((eq b) (x0 x1)) (x' x1))
% Found (fun (x1:iS)=> ((eq_ref b) (x0 x1))) as proof of (forall (x:iS), (((eq b) (x0 x)) (x' x)))
% Found (functional_extensionality_dep0000 (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of (((eq (iS->b)) x0) x')
% Found ((functional_extensionality_dep000 x') (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of (((eq (iS->b)) x0) x')
% Found (((functional_extensionality_dep00 x0) x') (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of (((eq (iS->b)) x0) x')
% Found ((((functional_extensionality_dep0 (fun (x3:iS)=> b)) x0) x') (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of (((eq (iS->b)) x0) x')
% Found (((((functional_extensionality_dep iS) (fun (x3:iS)=> b)) x0) x') (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((((functional_extensionality_dep iS) (fun (x3:iS)=> b)) x0) x') (fun (x1:iS)=> ((eq_ref b) (x0 x1))))) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq b) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq b) (x0 x1)) (x' x1))
% Found ((eq_ref b) (x0 x1)) as proof of (((eq b) (x0 x1)) (x' x1))
% Found ((eq_ref b) (x0 x1)) as proof of (((eq b) (x0 x1)) (x' x1))
% Found (fun (x1:iS)=> ((eq_ref b) (x0 x1))) as proof of (((eq b) (x0 x1)) (x' x1))
% Found (fun (x1:iS)=> ((eq_ref b) (x0 x1))) as proof of (forall (x:iS), (((eq b) (x0 x)) (x' x)))
% Found (functional_extensionality0000 (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of (((eq (iS->b)) x0) x')
% Found ((functional_extensionality000 x') (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of (((eq (iS->b)) x0) x')
% Found (((functional_extensionality00 x0) x') (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of (((eq (iS->b)) x0) x')
% Found ((((functional_extensionality0 b) x0) x') (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of (((eq (iS->b)) x0) x')
% Found (((((functional_extensionality iS) b) x0) x') (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((((functional_extensionality iS) b) x0) x') (fun (x1:iS)=> ((eq_ref b) (x0 x1))))) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (((unique (iS->b)) (fun (x1:(iS->b))=> ((and (((eq b) (x1 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x1 ((cP Xx) Xy))) ((cP2 (x1 Xx)) (x1 Xy))))))) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((unique (iS->b)) (fun (x1:(iS->b))=> ((and (((eq b) (x1 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x1 ((cP Xx) Xy))) ((cP2 (x1 Xx)) (x1 Xy))))))) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((unique (iS->b)) (fun (x1:(iS->b))=> ((and (((eq b) (x1 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x1 ((cP Xx) Xy))) ((cP2 (x1 Xx)) (x1 Xy))))))) x0))
% Found (fun (x0:(iS->b))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((unique (iS->b)) (fun (x1:(iS->b))=> ((and (((eq b) (x1 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x1 ((cP Xx) Xy))) ((cP2 (x1 Xx)) (x1 Xy))))))) x0))
% Found (fun (x0:(iS->b))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(iS->b)), (((eq Prop) (f x)) (((unique (iS->b)) (fun (x1:(iS->b))=> ((and (((eq b) (x1 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x1 ((cP Xx) Xy))) ((cP2 (x1 Xx)) (x1 Xy))))))) x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))):(((eq ((iS->b)->Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) (fun (x:(iS->b))=> ((and ((and (((eq b) (x c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x ((cP Xx) Xy))) ((cP2 (x Xx)) (x Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x) Xg))))))
% Found (eta_expansion_dep00 (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) as proof of (((eq ((iS->b)->Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) b0)
% Found ((eta_expansion_dep0 (fun (x3:(iS->b))=> Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) as proof of (((eq ((iS->b)->Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) b0)
% Found (((eta_expansion_dep (iS->b)) (fun (x3:(iS->b))=> Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) as proof of (((eq ((iS->b)->Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) b0)
% Found (((eta_expansion_dep (iS->b)) (fun (x3:(iS->b))=> Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) as proof of (((eq ((iS->b)->Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) b0)
% Found (((eta_expansion_dep (iS->b)) (fun (x3:(iS->b))=> Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) as proof of (((eq ((iS->b)->Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) b0)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (((unique (iS->b)) (fun (x1:(iS->b))=> ((and (((eq b) (x1 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x1 ((cP Xx) Xy))) ((cP2 (x1 Xx)) (x1 Xy))))))) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((unique (iS->b)) (fun (x1:(iS->b))=> ((and (((eq b) (x1 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x1 ((cP Xx) Xy))) ((cP2 (x1 Xx)) (x1 Xy))))))) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((unique (iS->b)) (fun (x1:(iS->b))=> ((and (((eq b) (x1 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x1 ((cP Xx) Xy))) ((cP2 (x1 Xx)) (x1 Xy))))))) x0))
% Found (fun (x0:(iS->b))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((unique (iS->b)) (fun (x1:(iS->b))=> ((and (((eq b) (x1 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x1 ((cP Xx) Xy))) ((cP2 (x1 Xx)) (x1 Xy))))))) x0))
% Found (fun (x0:(iS->b))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(iS->b)), (((eq Prop) (f x)) (((unique (iS->b)) (fun (x1:(iS->b))=> ((and (((eq b) (x1 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x1 ((cP Xx) Xy))) ((cP2 (x1 Xx)) (x1 Xy))))))) x)))
% Found eq_ref00:=(eq_ref0 (x0 c0)):(((eq b) (x0 c0)) (x0 c0))
% Found (eq_ref0 (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found eq_ref00:=(eq_ref0 x6):(((eq (iS->b)) x6) x6)
% Found (eq_ref0 x6) as proof of (((eq (iS->b)) x6) Xg)
% Found ((eq_ref (iS->b)) x6) as proof of (((eq (iS->b)) x6) Xg)
% Found ((eq_ref (iS->b)) x6) as proof of (((eq (iS->b)) x6) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((eq_ref (iS->b)) x6)) as proof of (((eq (iS->b)) x6) Xg)
% Found eq_ref00:=(eq_ref0 x0):(((eq (iS->b)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (iS->b)) x0) Xg)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) Xg)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x4:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x0)) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x3:(((eq b) (Xg c0)) c02)) (x4:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x0)) as proof of ((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->(((eq (iS->b)) x0) Xg))
% Found (fun (x3:(((eq b) (Xg c0)) c02)) (x4:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x0)) as proof of ((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->(((eq (iS->b)) x0) Xg)))
% Found (and_rect10 (fun (x3:(((eq b) (Xg c0)) c02)) (x4:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) Xg)
% Found ((and_rect1 (((eq (iS->b)) x0) Xg)) (fun (x3:(((eq b) (Xg c0)) c02)) (x4:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) Xg)
% Found (((fun (P:Type) (x3:((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->P)))=> (((((and_rect (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))) P) x3) x00)) (((eq (iS->b)) x0) Xg)) (fun (x3:(((eq b) (Xg c0)) c02)) (x4:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((fun (P:Type) (x3:((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->P)))=> (((((and_rect (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))) P) x3) x00)) (((eq (iS->b)) x0) Xg)) (fun (x3:(((eq b) (Xg c0)) c02)) (x4:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x0)))) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 x2):(((eq (iS->b)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (iS->b)) x2) x')
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) x')
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) x')
% Found (fun (x4:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x2)) as proof of (((eq (iS->b)) x2) x')
% Found (fun (x3:(((eq b) (x' c0)) c02)) (x4:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x2)) as proof of ((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->(((eq (iS->b)) x2) x'))
% Found (fun (x3:(((eq b) (x' c0)) c02)) (x4:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x2)) as proof of ((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->(((eq (iS->b)) x2) x')))
% Found (and_rect10 (fun (x3:(((eq b) (x' c0)) c02)) (x4:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) x')
% Found ((and_rect1 (((eq (iS->b)) x2) x')) (fun (x3:(((eq b) (x' c0)) c02)) (x4:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) x')
% Found (((fun (P:Type) (x3:((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->P)))=> (((((and_rect (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))) P) x3) x00)) (((eq (iS->b)) x2) x')) (fun (x3:(((eq b) (x' c0)) c02)) (x4:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((fun (P:Type) (x3:((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->P)))=> (((((and_rect (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))) P) x3) x00)) (((eq (iS->b)) x2) x')) (fun (x3:(((eq b) (x' c0)) c02)) (x4:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x2)))) as proof of (((eq (iS->b)) x2) x')
% Found eta_expansion000:=(eta_expansion00 Xg):(((eq (iS->b)) Xg) (fun (x:iS)=> (Xg x)))
% Found (eta_expansion00 Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found ((eta_expansion0 b) Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found (((eta_expansion iS) b) Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found (((eta_expansion iS) b) Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found (((eta_expansion iS) b) Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found (eq_sym000 (((eta_expansion iS) b) Xg)) as proof of (((eq (iS->b)) x0) Xg)
% Found ((eq_sym00 x0) (((eta_expansion iS) b) Xg)) as proof of (((eq (iS->b)) x0) Xg)
% Found (((eq_sym0 Xg) x0) (((eta_expansion iS) b) Xg)) as proof of (((eq (iS->b)) x0) Xg)
% Found ((((eq_sym (iS->b)) Xg) x0) (((eta_expansion iS) b) Xg)) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((((eq_sym (iS->b)) Xg) x0) (((eta_expansion iS) b) Xg))) as proof of (((eq (iS->b)) x0) Xg)
% Found eta_expansion000:=(eta_expansion00 ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))):(((eq ((iS->b)->Prop)) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) (fun (x:(iS->b))=> (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x)))
% Found (eta_expansion00 ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) as proof of (((eq ((iS->b)->Prop)) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) b0)
% Found ((eta_expansion0 Prop) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) as proof of (((eq ((iS->b)->Prop)) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) b0)
% Found (((eta_expansion (iS->b)) Prop) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) as proof of (((eq ((iS->b)->Prop)) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) b0)
% Found (((eta_expansion (iS->b)) Prop) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) as proof of (((eq ((iS->b)->Prop)) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) b0)
% Found (((eta_expansion (iS->b)) Prop) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) as proof of (((eq ((iS->b)->Prop)) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) b0)
% Found eq_ref00:=(eq_ref0 x0):(((eq (iS->b)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (iS->b)) x0) Xg)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) Xg)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((eq_ref (iS->b)) x0)) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 x'):(((eq (iS->b)) x') x')
% Found (eq_ref0 x') as proof of (((eq (iS->b)) x') x2)
% Found ((eq_ref (iS->b)) x') as proof of (((eq (iS->b)) x') x2)
% Found ((eq_ref (iS->b)) x') as proof of (((eq (iS->b)) x') x2)
% Found ((eq_ref (iS->b)) x') as proof of (((eq (iS->b)) x') x2)
% Found (eq_sym000 ((eq_ref (iS->b)) x')) as proof of (((eq (iS->b)) x2) x')
% Found ((eq_sym00 x2) ((eq_ref (iS->b)) x')) as proof of (((eq (iS->b)) x2) x')
% Found (((eq_sym0 x') x2) ((eq_ref (iS->b)) x')) as proof of (((eq (iS->b)) x2) x')
% Found ((((eq_sym (iS->b)) x') x2) ((eq_ref (iS->b)) x')) as proof of (((eq (iS->b)) x2) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((((eq_sym (iS->b)) x') x2) ((eq_ref (iS->b)) x'))) as proof of (((eq (iS->b)) x2) x')
% Found eq_ref00:=(eq_ref0 x2):(((eq (iS->b)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (iS->b)) x2) Xg)
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) Xg)
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((eq_ref (iS->b)) x2)) as proof of (((eq (iS->b)) x2) Xg)
% Found eq_ref00:=(eq_ref0 x4):(((eq (iS->b)) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq (iS->b)) x4) Xg)
% Found ((eq_ref (iS->b)) x4) as proof of (((eq (iS->b)) x4) Xg)
% Found ((eq_ref (iS->b)) x4) as proof of (((eq (iS->b)) x4) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((eq_ref (iS->b)) x4)) as proof of (((eq (iS->b)) x4) Xg)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and ((and (((eq b) (x2 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x2 ((cP Xx) Xy))) ((cP2 (x2 Xx)) (x2 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x2) Xg)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and ((and (((eq b) (x2 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x2 ((cP Xx) Xy))) ((cP2 (x2 Xx)) (x2 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x2) Xg)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and ((and (((eq b) (x2 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x2 ((cP Xx) Xy))) ((cP2 (x2 Xx)) (x2 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x2) Xg)))))
% Found (fun (x2:(iS->b))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and ((and (((eq b) (x2 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x2 ((cP Xx) Xy))) ((cP2 (x2 Xx)) (x2 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x2) Xg)))))
% Found (fun (x2:(iS->b))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(iS->b)), (((eq Prop) (f x)) ((and ((and (((eq b) (x c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x ((cP Xx) Xy))) ((cP2 (x Xx)) (x Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x) Xg))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and ((and (((eq b) (x2 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x2 ((cP Xx) Xy))) ((cP2 (x2 Xx)) (x2 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x2) Xg)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and ((and (((eq b) (x2 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x2 ((cP Xx) Xy))) ((cP2 (x2 Xx)) (x2 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x2) Xg)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and ((and (((eq b) (x2 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x2 ((cP Xx) Xy))) ((cP2 (x2 Xx)) (x2 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x2) Xg)))))
% Found (fun (x2:(iS->b))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and ((and (((eq b) (x2 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x2 ((cP Xx) Xy))) ((cP2 (x2 Xx)) (x2 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x2) Xg)))))
% Found (fun (x2:(iS->b))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(iS->b)), (((eq Prop) (f x)) ((and ((and (((eq b) (x c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x ((cP Xx) Xy))) ((cP2 (x Xx)) (x Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x) 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 (iS->b)) x6) P) as proof of ((P x6)->(P Xg))
% Found (((eq_ref (iS->b)) x6) P) as proof of ((P x6)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x6) P)) as proof of ((P x6)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x6) P)) as proof of (((eq (iS->b)) x6) Xg)
% Found x7:(P x6)
% Instantiate: x6:=Xg:(iS->b)
% Found (fun (x7:(P x6))=> x7) as proof of (P Xg)
% Found (fun (P:((iS->b)->Prop)) (x7:(P x6))=> x7) as proof of ((P x6)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop)) (x7:(P x6))=> x7) as proof of (((eq (iS->b)) x6) Xg)
% Found eq_ref00:=(eq_ref0 x0):(((eq (iS->b)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (iS->b)) x0) Xg)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) Xg)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x4:(forall (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))=> ((eq_ref (iS->b)) x0)) as proof of (((eq (iS->b)) 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))=> ((eq_ref (iS->b)) x0)) as proof of ((forall (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv))))->(((eq (iS->b)) 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))=> ((eq_ref (iS->b)) 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv))))->(((eq (iS->b)) 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) Xg)
% Found ((and_rect1 (((eq (iS->b)) 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv))))->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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv))))) P) x3) x1)) (((eq (iS->b)) 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv))))->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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv))))) P) x3) x1)) (((eq (iS->b)) 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))=> ((eq_ref (iS->b)) x0)))) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 x2):(((eq (iS->b)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (iS->b)) x2) Xg)
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) Xg)
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) Xg)
% Found (fun (x4:(forall (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))=> ((eq_ref (iS->b)) x2)) as proof of (((eq (iS->b)) 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))=> ((eq_ref (iS->b)) x2)) as proof of ((forall (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv))))->(((eq (iS->b)) 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))=> ((eq_ref (iS->b)) 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv))))->(((eq (iS->b)) 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) Xg)
% Found ((and_rect1 (((eq (iS->b)) 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv))))->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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv))))) P) x3) x0)) (((eq (iS->b)) 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv))))->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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv))))) P) x3) x0)) (((eq (iS->b)) 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))=> ((eq_ref (iS->b)) x2)))) as proof of (((eq (iS->b)) x2) Xg)
% Found eq_ref00:=(eq_ref0 x0):(((eq (iS->b)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (iS->b)) x0) x')
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) x')
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x4:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x0)) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x3:(((eq b) (x' c0)) c02)) (x4:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x0)) as proof of ((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->(((eq (iS->b)) x0) x'))
% Found (fun (x3:(((eq b) (x' c0)) c02)) (x4:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x0)) as proof of ((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->(((eq (iS->b)) x0) x')))
% Found (and_rect10 (fun (x3:(((eq b) (x' c0)) c02)) (x4:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) x')
% Found ((and_rect1 (((eq (iS->b)) x0) x')) (fun (x3:(((eq b) (x' c0)) c02)) (x4:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) x')
% Found (((fun (P:Type) (x3:((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->P)))=> (((((and_rect (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))) P) x3) x00)) (((eq (iS->b)) x0) x')) (fun (x3:(((eq b) (x' c0)) c02)) (x4:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((fun (P:Type) (x3:((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->P)))=> (((((and_rect (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))) P) x3) x00)) (((eq (iS->b)) x0) x')) (fun (x3:(((eq b) (x' c0)) c02)) (x4:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x0)))) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 x6):(((eq (iS->b)) x6) x6)
% Found (eq_ref0 x6) as proof of (((eq (iS->b)) x6) x')
% Found ((eq_ref (iS->b)) x6) as proof of (((eq (iS->b)) x6) x')
% Found ((eq_ref (iS->b)) x6) as proof of (((eq (iS->b)) x6) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((eq_ref (iS->b)) x6)) as proof of (((eq (iS->b)) x6) x')
% Found eq_ref00:=(eq_ref0 (x4 c0)):(((eq b) (x4 c0)) (x4 c0))
% Found (eq_ref0 (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found ((eq_ref b) (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found ((eq_ref b) (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found ((eq_ref b) (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found eta_expansion_dep000:=(eta_expansion_dep00 x'):(((eq (iS->b)) x') (fun (x:iS)=> (x' x)))
% Found (eta_expansion_dep00 x') as proof of (((eq (iS->b)) x') x0)
% Found ((eta_expansion_dep0 (fun (x4:iS)=> b)) x') as proof of (((eq (iS->b)) x') x0)
% Found (((eta_expansion_dep iS) (fun (x4:iS)=> b)) x') as proof of (((eq (iS->b)) x') x0)
% Found (((eta_expansion_dep iS) (fun (x4:iS)=> b)) x') as proof of (((eq (iS->b)) x') x0)
% Found (((eta_expansion_dep iS) (fun (x4:iS)=> b)) x') as proof of (((eq (iS->b)) x') x0)
% Found (eq_sym000 (((eta_expansion_dep iS) (fun (x4:iS)=> b)) x')) as proof of (((eq (iS->b)) x0) x')
% Found ((eq_sym00 x0) (((eta_expansion_dep iS) (fun (x4:iS)=> b)) x')) as proof of (((eq (iS->b)) x0) x')
% Found (((eq_sym0 x') x0) (((eta_expansion_dep iS) (fun (x4:iS)=> b)) x')) as proof of (((eq (iS->b)) x0) x')
% Found ((((eq_sym (iS->b)) x') x0) (((eta_expansion_dep iS) (fun (x4:iS)=> b)) x')) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((((eq_sym (iS->b)) x') x0) (((eta_expansion_dep iS) (fun (x4:iS)=> b)) x'))) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 (x2 x3)):(((eq b) (x2 x3)) (x2 x3))
% Found (eq_ref0 (x2 x3)) as proof of (((eq b) (x2 x3)) (Xg x3))
% Found ((eq_ref b) (x2 x3)) as proof of (((eq b) (x2 x3)) (Xg x3))
% Found ((eq_ref b) (x2 x3)) as proof of (((eq b) (x2 x3)) (Xg x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x2 x3))) as proof of (((eq b) (x2 x3)) (Xg x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x2 x3))) as proof of (forall (x:iS), (((eq b) (x2 x)) (Xg x)))
% Found (functional_extensionality_dep0000 (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of (((eq (iS->b)) x2) Xg)
% Found ((functional_extensionality_dep000 Xg) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of (((eq (iS->b)) x2) Xg)
% Found (((functional_extensionality_dep00 x2) Xg) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of (((eq (iS->b)) x2) Xg)
% Found ((((functional_extensionality_dep0 (fun (x5:iS)=> b)) x2) Xg) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of (((eq (iS->b)) x2) Xg)
% Found (((((functional_extensionality_dep iS) (fun (x5:iS)=> b)) x2) Xg) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of (((eq (iS->b)) x2) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((((functional_extensionality_dep iS) (fun (x5:iS)=> b)) x2) Xg) (fun (x3:iS)=> ((eq_ref b) (x2 x3))))) as proof of (((eq (iS->b)) x2) Xg)
% Found eq_ref00:=(eq_ref0 (x2 x3)):(((eq b) (x2 x3)) (x2 x3))
% Found (eq_ref0 (x2 x3)) as proof of (((eq b) (x2 x3)) (Xg x3))
% Found ((eq_ref b) (x2 x3)) as proof of (((eq b) (x2 x3)) (Xg x3))
% Found ((eq_ref b) (x2 x3)) as proof of (((eq b) (x2 x3)) (Xg x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x2 x3))) as proof of (((eq b) (x2 x3)) (Xg x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x2 x3))) as proof of (forall (x:iS), (((eq b) (x2 x)) (Xg x)))
% Found (functional_extensionality0000 (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of (((eq (iS->b)) x2) Xg)
% Found ((functional_extensionality000 Xg) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of (((eq (iS->b)) x2) Xg)
% Found (((functional_extensionality00 x2) Xg) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of (((eq (iS->b)) x2) Xg)
% Found ((((functional_extensionality0 b) x2) Xg) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of (((eq (iS->b)) x2) Xg)
% Found (((((functional_extensionality iS) b) x2) Xg) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of (((eq (iS->b)) x2) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((((functional_extensionality iS) b) x2) Xg) (fun (x3:iS)=> ((eq_ref b) (x2 x3))))) as proof of (((eq (iS->b)) x2) Xg)
% Found x7:(P x0)
% Instantiate: x0:=Xg:(iS->b)
% Found (fun (x7:(P x0))=> x7) as proof of (P Xg)
% Found (fun (P:((iS->b)->Prop)) (x7:(P x0))=> x7) as proof of ((P x0)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop)) (x7:(P x0))=> x7) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x2))
% Found (fun (x2:(iS->b))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x2))
% Found (fun (x2:(iS->b))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(iS->b)), (((eq Prop) (f x)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x)))
% 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 (iS->b)) x0) P) as proof of ((P x0)->(P Xg))
% Found (((eq_ref (iS->b)) x0) P) as proof of ((P x0)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x0) P)) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x2))
% Found (fun (x2:(iS->b))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x2))
% Found (fun (x2:(iS->b))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(iS->b)), (((eq Prop) (f x)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x)))
% 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 (iS->b)) x2) P) as proof of ((P x2)->(P Xg))
% Found (((eq_ref (iS->b)) x2) P) as proof of ((P x2)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x2) P)) as proof of (((eq (iS->b)) x2) Xg)
% Found x7:(P x2)
% Instantiate: x2:=Xg:(iS->b)
% Found (fun (x7:(P x2))=> x7) as proof of (P Xg)
% Found (fun (P:((iS->b)->Prop)) (x7:(P x2))=> x7) as proof of ((P x2)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop)) (x7:(P x2))=> x7) as proof of (((eq (iS->b)) x2) Xg)
% Found x7:(P x4)
% Instantiate: x4:=Xg:(iS->b)
% Found (fun (x7:(P x4))=> x7) as proof of (P Xg)
% Found (fun (P:((iS->b)->Prop)) (x7:(P x4))=> x7) as proof of ((P x4)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop)) (x7:(P x4))=> x7) as proof of (((eq (iS->b)) x4) 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 (iS->b)) x4) P) as proof of ((P x4)->(P Xg))
% Found (((eq_ref (iS->b)) x4) P) as proof of ((P x4)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x4) P)) as proof of ((P x4)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x4) P)) as proof of (((eq (iS->b)) x4) Xg)
% Found eq_ref00:=(eq_ref0 x0):(((eq (iS->b)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (iS->b)) x0) x')
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) x')
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((eq_ref (iS->b)) x0)) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 x2):(((eq (iS->b)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (iS->b)) x2) x')
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) x')
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((eq_ref (iS->b)) x2)) as proof of (((eq (iS->b)) x2) x')
% Found eq_ref00:=(eq_ref0 (x0 c0)):(((eq b) (x0 c0)) (x0 c0))
% Found (eq_ref0 (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found eq_ref00:=(eq_ref0 x4):(((eq (iS->b)) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq (iS->b)) x4) x')
% Found ((eq_ref (iS->b)) x4) as proof of (((eq (iS->b)) x4) x')
% Found ((eq_ref (iS->b)) x4) as proof of (((eq (iS->b)) x4) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((eq_ref (iS->b)) x4)) as proof of (((eq (iS->b)) x4) x')
% Found eq_ref00:=(eq_ref0 (x2 c0)):(((eq b) (x2 c0)) (x2 c0))
% Found (eq_ref0 (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found eq_ref00:=(eq_ref0 (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))):(((eq ((iS->b)->Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg))))))
% Found (eq_ref0 (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) as proof of (((eq ((iS->b)->Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) b0)
% Found ((eq_ref ((iS->b)->Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) as proof of (((eq ((iS->b)->Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) b0)
% Found ((eq_ref ((iS->b)->Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) as proof of (((eq ((iS->b)->Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) b0)
% Found ((eq_ref ((iS->b)->Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) as proof of (((eq ((iS->b)->Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) b0)
% Found eq_ref00:=(eq_ref0 x0):(((eq (iS->b)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (iS->b)) x0) x')
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) x')
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x4:(forall (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))=> ((eq_ref (iS->b)) x0)) as proof of (((eq (iS->b)) x0) x')
% 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))=> ((eq_ref (iS->b)) x0)) as proof of ((forall (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv))))->(((eq (iS->b)) x0) x'))
% 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))=> ((eq_ref (iS->b)) 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv))))->(((eq (iS->b)) x0) x')))
% 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) x')
% Found ((and_rect1 (((eq (iS->b)) x0) x')) (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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) x')
% 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv))))->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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv))))) P) x3) x1)) (((eq (iS->b)) x0) x')) (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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv))))->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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv))))) P) x3) x1)) (((eq (iS->b)) x0) x')) (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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))=> ((eq_ref (iS->b)) x0)))) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 x2):(((eq (iS->b)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (iS->b)) x2) x')
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) x')
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) x')
% Found (fun (x4:(forall (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))=> ((eq_ref (iS->b)) x2)) as proof of (((eq (iS->b)) x2) x')
% 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))=> ((eq_ref (iS->b)) x2)) as proof of ((forall (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv))))->(((eq (iS->b)) x2) x'))
% 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))=> ((eq_ref (iS->b)) 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv))))->(((eq (iS->b)) x2) x')))
% 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) x')
% Found ((and_rect1 (((eq (iS->b)) x2) x')) (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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) x')
% 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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv))))->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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv))))) P) x3) x0)) (((eq (iS->b)) x2) x')) (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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv))))->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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv))))) P) x3) x0)) (((eq (iS->b)) x2) x')) (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 (Xx:b) (Xy:b) (Xu:b) (Xv:b), ((((eq b) ((cP2 Xx) Xu)) ((cP2 Xy) Xv))->((and (((eq b) Xx) Xy)) (((eq b) Xu) Xv)))))=> ((eq_ref (iS->b)) x2)))) as proof of (((eq (iS->b)) x2) x')
% Found eq_ref00:=(eq_ref0 (x0 x3)):(((eq b) (x0 x3)) (x0 x3))
% Found (eq_ref0 (x0 x3)) as proof of (((eq b) (x0 x3)) (Xg x3))
% Found ((eq_ref b) (x0 x3)) as proof of (((eq b) (x0 x3)) (Xg x3))
% Found ((eq_ref b) (x0 x3)) as proof of (((eq b) (x0 x3)) (Xg x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x0 x3))) as proof of (((eq b) (x0 x3)) (Xg x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x0 x3))) as proof of (forall (x:iS), (((eq b) (x0 x)) (Xg x)))
% Found (functional_extensionality0000 (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of (((eq (iS->b)) x0) Xg)
% Found ((functional_extensionality000 Xg) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of (((eq (iS->b)) x0) Xg)
% Found (((functional_extensionality00 x0) Xg) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of (((eq (iS->b)) x0) Xg)
% Found ((((functional_extensionality0 b) x0) Xg) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of (((eq (iS->b)) x0) Xg)
% Found (((((functional_extensionality iS) b) x0) Xg) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((((functional_extensionality iS) b) x0) Xg) (fun (x3:iS)=> ((eq_ref b) (x0 x3))))) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 (x0 x3)):(((eq b) (x0 x3)) (x0 x3))
% Found (eq_ref0 (x0 x3)) as proof of (((eq b) (x0 x3)) (Xg x3))
% Found ((eq_ref b) (x0 x3)) as proof of (((eq b) (x0 x3)) (Xg x3))
% Found ((eq_ref b) (x0 x3)) as proof of (((eq b) (x0 x3)) (Xg x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x0 x3))) as proof of (((eq b) (x0 x3)) (Xg x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x0 x3))) as proof of (forall (x:iS), (((eq b) (x0 x)) (Xg x)))
% Found (functional_extensionality_dep0000 (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of (((eq (iS->b)) x0) Xg)
% Found ((functional_extensionality_dep000 Xg) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of (((eq (iS->b)) x0) Xg)
% Found (((functional_extensionality_dep00 x0) Xg) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of (((eq (iS->b)) x0) Xg)
% Found ((((functional_extensionality_dep0 (fun (x5:iS)=> b)) x0) Xg) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of (((eq (iS->b)) x0) Xg)
% Found (((((functional_extensionality_dep iS) (fun (x5:iS)=> b)) x0) Xg) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((((functional_extensionality_dep iS) (fun (x5:iS)=> b)) x0) Xg) (fun (x3:iS)=> ((eq_ref b) (x0 x3))))) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref000:=(eq_ref00 P):((P x6)->(P x6))
% Found (eq_ref00 P) as proof of ((P x6)->(P x'))
% Found ((eq_ref0 x6) P) as proof of ((P x6)->(P x'))
% Found (((eq_ref (iS->b)) x6) P) as proof of ((P x6)->(P x'))
% Found (((eq_ref (iS->b)) x6) P) as proof of ((P x6)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x6) P)) as proof of ((P x6)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x6) P)) as proof of (((eq (iS->b)) x6) x')
% Found x7:(P x6)
% Instantiate: x6:=x':(iS->b)
% Found (fun (x7:(P x6))=> x7) as proof of (P x')
% Found (fun (P:((iS->b)->Prop)) (x7:(P x6))=> x7) as proof of ((P x6)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop)) (x7:(P x6))=> x7) as proof of (((eq (iS->b)) x6) x')
% Found eq_ref00:=(eq_ref0 (x4 c0)):(((eq b) (x4 c0)) (x4 c0))
% Found (eq_ref0 (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found ((eq_ref b) (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found ((eq_ref b) (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found ((eq_ref b) (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found eq_ref00:=(eq_ref0 (x0 c0)):(((eq b) (x0 c0)) (x0 c0))
% Found (eq_ref0 (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found eq_ref00:=(eq_ref0 (x2 x3)):(((eq b) (x2 x3)) (x2 x3))
% Found (eq_ref0 (x2 x3)) as proof of (((eq b) (x2 x3)) (x' x3))
% Found ((eq_ref b) (x2 x3)) as proof of (((eq b) (x2 x3)) (x' x3))
% Found ((eq_ref b) (x2 x3)) as proof of (((eq b) (x2 x3)) (x' x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x2 x3))) as proof of (((eq b) (x2 x3)) (x' x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x2 x3))) as proof of (forall (x:iS), (((eq b) (x2 x)) (x' x)))
% Found (functional_extensionality0000 (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of (((eq (iS->b)) x2) x')
% Found ((functional_extensionality000 x') (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of (((eq (iS->b)) x2) x')
% Found (((functional_extensionality00 x2) x') (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of (((eq (iS->b)) x2) x')
% Found ((((functional_extensionality0 b) x2) x') (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of (((eq (iS->b)) x2) x')
% Found (((((functional_extensionality iS) b) x2) x') (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of (((eq (iS->b)) x2) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((((functional_extensionality iS) b) x2) x') (fun (x3:iS)=> ((eq_ref b) (x2 x3))))) as proof of (((eq (iS->b)) x2) x')
% Found eq_ref00:=(eq_ref0 (x2 x3)):(((eq b) (x2 x3)) (x2 x3))
% Found (eq_ref0 (x2 x3)) as proof of (((eq b) (x2 x3)) (x' x3))
% Found ((eq_ref b) (x2 x3)) as proof of (((eq b) (x2 x3)) (x' x3))
% Found ((eq_ref b) (x2 x3)) as proof of (((eq b) (x2 x3)) (x' x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x2 x3))) as proof of (((eq b) (x2 x3)) (x' x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x2 x3))) as proof of (forall (x:iS), (((eq b) (x2 x)) (x' x)))
% Found (functional_extensionality_dep0000 (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of (((eq (iS->b)) x2) x')
% Found ((functional_extensionality_dep000 x') (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of (((eq (iS->b)) x2) x')
% Found (((functional_extensionality_dep00 x2) x') (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of (((eq (iS->b)) x2) x')
% Found ((((functional_extensionality_dep0 (fun (x5:iS)=> b)) x2) x') (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of (((eq (iS->b)) x2) x')
% Found (((((functional_extensionality_dep iS) (fun (x5:iS)=> b)) x2) x') (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of (((eq (iS->b)) x2) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((((functional_extensionality_dep iS) (fun (x5:iS)=> b)) x2) x') (fun (x3:iS)=> ((eq_ref b) (x2 x3))))) as proof of (((eq (iS->b)) x2) x')
% Found eq_ref00:=(eq_ref0 x4):(((eq (iS->b)) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq (iS->b)) x4) Xg)
% Found ((eq_ref (iS->b)) x4) as proof of (((eq (iS->b)) x4) Xg)
% Found ((eq_ref (iS->b)) x4) as proof of (((eq (iS->b)) x4) Xg)
% Found (fun (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x4)) as proof of (((eq (iS->b)) x4) Xg)
% Found (fun (x5:(((eq b) (Xg c0)) c02)) (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x4)) as proof of ((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->(((eq (iS->b)) x4) Xg))
% Found (fun (x5:(((eq b) (Xg c0)) c02)) (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x4)) as proof of ((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->(((eq (iS->b)) x4) Xg)))
% Found (and_rect20 (fun (x5:(((eq b) (Xg c0)) c02)) (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x4))) as proof of (((eq (iS->b)) x4) Xg)
% Found ((and_rect2 (((eq (iS->b)) x4) Xg)) (fun (x5:(((eq b) (Xg c0)) c02)) (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x4))) as proof of (((eq (iS->b)) x4) Xg)
% Found (((fun (P:Type) (x5:((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->P)))=> (((((and_rect (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))) P) x5) x00)) (((eq (iS->b)) x4) Xg)) (fun (x5:(((eq b) (Xg c0)) c02)) (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x4))) as proof of (((eq (iS->b)) x4) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((fun (P:Type) (x5:((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->P)))=> (((((and_rect (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))) P) x5) x00)) (((eq (iS->b)) x4) Xg)) (fun (x5:(((eq b) (Xg c0)) c02)) (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x4)))) as proof of (((eq (iS->b)) x4) Xg)
% Found eq_ref00:=(eq_ref0 Xg):(((eq (iS->b)) Xg) Xg)
% Found (eq_ref0 Xg) as proof of (((eq (iS->b)) Xg) x4)
% Found ((eq_ref (iS->b)) Xg) as proof of (((eq (iS->b)) Xg) x4)
% Found ((eq_ref (iS->b)) Xg) as proof of (((eq (iS->b)) Xg) x4)
% Found ((eq_ref (iS->b)) Xg) as proof of (((eq (iS->b)) Xg) x4)
% Found (eq_sym000 ((eq_ref (iS->b)) Xg)) as proof of (((eq (iS->b)) x4) Xg)
% Found ((eq_sym00 x4) ((eq_ref (iS->b)) Xg)) as proof of (((eq (iS->b)) x4) Xg)
% Found (((eq_sym0 Xg) x4) ((eq_ref (iS->b)) Xg)) as proof of (((eq (iS->b)) x4) Xg)
% Found ((((eq_sym (iS->b)) Xg) x4) ((eq_ref (iS->b)) Xg)) as proof of (((eq (iS->b)) x4) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((((eq_sym (iS->b)) Xg) x4) ((eq_ref (iS->b)) Xg))) as proof of (((eq (iS->b)) x4) Xg)
% Found x7:(P x0)
% Instantiate: x0:=x':(iS->b)
% Found (fun (x7:(P x0))=> x7) as proof of (P x')
% Found (fun (P:((iS->b)->Prop)) (x7:(P x0))=> x7) as proof of ((P x0)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop)) (x7:(P x0))=> x7) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of ((P x0)->(P x'))
% Found ((eq_ref0 x0) P) as proof of ((P x0)->(P x'))
% Found (((eq_ref (iS->b)) x0) P) as proof of ((P x0)->(P x'))
% Found (((eq_ref (iS->b)) x0) P) as proof of ((P x0)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x0) P)) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% Found (eq_ref00 P) as proof of ((P x2)->(P x'))
% Found ((eq_ref0 x2) P) as proof of ((P x2)->(P x'))
% Found (((eq_ref (iS->b)) x2) P) as proof of ((P x2)->(P x'))
% Found (((eq_ref (iS->b)) x2) P) as proof of ((P x2)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x2) P)) as proof of (((eq (iS->b)) x2) x')
% Found x7:(P x2)
% Instantiate: x2:=x':(iS->b)
% Found (fun (x7:(P x2))=> x7) as proof of (P x')
% Found (fun (P:((iS->b)->Prop)) (x7:(P x2))=> x7) as proof of ((P x2)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop)) (x7:(P x2))=> x7) as proof of (((eq (iS->b)) x2) x')
% Found x7:(P x4)
% Instantiate: x4:=x':(iS->b)
% Found (fun (x7:(P x4))=> x7) as proof of (P x')
% Found (fun (P:((iS->b)->Prop)) (x7:(P x4))=> x7) as proof of ((P x4)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop)) (x7:(P x4))=> x7) as proof of (((eq (iS->b)) x4) x')
% Found eq_ref000:=(eq_ref00 P):((P x4)->(P x4))
% Found (eq_ref00 P) as proof of ((P x4)->(P x'))
% Found ((eq_ref0 x4) P) as proof of ((P x4)->(P x'))
% Found (((eq_ref (iS->b)) x4) P) as proof of ((P x4)->(P x'))
% Found (((eq_ref (iS->b)) x4) P) as proof of ((P x4)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x4) P)) as proof of ((P x4)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x4) P)) as proof of (((eq (iS->b)) x4) x')
% Found eq_ref00:=(eq_ref0 (x0 c0)):(((eq b) (x0 c0)) (x0 c0))
% Found (eq_ref0 (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found eq_ref00:=(eq_ref0 (x2 c0)):(((eq b) (x2 c0)) (x2 c0))
% Found (eq_ref0 (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))):(((eq ((iS->b)->Prop)) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) (fun (x:(iS->b))=> (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x)))
% Found (eta_expansion_dep00 ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) as proof of (((eq ((iS->b)->Prop)) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) b0)
% Found ((eta_expansion_dep0 (fun (x5:(iS->b))=> Prop)) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) as proof of (((eq ((iS->b)->Prop)) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) b0)
% Found (((eta_expansion_dep (iS->b)) (fun (x5:(iS->b))=> Prop)) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) as proof of (((eq ((iS->b)->Prop)) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) b0)
% Found (((eta_expansion_dep (iS->b)) (fun (x5:(iS->b))=> Prop)) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) as proof of (((eq ((iS->b)->Prop)) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) b0)
% Found (((eta_expansion_dep (iS->b)) (fun (x5:(iS->b))=> Prop)) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) as proof of (((eq ((iS->b)->Prop)) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) b0)
% Found eq_ref00:=(eq_ref0 (x0 x3)):(((eq b) (x0 x3)) (x0 x3))
% Found (eq_ref0 (x0 x3)) as proof of (((eq b) (x0 x3)) (x' x3))
% Found ((eq_ref b) (x0 x3)) as proof of (((eq b) (x0 x3)) (x' x3))
% Found ((eq_ref b) (x0 x3)) as proof of (((eq b) (x0 x3)) (x' x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x0 x3))) as proof of (((eq b) (x0 x3)) (x' x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x0 x3))) as proof of (forall (x:iS), (((eq b) (x0 x)) (x' x)))
% Found (functional_extensionality0000 (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of (((eq (iS->b)) x0) x')
% Found ((functional_extensionality000 x') (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of (((eq (iS->b)) x0) x')
% Found (((functional_extensionality00 x0) x') (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of (((eq (iS->b)) x0) x')
% Found ((((functional_extensionality0 b) x0) x') (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of (((eq (iS->b)) x0) x')
% Found (((((functional_extensionality iS) b) x0) x') (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((((functional_extensionality iS) b) x0) x') (fun (x3:iS)=> ((eq_ref b) (x0 x3))))) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 (x0 x3)):(((eq b) (x0 x3)) (x0 x3))
% Found (eq_ref0 (x0 x3)) as proof of (((eq b) (x0 x3)) (x' x3))
% Found ((eq_ref b) (x0 x3)) as proof of (((eq b) (x0 x3)) (x' x3))
% Found ((eq_ref b) (x0 x3)) as proof of (((eq b) (x0 x3)) (x' x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x0 x3))) as proof of (((eq b) (x0 x3)) (x' x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x0 x3))) as proof of (forall (x:iS), (((eq b) (x0 x)) (x' x)))
% Found (functional_extensionality_dep0000 (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of (((eq (iS->b)) x0) x')
% Found ((functional_extensionality_dep000 x') (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of (((eq (iS->b)) x0) x')
% Found (((functional_extensionality_dep00 x0) x') (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of (((eq (iS->b)) x0) x')
% Found ((((functional_extensionality_dep0 (fun (x5:iS)=> b)) x0) x') (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of (((eq (iS->b)) x0) x')
% Found (((((functional_extensionality_dep iS) (fun (x5:iS)=> b)) x0) x') (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((((functional_extensionality_dep iS) (fun (x5:iS)=> b)) x0) x') (fun (x3:iS)=> ((eq_ref b) (x0 x3))))) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 (x0 c0)):(((eq b) (x0 c0)) (x0 c0))
% Found (eq_ref0 (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and ((and (((eq b) (x4 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x4 ((cP Xx) Xy))) ((cP2 (x4 Xx)) (x4 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x4) Xg)))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((and (((eq b) (x4 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x4 ((cP Xx) Xy))) ((cP2 (x4 Xx)) (x4 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x4) Xg)))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((and (((eq b) (x4 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x4 ((cP Xx) Xy))) ((cP2 (x4 Xx)) (x4 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x4) Xg)))))
% Found (fun (x4:(iS->b))=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((and (((eq b) (x4 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x4 ((cP Xx) Xy))) ((cP2 (x4 Xx)) (x4 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x4) Xg)))))
% Found (fun (x4:(iS->b))=> ((eq_ref Prop) (f x4))) as proof of (forall (x:(iS->b)), (((eq Prop) (f x)) ((and ((and (((eq b) (x c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x ((cP Xx) Xy))) ((cP2 (x Xx)) (x Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x) Xg))))))
% Found eq_ref00:=(eq_ref0 x0):(((eq (iS->b)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (iS->b)) x0) Xg)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) Xg)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x0)) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x5:(((eq b) (Xg c0)) c02)) (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x0)) as proof of ((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->(((eq (iS->b)) x0) Xg))
% Found (fun (x5:(((eq b) (Xg c0)) c02)) (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x0)) as proof of ((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->(((eq (iS->b)) x0) Xg)))
% Found (and_rect20 (fun (x5:(((eq b) (Xg c0)) c02)) (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) Xg)
% Found ((and_rect2 (((eq (iS->b)) x0) Xg)) (fun (x5:(((eq b) (Xg c0)) c02)) (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) Xg)
% Found (((fun (P:Type) (x5:((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->P)))=> (((((and_rect (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))) P) x5) x00)) (((eq (iS->b)) x0) Xg)) (fun (x5:(((eq b) (Xg c0)) c02)) (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((fun (P:Type) (x5:((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->P)))=> (((((and_rect (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))) P) x5) x00)) (((eq (iS->b)) x0) Xg)) (fun (x5:(((eq b) (Xg c0)) c02)) (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x0)))) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and ((and (((eq b) (x4 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x4 ((cP Xx) Xy))) ((cP2 (x4 Xx)) (x4 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x4) Xg)))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((and (((eq b) (x4 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x4 ((cP Xx) Xy))) ((cP2 (x4 Xx)) (x4 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x4) Xg)))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((and (((eq b) (x4 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x4 ((cP Xx) Xy))) ((cP2 (x4 Xx)) (x4 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x4) Xg)))))
% Found (fun (x4:(iS->b))=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((and (((eq b) (x4 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x4 ((cP Xx) Xy))) ((cP2 (x4 Xx)) (x4 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x4) Xg)))))
% Found (fun (x4:(iS->b))=> ((eq_ref Prop) (f x4))) as proof of (forall (x:(iS->b)), (((eq Prop) (f x)) ((and ((and (((eq b) (x c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x ((cP Xx) Xy))) ((cP2 (x Xx)) (x Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x) Xg))))))
% Found eq_ref00:=(eq_ref0 x2):(((eq (iS->b)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (iS->b)) x2) Xg)
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) Xg)
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) Xg)
% Found (fun (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x2)) as proof of (((eq (iS->b)) x2) Xg)
% Found (fun (x5:(((eq b) (Xg c0)) c02)) (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x2)) as proof of ((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->(((eq (iS->b)) x2) Xg))
% Found (fun (x5:(((eq b) (Xg c0)) c02)) (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x2)) as proof of ((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->(((eq (iS->b)) x2) Xg)))
% Found (and_rect20 (fun (x5:(((eq b) (Xg c0)) c02)) (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) Xg)
% Found ((and_rect2 (((eq (iS->b)) x2) Xg)) (fun (x5:(((eq b) (Xg c0)) c02)) (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) Xg)
% Found (((fun (P:Type) (x5:((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->P)))=> (((((and_rect (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))) P) x5) x00)) (((eq (iS->b)) x2) Xg)) (fun (x5:(((eq b) (Xg c0)) c02)) (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((fun (P:Type) (x5:((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->P)))=> (((((and_rect (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))) P) x5) x00)) (((eq (iS->b)) x2) Xg)) (fun (x5:(((eq b) (Xg c0)) c02)) (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x2)))) as proof of (((eq (iS->b)) x2) Xg)
% Found eta_expansion000:=(eta_expansion00 Xg):(((eq (iS->b)) Xg) (fun (x:iS)=> (Xg x)))
% Found (eta_expansion00 Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found ((eta_expansion0 b) Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found (((eta_expansion iS) b) Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found (((eta_expansion iS) b) Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found (((eta_expansion iS) b) Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found (eq_sym000 (((eta_expansion iS) b) Xg)) as proof of (((eq (iS->b)) x0) Xg)
% Found ((eq_sym00 x0) (((eta_expansion iS) b) Xg)) as proof of (((eq (iS->b)) x0) Xg)
% Found (((eq_sym0 Xg) x0) (((eta_expansion iS) b) Xg)) as proof of (((eq (iS->b)) x0) Xg)
% Found ((((eq_sym (iS->b)) Xg) x0) (((eta_expansion iS) b) Xg)) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((((eq_sym (iS->b)) Xg) x0) (((eta_expansion iS) b) Xg))) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 Xg):(((eq (iS->b)) Xg) Xg)
% Found (eq_ref0 Xg) as proof of (((eq (iS->b)) Xg) x2)
% Found ((eq_ref (iS->b)) Xg) as proof of (((eq (iS->b)) Xg) x2)
% Found ((eq_ref (iS->b)) Xg) as proof of (((eq (iS->b)) Xg) x2)
% Found ((eq_ref (iS->b)) Xg) as proof of (((eq (iS->b)) Xg) x2)
% Found (eq_sym000 ((eq_ref (iS->b)) Xg)) as proof of (((eq (iS->b)) x2) Xg)
% Found ((eq_sym00 x2) ((eq_ref (iS->b)) Xg)) as proof of (((eq (iS->b)) x2) Xg)
% Found (((eq_sym0 Xg) x2) ((eq_ref (iS->b)) Xg)) as proof of (((eq (iS->b)) x2) Xg)
% Found ((((eq_sym (iS->b)) Xg) x2) ((eq_ref (iS->b)) Xg)) as proof of (((eq (iS->b)) x2) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((((eq_sym (iS->b)) Xg) x2) ((eq_ref (iS->b)) Xg))) as proof of (((eq (iS->b)) x2) Xg)
% Found eq_ref00:=(eq_ref0 x4):(((eq (iS->b)) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq (iS->b)) x4) x')
% Found ((eq_ref (iS->b)) x4) as proof of (((eq (iS->b)) x4) x')
% Found ((eq_ref (iS->b)) x4) as proof of (((eq (iS->b)) x4) x')
% Found (fun (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x4)) as proof of (((eq (iS->b)) x4) x')
% Found (fun (x5:(((eq b) (x' c0)) c02)) (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x4)) as proof of ((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->(((eq (iS->b)) x4) x'))
% Found (fun (x5:(((eq b) (x' c0)) c02)) (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x4)) as proof of ((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->(((eq (iS->b)) x4) x')))
% Found (and_rect20 (fun (x5:(((eq b) (x' c0)) c02)) (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x4))) as proof of (((eq (iS->b)) x4) x')
% Found ((and_rect2 (((eq (iS->b)) x4) x')) (fun (x5:(((eq b) (x' c0)) c02)) (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x4))) as proof of (((eq (iS->b)) x4) x')
% Found (((fun (P:Type) (x5:((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->P)))=> (((((and_rect (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))) P) x5) x00)) (((eq (iS->b)) x4) x')) (fun (x5:(((eq b) (x' c0)) c02)) (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x4))) as proof of (((eq (iS->b)) x4) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((fun (P:Type) (x5:((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->P)))=> (((((and_rect (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))) P) x5) x00)) (((eq (iS->b)) x4) x')) (fun (x5:(((eq b) (x' c0)) c02)) (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x4)))) as proof of (((eq (iS->b)) x4) x')
% Found eq_ref00:=(eq_ref0 x8):(((eq (iS->b)) x8) x8)
% Found (eq_ref0 x8) as proof of (((eq (iS->b)) x8) Xg)
% Found ((eq_ref (iS->b)) x8) as proof of (((eq (iS->b)) x8) Xg)
% Found ((eq_ref (iS->b)) x8) as proof of (((eq (iS->b)) x8) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((eq_ref (iS->b)) x8)) as proof of (((eq (iS->b)) x8) Xg)
% Found eta_expansion000:=(eta_expansion00 x'):(((eq (iS->b)) x') (fun (x:iS)=> (x' x)))
% Found (eta_expansion00 x') as proof of (((eq (iS->b)) x') x4)
% Found ((eta_expansion0 b) x') as proof of (((eq (iS->b)) x') x4)
% Found (((eta_expansion iS) b) x') as proof of (((eq (iS->b)) x') x4)
% Found (((eta_expansion iS) b) x') as proof of (((eq (iS->b)) x') x4)
% Found (((eta_expansion iS) b) x') as proof of (((eq (iS->b)) x') x4)
% Found (eq_sym000 (((eta_expansion iS) b) x')) as proof of (((eq (iS->b)) x4) x')
% Found ((eq_sym00 x4) (((eta_expansion iS) b) x')) as proof of (((eq (iS->b)) x4) x')
% Found (((eq_sym0 x') x4) (((eta_expansion iS) b) x')) as proof of (((eq (iS->b)) x4) x')
% Found ((((eq_sym (iS->b)) x') x4) (((eta_expansion iS) b) x')) as proof of (((eq (iS->b)) x4) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((((eq_sym (iS->b)) x') x4) (((eta_expansion iS) b) x'))) as proof of (((eq (iS->b)) x4) x')
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x4))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x4))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x4))
% Found (fun (x4:(iS->b))=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x4))
% Found (fun (x4:(iS->b))=> ((eq_ref Prop) (f x4))) as proof of (forall (x:(iS->b)), (((eq Prop) (f x)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x)))
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x4))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x4))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x4))
% Found (fun (x4:(iS->b))=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x4))
% Found (fun (x4:(iS->b))=> ((eq_ref Prop) (f x4))) as proof of (forall (x:(iS->b)), (((eq Prop) (f x)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x)))
% Found eq_ref00:=(eq_ref0 x0):(((eq (iS->b)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (iS->b)) x0) Xg)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) Xg)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x6:(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)))))=> ((eq_ref (iS->b)) x0)) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x5:((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)))))) (x6:(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)))))=> ((eq_ref (iS->b)) x0)) as proof of ((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))))->(((eq (iS->b)) x0) Xg))
% Found (fun (x5:((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)))))) (x6:(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)))))=> ((eq_ref (iS->b)) x0)) as proof of (((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))))->(((eq (iS->b)) x0) Xg)))
% Found (and_rect20 (fun (x5:((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)))))) (x6:(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)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) Xg)
% Found ((and_rect2 (((eq (iS->b)) x0) Xg)) (fun (x5:((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)))))) (x6:(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)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) Xg)
% Found (((fun (P:Type) (x5:(((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))))->P)))=> (((((and_rect ((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))))) P) x5) x3)) (((eq (iS->b)) x0) Xg)) (fun (x5:((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)))))) (x6:(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)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((fun (P:Type) (x5:(((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))))->P)))=> (((((and_rect ((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))))) P) x5) x3)) (((eq (iS->b)) x0) Xg)) (fun (x5:((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)))))) (x6:(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)))))=> ((eq_ref (iS->b)) x0)))) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 x2):(((eq (iS->b)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (iS->b)) x2) Xg)
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) Xg)
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) Xg)
% Found (fun (x6:(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)))))=> ((eq_ref (iS->b)) x2)) as proof of (((eq (iS->b)) x2) Xg)
% Found (fun (x5:((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)))))) (x6:(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)))))=> ((eq_ref (iS->b)) x2)) as proof of ((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))))->(((eq (iS->b)) x2) Xg))
% Found (fun (x5:((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)))))) (x6:(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)))))=> ((eq_ref (iS->b)) x2)) as proof of (((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))))->(((eq (iS->b)) x2) Xg)))
% Found (and_rect20 (fun (x5:((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)))))) (x6:(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)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) Xg)
% Found ((and_rect2 (((eq (iS->b)) x2) Xg)) (fun (x5:((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)))))) (x6:(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)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) Xg)
% Found (((fun (P:Type) (x5:(((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))))->P)))=> (((((and_rect ((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))))) P) x5) x3)) (((eq (iS->b)) x2) Xg)) (fun (x5:((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)))))) (x6:(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)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((fun (P:Type) (x5:(((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))))->P)))=> (((((and_rect ((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))))) P) x5) x3)) (((eq (iS->b)) x2) Xg)) (fun (x5:((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)))))) (x6:(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)))))=> ((eq_ref (iS->b)) x2)))) as proof of (((eq (iS->b)) x2) Xg)
% Found eq_ref00:=(eq_ref0 x4):(((eq (iS->b)) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq (iS->b)) x4) Xg)
% Found ((eq_ref (iS->b)) x4) as proof of (((eq (iS->b)) x4) Xg)
% Found ((eq_ref (iS->b)) x4) as proof of (((eq (iS->b)) x4) Xg)
% Found (fun (x6:(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)))))=> ((eq_ref (iS->b)) x4)) as proof of (((eq (iS->b)) x4) Xg)
% Found (fun (x5:((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)))))) (x6:(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)))))=> ((eq_ref (iS->b)) x4)) as proof of ((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))))->(((eq (iS->b)) x4) Xg))
% Found (fun (x5:((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)))))) (x6:(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)))))=> ((eq_ref (iS->b)) x4)) as proof of (((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))))->(((eq (iS->b)) x4) Xg)))
% Found (and_rect20 (fun (x5:((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)))))) (x6:(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)))))=> ((eq_ref (iS->b)) x4))) as proof of (((eq (iS->b)) x4) Xg)
% Found ((and_rect2 (((eq (iS->b)) x4) Xg)) (fun (x5:((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)))))) (x6:(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)))))=> ((eq_ref (iS->b)) x4))) as proof of (((eq (iS->b)) x4) Xg)
% Found (((fun (P:Type) (x5:(((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))))->P)))=> (((((and_rect ((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))))) P) x5) x2)) (((eq (iS->b)) x4) Xg)) (fun (x5:((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)))))) (x6:(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)))))=> ((eq_ref (iS->b)) x4))) as proof of (((eq (iS->b)) x4) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((fun (P:Type) (x5:(((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))))->P)))=> (((((and_rect ((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))))) P) x5) x2)) (((eq (iS->b)) x4) Xg)) (fun (x5:((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)))))) (x6:(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)))))=> ((eq_ref (iS->b)) x4)))) as proof of (((eq (iS->b)) x4) Xg)
% Found eq_ref00:=(eq_ref0 x0):(((eq (iS->b)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (iS->b)) x0) Xg)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) Xg)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((eq_ref (iS->b)) x0)) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 x0):(((eq (iS->b)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (iS->b)) x0) x')
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) x')
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x0)) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x5:(((eq b) (x' c0)) c02)) (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x0)) as proof of ((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->(((eq (iS->b)) x0) x'))
% Found (fun (x5:(((eq b) (x' c0)) c02)) (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x0)) as proof of ((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->(((eq (iS->b)) x0) x')))
% Found (and_rect20 (fun (x5:(((eq b) (x' c0)) c02)) (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) x')
% Found ((and_rect2 (((eq (iS->b)) x0) x')) (fun (x5:(((eq b) (x' c0)) c02)) (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) x')
% Found (((fun (P:Type) (x5:((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->P)))=> (((((and_rect (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))) P) x5) x00)) (((eq (iS->b)) x0) x')) (fun (x5:(((eq b) (x' c0)) c02)) (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((fun (P:Type) (x5:((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->P)))=> (((((and_rect (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))) P) x5) x00)) (((eq (iS->b)) x0) x')) (fun (x5:(((eq b) (x' c0)) c02)) (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x0)))) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 x2):(((eq (iS->b)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (iS->b)) x2) Xg)
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) Xg)
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((eq_ref (iS->b)) x2)) as proof of (((eq (iS->b)) x2) Xg)
% Found eq_ref00:=(eq_ref0 x2):(((eq (iS->b)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (iS->b)) x2) x')
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) x')
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) x')
% Found (fun (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x2)) as proof of (((eq (iS->b)) x2) x')
% Found (fun (x5:(((eq b) (x' c0)) c02)) (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x2)) as proof of ((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->(((eq (iS->b)) x2) x'))
% Found (fun (x5:(((eq b) (x' c0)) c02)) (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x2)) as proof of ((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->(((eq (iS->b)) x2) x')))
% Found (and_rect20 (fun (x5:(((eq b) (x' c0)) c02)) (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) x')
% Found ((and_rect2 (((eq (iS->b)) x2) x')) (fun (x5:(((eq b) (x' c0)) c02)) (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) x')
% Found (((fun (P:Type) (x5:((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->P)))=> (((((and_rect (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))) P) x5) x00)) (((eq (iS->b)) x2) x')) (fun (x5:(((eq b) (x' c0)) c02)) (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((fun (P:Type) (x5:((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->P)))=> (((((and_rect (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))) P) x5) x00)) (((eq (iS->b)) x2) x')) (fun (x5:(((eq b) (x' c0)) c02)) (x6:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x2)))) as proof of (((eq (iS->b)) x2) x')
% Found eq_ref00:=(eq_ref0 x4):(((eq (iS->b)) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq (iS->b)) x4) Xg)
% Found ((eq_ref (iS->b)) x4) as proof of (((eq (iS->b)) x4) Xg)
% Found ((eq_ref (iS->b)) x4) as proof of (((eq (iS->b)) x4) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((eq_ref (iS->b)) x4)) as proof of (((eq (iS->b)) x4) Xg)
% Found eq_ref00:=(eq_ref0 x6):(((eq (iS->b)) x6) x6)
% Found (eq_ref0 x6) as proof of (((eq (iS->b)) x6) Xg)
% Found ((eq_ref (iS->b)) x6) as proof of (((eq (iS->b)) x6) Xg)
% Found ((eq_ref (iS->b)) x6) as proof of (((eq (iS->b)) x6) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((eq_ref (iS->b)) x6)) as proof of (((eq (iS->b)) x6) Xg)
% Found eq_ref00:=(eq_ref0 (x6 c0)):(((eq b) (x6 c0)) (x6 c0))
% Found (eq_ref0 (x6 c0)) as proof of (((eq b) (x6 c0)) c02)
% Found ((eq_ref b) (x6 c0)) as proof of (((eq b) (x6 c0)) c02)
% Found ((eq_ref b) (x6 c0)) as proof of (((eq b) (x6 c0)) c02)
% Found ((eq_ref b) (x6 c0)) as proof of (((eq b) (x6 c0)) c02)
% Found eq_ref00:=(eq_ref0 x'):(((eq (iS->b)) x') x')
% Found (eq_ref0 x') as proof of (((eq (iS->b)) x') x0)
% Found ((eq_ref (iS->b)) x') as proof of (((eq (iS->b)) x') x0)
% Found ((eq_ref (iS->b)) x') as proof of (((eq (iS->b)) x') x0)
% Found ((eq_ref (iS->b)) x') as proof of (((eq (iS->b)) x') x0)
% Found (eq_sym000 ((eq_ref (iS->b)) x')) as proof of (((eq (iS->b)) x0) x')
% Found ((eq_sym00 x0) ((eq_ref (iS->b)) x')) as proof of (((eq (iS->b)) x0) x')
% Found (((eq_sym0 x') x0) ((eq_ref (iS->b)) x')) as proof of (((eq (iS->b)) x0) x')
% Found ((((eq_sym (iS->b)) x') x0) ((eq_ref (iS->b)) x')) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((((eq_sym (iS->b)) x') x0) ((eq_ref (iS->b)) x'))) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 x'):(((eq (iS->b)) x') x')
% Found (eq_ref0 x') as proof of (((eq (iS->b)) x') x2)
% Found ((eq_ref (iS->b)) x') as proof of (((eq (iS->b)) x') x2)
% Found ((eq_ref (iS->b)) x') as proof of (((eq (iS->b)) x') x2)
% Found ((eq_ref (iS->b)) x') as proof of (((eq (iS->b)) x') x2)
% Found (eq_sym000 ((eq_ref (iS->b)) x')) as proof of (((eq (iS->b)) x2) x')
% Found ((eq_sym00 x2) ((eq_ref (iS->b)) x')) as proof of (((eq (iS->b)) x2) x')
% Found (((eq_sym0 x') x2) ((eq_ref (iS->b)) x')) as proof of (((eq (iS->b)) x2) x')
% Found ((((eq_sym (iS->b)) x') x2) ((eq_ref (iS->b)) x')) as proof of (((eq (iS->b)) x2) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((((eq_sym (iS->b)) x') x2) ((eq_ref (iS->b)) x'))) as proof of (((eq (iS->b)) x2) x')
% Found eq_ref00:=(eq_ref0 (x4 x5)):(((eq b) (x4 x5)) (x4 x5))
% Found (eq_ref0 (x4 x5)) as proof of (((eq b) (x4 x5)) (Xg x5))
% Found ((eq_ref b) (x4 x5)) as proof of (((eq b) (x4 x5)) (Xg x5))
% Found ((eq_ref b) (x4 x5)) as proof of (((eq b) (x4 x5)) (Xg x5))
% Found (fun (x5:iS)=> ((eq_ref b) (x4 x5))) as proof of (((eq b) (x4 x5)) (Xg x5))
% Found (fun (x5:iS)=> ((eq_ref b) (x4 x5))) as proof of (forall (x:iS), (((eq b) (x4 x)) (Xg x)))
% Found (functional_extensionality_dep0000 (fun (x5:iS)=> ((eq_ref b) (x4 x5)))) as proof of (((eq (iS->b)) x4) Xg)
% Found ((functional_extensionality_dep000 Xg) (fun (x5:iS)=> ((eq_ref b) (x4 x5)))) as proof of (((eq (iS->b)) x4) Xg)
% Found (((functional_extensionality_dep00 x4) Xg) (fun (x5:iS)=> ((eq_ref b) (x4 x5)))) as proof of (((eq (iS->b)) x4) Xg)
% Found ((((functional_extensionality_dep0 (fun (x7:iS)=> b)) x4) Xg) (fun (x5:iS)=> ((eq_ref b) (x4 x5)))) as proof of (((eq (iS->b)) x4) Xg)
% Found (((((functional_extensionality_dep iS) (fun (x7:iS)=> b)) x4) Xg) (fun (x5:iS)=> ((eq_ref b) (x4 x5)))) as proof of (((eq (iS->b)) x4) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((((functional_extensionality_dep iS) (fun (x7:iS)=> b)) x4) Xg) (fun (x5:iS)=> ((eq_ref b) (x4 x5))))) as proof of (((eq (iS->b)) x4) Xg)
% Found eq_ref00:=(eq_ref0 (x4 x5)):(((eq b) (x4 x5)) (x4 x5))
% Found (eq_ref0 (x4 x5)) as proof of (((eq b) (x4 x5)) (Xg x5))
% Found ((eq_ref b) (x4 x5)) as proof of (((eq b) (x4 x5)) (Xg x5))
% Found ((eq_ref b) (x4 x5)) as proof of (((eq b) (x4 x5)) (Xg x5))
% Found (fun (x5:iS)=> ((eq_ref b) (x4 x5))) as proof of (((eq b) (x4 x5)) (Xg x5))
% Found (fun (x5:iS)=> ((eq_ref b) (x4 x5))) as proof of (forall (x:iS), (((eq b) (x4 x)) (Xg x)))
% Found (functional_extensionality0000 (fun (x5:iS)=> ((eq_ref b) (x4 x5)))) as proof of (((eq (iS->b)) x4) Xg)
% Found ((functional_extensionality000 Xg) (fun (x5:iS)=> ((eq_ref b) (x4 x5)))) as proof of (((eq (iS->b)) x4) Xg)
% Found (((functional_extensionality00 x4) Xg) (fun (x5:iS)=> ((eq_ref b) (x4 x5)))) as proof of (((eq (iS->b)) x4) Xg)
% Found ((((functional_extensionality0 b) x4) Xg) (fun (x5:iS)=> ((eq_ref b) (x4 x5)))) as proof of (((eq (iS->b)) x4) Xg)
% Found (((((functional_extensionality iS) b) x4) Xg) (fun (x5:iS)=> ((eq_ref b) (x4 x5)))) as proof of (((eq (iS->b)) x4) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((((functional_extensionality iS) b) x4) Xg) (fun (x5:iS)=> ((eq_ref b) (x4 x5))))) as proof of (((eq (iS->b)) x4) Xg)
% Found x9:(P x8)
% Instantiate: x8:=Xg:(iS->b)
% Found (fun (x9:(P x8))=> x9) as proof of (P Xg)
% Found (fun (P:((iS->b)->Prop)) (x9:(P x8))=> x9) as proof of ((P x8)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop)) (x9:(P x8))=> x9) as proof of (((eq (iS->b)) x8) Xg)
% Found eq_ref000:=(eq_ref00 P):((P x8)->(P x8))
% Found (eq_ref00 P) as proof of ((P x8)->(P Xg))
% Found ((eq_ref0 x8) P) as proof of ((P x8)->(P Xg))
% Found (((eq_ref (iS->b)) x8) P) as proof of ((P x8)->(P Xg))
% Found (((eq_ref (iS->b)) x8) P) as proof of ((P x8)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x8) P)) as proof of ((P x8)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x8) P)) as proof of (((eq (iS->b)) x8) Xg)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))):(((eq ((iS->b)->Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) (fun (x:(iS->b))=> ((and ((and (((eq b) (x c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x ((cP Xx) Xy))) ((cP2 (x Xx)) (x Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x) Xg))))))
% Found (eta_expansion_dep00 (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) as proof of (((eq ((iS->b)->Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) b0)
% Found ((eta_expansion_dep0 (fun (x7:(iS->b))=> Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) as proof of (((eq ((iS->b)->Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) b0)
% Found (((eta_expansion_dep (iS->b)) (fun (x7:(iS->b))=> Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) as proof of (((eq ((iS->b)->Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) b0)
% Found (((eta_expansion_dep (iS->b)) (fun (x7:(iS->b))=> Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) as proof of (((eq ((iS->b)->Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) b0)
% Found (((eta_expansion_dep (iS->b)) (fun (x7:(iS->b))=> Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) as proof of (((eq ((iS->b)->Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) b0)
% Found eq_ref00:=(eq_ref0 x8):(((eq (iS->b)) x8) x8)
% Found (eq_ref0 x8) as proof of (((eq (iS->b)) x8) x')
% Found ((eq_ref (iS->b)) x8) as proof of (((eq (iS->b)) x8) x')
% Found ((eq_ref (iS->b)) x8) as proof of (((eq (iS->b)) x8) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((eq_ref (iS->b)) x8)) as proof of (((eq (iS->b)) x8) x')
% Found eq_ref00:=(eq_ref0 (x0 c0)):(((eq b) (x0 c0)) (x0 c0))
% Found (eq_ref0 (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found eq_ref00:=(eq_ref0 (x2 c0)):(((eq b) (x2 c0)) (x2 c0))
% Found (eq_ref0 (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found eq_ref00:=(eq_ref0 (x0 c0)):(((eq b) (x0 c0)) (x0 c0))
% Found (eq_ref0 (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found eq_ref00:=(eq_ref0 (x2 c0)):(((eq b) (x2 c0)) (x2 c0))
% Found (eq_ref0 (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found eq_ref00:=(eq_ref0 (x4 c0)):(((eq b) (x4 c0)) (x4 c0))
% Found (eq_ref0 (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found ((eq_ref b) (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found ((eq_ref b) (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found ((eq_ref b) (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found eq_ref00:=(eq_ref0 x0):(((eq (iS->b)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (iS->b)) x0) x')
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) x')
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x6:(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)))))=> ((eq_ref (iS->b)) x0)) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x5:((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)))))) (x6:(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)))))=> ((eq_ref (iS->b)) x0)) as proof of ((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))))->(((eq (iS->b)) x0) x'))
% Found (fun (x5:((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)))))) (x6:(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)))))=> ((eq_ref (iS->b)) x0)) as proof of (((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))))->(((eq (iS->b)) x0) x')))
% Found (and_rect20 (fun (x5:((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)))))) (x6:(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)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) x')
% Found ((and_rect2 (((eq (iS->b)) x0) x')) (fun (x5:((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)))))) (x6:(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)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) x')
% Found (((fun (P:Type) (x5:(((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))))->P)))=> (((((and_rect ((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))))) P) x5) x3)) (((eq (iS->b)) x0) x')) (fun (x5:((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)))))) (x6:(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)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((fun (P:Type) (x5:(((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))))->P)))=> (((((and_rect ((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))))) P) x5) x3)) (((eq (iS->b)) x0) x')) (fun (x5:((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)))))) (x6:(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)))))=> ((eq_ref (iS->b)) x0)))) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 x2):(((eq (iS->b)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (iS->b)) x2) x')
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) x')
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) x')
% Found (fun (x6:(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)))))=> ((eq_ref (iS->b)) x2)) as proof of (((eq (iS->b)) x2) x')
% Found (fun (x5:((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)))))) (x6:(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)))))=> ((eq_ref (iS->b)) x2)) as proof of ((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))))->(((eq (iS->b)) x2) x'))
% Found (fun (x5:((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)))))) (x6:(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)))))=> ((eq_ref (iS->b)) x2)) as proof of (((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))))->(((eq (iS->b)) x2) x')))
% Found (and_rect20 (fun (x5:((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)))))) (x6:(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)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) x')
% Found ((and_rect2 (((eq (iS->b)) x2) x')) (fun (x5:((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)))))) (x6:(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)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) x')
% Found (((fun (P:Type) (x5:(((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))))->P)))=> (((((and_rect ((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))))) P) x5) x3)) (((eq (iS->b)) x2) x')) (fun (x5:((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)))))) (x6:(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)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((fun (P:Type) (x5:(((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))))->P)))=> (((((and_rect ((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))))) P) x5) x3)) (((eq (iS->b)) x2) x')) (fun (x5:((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)))))) (x6:(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)))))=> ((eq_ref (iS->b)) x2)))) as proof of (((eq (iS->b)) x2) x')
% Found eq_ref00:=(eq_ref0 (x0 x5)):(((eq b) (x0 x5)) (x0 x5))
% Found (eq_ref0 (x0 x5)) as proof of (((eq b) (x0 x5)) (Xg x5))
% Found ((eq_ref b) (x0 x5)) as proof of (((eq b) (x0 x5)) (Xg x5))
% Found ((eq_ref b) (x0 x5)) as proof of (((eq b) (x0 x5)) (Xg x5))
% Found (fun (x5:iS)=> ((eq_ref b) (x0 x5))) as proof of (((eq b) (x0 x5)) (Xg x5))
% Found (fun (x5:iS)=> ((eq_ref b) (x0 x5))) as proof of (forall (x:iS), (((eq b) (x0 x)) (Xg x)))
% Found (functional_extensionality0000 (fun (x5:iS)=> ((eq_ref b) (x0 x5)))) as proof of (((eq (iS->b)) x0) Xg)
% Found ((functional_extensionality000 Xg) (fun (x5:iS)=> ((eq_ref b) (x0 x5)))) as proof of (((eq (iS->b)) x0) Xg)
% Found (((functional_extensionality00 x0) Xg) (fun (x5:iS)=> ((eq_ref b) (x0 x5)))) as proof of (((eq (iS->b)) x0) Xg)
% Found ((((functional_extensionality0 b) x0) Xg) (fun (x5:iS)=> ((eq_ref b) (x0 x5)))) as proof of (((eq (iS->b)) x0) Xg)
% Found (((((functional_extensionality iS) b) x0) Xg) (fun (x5:iS)=> ((eq_ref b) (x0 x5)))) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((((functional_extensionality iS) b) x0) Xg) (fun (x5:iS)=> ((eq_ref b) (x0 x5))))) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 (x0 x5)):(((eq b) (x0 x5)) (x0 x5))
% Found (eq_ref0 (x0 x5)) as proof of (((eq b) (x0 x5)) (Xg x5))
% Found ((eq_ref b) (x0 x5)) as proof of (((eq b) (x0 x5)) (Xg x5))
% Found ((eq_ref b) (x0 x5)) as proof of (((eq b) (x0 x5)) (Xg x5))
% Found (fun (x5:iS)=> ((eq_ref b) (x0 x5))) as proof of (((eq b) (x0 x5)) (Xg x5))
% Found (fun (x5:iS)=> ((eq_ref b) (x0 x5))) as proof of (forall (x:iS), (((eq b) (x0 x)) (Xg x)))
% Found (functional_extensionality_dep0000 (fun (x5:iS)=> ((eq_ref b) (x0 x5)))) as proof of (((eq (iS->b)) x0) Xg)
% Found ((functional_extensionality_dep000 Xg) (fun (x5:iS)=> ((eq_ref b) (x0 x5)))) as proof of (((eq (iS->b)) x0) Xg)
% Found (((functional_extensionality_dep00 x0) Xg) (fun (x5:iS)=> ((eq_ref b) (x0 x5)))) as proof of (((eq (iS->b)) x0) Xg)
% Found ((((functional_extensionality_dep0 (fun (x7:iS)=> b)) x0) Xg) (fun (x5:iS)=> ((eq_ref b) (x0 x5)))) as proof of (((eq (iS->b)) x0) Xg)
% Found (((((functional_extensionality_dep iS) (fun (x7:iS)=> b)) x0) Xg) (fun (x5:iS)=> ((eq_ref b) (x0 x5)))) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((((functional_extensionality_dep iS) (fun (x7:iS)=> b)) x0) Xg) (fun (x5:iS)=> ((eq_ref b) (x0 x5))))) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 x4):(((eq (iS->b)) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq (iS->b)) x4) x')
% Found ((eq_ref (iS->b)) x4) as proof of (((eq (iS->b)) x4) x')
% Found ((eq_ref (iS->b)) x4) as proof of (((eq (iS->b)) x4) x')
% Found (fun (x6:(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)))))=> ((eq_ref (iS->b)) x4)) as proof of (((eq (iS->b)) x4) x')
% Found (fun (x5:((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)))))) (x6:(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)))))=> ((eq_ref (iS->b)) x4)) as proof of ((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))))->(((eq (iS->b)) x4) x'))
% Found (fun (x5:((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)))))) (x6:(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)))))=> ((eq_ref (iS->b)) x4)) as proof of (((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))))->(((eq (iS->b)) x4) x')))
% Found (and_rect20 (fun (x5:((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)))))) (x6:(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)))))=> ((eq_ref (iS->b)) x4))) as proof of (((eq (iS->b)) x4) x')
% Found ((and_rect2 (((eq (iS->b)) x4) x')) (fun (x5:((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)))))) (x6:(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)))))=> ((eq_ref (iS->b)) x4))) as proof of (((eq (iS->b)) x4) x')
% Found (((fun (P:Type) (x5:(((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))))->P)))=> (((((and_rect ((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))))) P) x5) x2)) (((eq (iS->b)) x4) x')) (fun (x5:((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)))))) (x6:(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)))))=> ((eq_ref (iS->b)) x4))) as proof of (((eq (iS->b)) x4) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((fun (P:Type) (x5:(((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))))->P)))=> (((((and_rect ((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))))) P) x5) x2)) (((eq (iS->b)) x4) x')) (fun (x5:((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)))))) (x6:(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)))))=> ((eq_ref (iS->b)) x4)))) as proof of (((eq (iS->b)) x4) x')
% Found eq_ref00:=(eq_ref0 (x2 x5)):(((eq b) (x2 x5)) (x2 x5))
% Found (eq_ref0 (x2 x5)) as proof of (((eq b) (x2 x5)) (Xg x5))
% Found ((eq_ref b) (x2 x5)) as proof of (((eq b) (x2 x5)) (Xg x5))
% Found ((eq_ref b) (x2 x5)) as proof of (((eq b) (x2 x5)) (Xg x5))
% Found (fun (x5:iS)=> ((eq_ref b) (x2 x5))) as proof of (((eq b) (x2 x5)) (Xg x5))
% Found (fun (x5:iS)=> ((eq_ref b) (x2 x5))) as proof of (forall (x:iS), (((eq b) (x2 x)) (Xg x)))
% Found (functional_extensionality_dep0000 (fun (x5:iS)=> ((eq_ref b) (x2 x5)))) as proof of (((eq (iS->b)) x2) Xg)
% Found ((functional_extensionality_dep000 Xg) (fun (x5:iS)=> ((eq_ref b) (x2 x5)))) as proof of (((eq (iS->b)) x2) Xg)
% Found (((functional_extensionality_dep00 x2) Xg) (fun (x5:iS)=> ((eq_ref b) (x2 x5)))) as proof of (((eq (iS->b)) x2) Xg)
% Found ((((functional_extensionality_dep0 (fun (x7:iS)=> b)) x2) Xg) (fun (x5:iS)=> ((eq_ref b) (x2 x5)))) as proof of (((eq (iS->b)) x2) Xg)
% Found (((((functional_extensionality_dep iS) (fun (x7:iS)=> b)) x2) Xg) (fun (x5:iS)=> ((eq_ref b) (x2 x5)))) as proof of (((eq (iS->b)) x2) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((((functional_extensionality_dep iS) (fun (x7:iS)=> b)) x2) Xg) (fun (x5:iS)=> ((eq_ref b) (x2 x5))))) as proof of (((eq (iS->b)) x2) Xg)
% Found eq_ref00:=(eq_ref0 (x2 x5)):(((eq b) (x2 x5)) (x2 x5))
% Found (eq_ref0 (x2 x5)) as proof of (((eq b) (x2 x5)) (Xg x5))
% Found ((eq_ref b) (x2 x5)) as proof of (((eq b) (x2 x5)) (Xg x5))
% Found ((eq_ref b) (x2 x5)) as proof of (((eq b) (x2 x5)) (Xg x5))
% Found (fun (x5:iS)=> ((eq_ref b) (x2 x5))) as proof of (((eq b) (x2 x5)) (Xg x5))
% Found (fun (x5:iS)=> ((eq_ref b) (x2 x5))) as proof of (forall (x:iS), (((eq b) (x2 x)) (Xg x)))
% Found (functional_extensionality0000 (fun (x5:iS)=> ((eq_ref b) (x2 x5)))) as proof of (((eq (iS->b)) x2) Xg)
% Found ((functional_extensionality000 Xg) (fun (x5:iS)=> ((eq_ref b) (x2 x5)))) as proof of (((eq (iS->b)) x2) Xg)
% Found (((functional_extensionality00 x2) Xg) (fun (x5:iS)=> ((eq_ref b) (x2 x5)))) as proof of (((eq (iS->b)) x2) Xg)
% Found ((((functional_extensionality0 b) x2) Xg) (fun (x5:iS)=> ((eq_ref b) (x2 x5)))) as proof of (((eq (iS->b)) x2) Xg)
% Found (((((functional_extensionality iS) b) x2) Xg) (fun (x5:iS)=> ((eq_ref b) (x2 x5)))) as proof of (((eq (iS->b)) x2) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((((functional_extensionality iS) b) x2) Xg) (fun (x5:iS)=> ((eq_ref b) (x2 x5))))) as proof of (((eq (iS->b)) 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 (iS->b)) x0) P) as proof of ((P x0)->(P Xg))
% Found (((eq_ref (iS->b)) x0) P) as proof of ((P x0)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x0) P)) as proof of (((eq (iS->b)) x0) Xg)
% Found x9:(P x0)
% Instantiate: x0:=Xg:(iS->b)
% Found (fun (x9:(P x0))=> x9) as proof of (P Xg)
% Found (fun (P:((iS->b)->Prop)) (x9:(P x0))=> x9) as proof of ((P x0)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop)) (x9:(P x0))=> x9) as proof of (((eq (iS->b)) 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 (iS->b)) x2) P) as proof of ((P x2)->(P Xg))
% Found (((eq_ref (iS->b)) x2) P) as proof of ((P x2)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x2) P)) as proof of (((eq (iS->b)) x2) Xg)
% Found x9:(P x2)
% Instantiate: x2:=Xg:(iS->b)
% Found (fun (x9:(P x2))=> x9) as proof of (P Xg)
% Found (fun (P:((iS->b)->Prop)) (x9:(P x2))=> x9) as proof of ((P x2)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop)) (x9:(P x2))=> x9) as proof of (((eq (iS->b)) x2) Xg)
% Found x9:(P x4)
% Instantiate: x4:=Xg:(iS->b)
% Found (fun (x9:(P x4))=> x9) as proof of (P Xg)
% Found (fun (P:((iS->b)->Prop)) (x9:(P x4))=> x9) as proof of ((P x4)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop)) (x9:(P x4))=> x9) as proof of (((eq (iS->b)) x4) 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 (iS->b)) x4) P) as proof of ((P x4)->(P Xg))
% Found (((eq_ref (iS->b)) x4) P) as proof of ((P x4)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x4) P)) as proof of ((P x4)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x4) P)) as proof of (((eq (iS->b)) 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 (iS->b)) x6) P) as proof of ((P x6)->(P Xg))
% Found (((eq_ref (iS->b)) x6) P) as proof of ((P x6)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x6) P)) as proof of ((P x6)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x6) P)) as proof of (((eq (iS->b)) x6) Xg)
% Found x9:(P x6)
% Instantiate: x6:=Xg:(iS->b)
% Found (fun (x9:(P x6))=> x9) as proof of (P Xg)
% Found (fun (P:((iS->b)->Prop)) (x9:(P x6))=> x9) as proof of ((P x6)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop)) (x9:(P x6))=> x9) as proof of (((eq (iS->b)) x6) Xg)
% Found eq_ref00:=(eq_ref0 (x6 c0)):(((eq b) (x6 c0)) (x6 c0))
% Found (eq_ref0 (x6 c0)) as proof of (((eq b) (x6 c0)) c02)
% Found ((eq_ref b) (x6 c0)) as proof of (((eq b) (x6 c0)) c02)
% Found ((eq_ref b) (x6 c0)) as proof of (((eq b) (x6 c0)) c02)
% Found ((eq_ref b) (x6 c0)) as proof of (((eq b) (x6 c0)) c02)
% Found eq_ref00:=(eq_ref0 x0):(((eq (iS->b)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (iS->b)) x0) x')
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) x')
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((eq_ref (iS->b)) x0)) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 x2):(((eq (iS->b)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (iS->b)) x2) x')
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) x')
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((eq_ref (iS->b)) x2)) as proof of (((eq (iS->b)) x2) x')
% Found eq_ref00:=(eq_ref0 x4):(((eq (iS->b)) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq (iS->b)) x4) x')
% Found ((eq_ref (iS->b)) x4) as proof of (((eq (iS->b)) x4) x')
% Found ((eq_ref (iS->b)) x4) as proof of (((eq (iS->b)) x4) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((eq_ref (iS->b)) x4)) as proof of (((eq (iS->b)) x4) x')
% Found eq_ref00:=(eq_ref0 x6):(((eq (iS->b)) x6) x6)
% Found (eq_ref0 x6) as proof of (((eq (iS->b)) x6) x')
% Found ((eq_ref (iS->b)) x6) as proof of (((eq (iS->b)) x6) x')
% Found ((eq_ref (iS->b)) x6) as proof of (((eq (iS->b)) x6) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((eq_ref (iS->b)) x6)) as proof of (((eq (iS->b)) x6) x')
% Found eq_ref00:=(eq_ref0 (x4 x5)):(((eq b) (x4 x5)) (x4 x5))
% Found (eq_ref0 (x4 x5)) as proof of (((eq b) (x4 x5)) (x' x5))
% Found ((eq_ref b) (x4 x5)) as proof of (((eq b) (x4 x5)) (x' x5))
% Found ((eq_ref b) (x4 x5)) as proof of (((eq b) (x4 x5)) (x' x5))
% Found (fun (x5:iS)=> ((eq_ref b) (x4 x5))) as proof of (((eq b) (x4 x5)) (x' x5))
% Found (fun (x5:iS)=> ((eq_ref b) (x4 x5))) as proof of (forall (x:iS), (((eq b) (x4 x)) (x' x)))
% Found (functional_extensionality0000 (fun (x5:iS)=> ((eq_ref b) (x4 x5)))) as proof of (((eq (iS->b)) x4) x')
% Found ((functional_extensionality000 x') (fun (x5:iS)=> ((eq_ref b) (x4 x5)))) as proof of (((eq (iS->b)) x4) x')
% Found (((functional_extensionality00 x4) x') (fun (x5:iS)=> ((eq_ref b) (x4 x5)))) as proof of (((eq (iS->b)) x4) x')
% Found ((((functional_extensionality0 b) x4) x') (fun (x5:iS)=> ((eq_ref b) (x4 x5)))) as proof of (((eq (iS->b)) x4) x')
% Found (((((functional_extensionality iS) b) x4) x') (fun (x5:iS)=> ((eq_ref b) (x4 x5)))) as proof of (((eq (iS->b)) x4) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((((functional_extensionality iS) b) x4) x') (fun (x5:iS)=> ((eq_ref b) (x4 x5))))) as proof of (((eq (iS->b)) x4) x')
% Found eq_ref00:=(eq_ref0 (x4 x5)):(((eq b) (x4 x5)) (x4 x5))
% Found (eq_ref0 (x4 x5)) as proof of (((eq b) (x4 x5)) (x' x5))
% Found ((eq_ref b) (x4 x5)) as proof of (((eq b) (x4 x5)) (x' x5))
% Found ((eq_ref b) (x4 x5)) as proof of (((eq b) (x4 x5)) (x' x5))
% Found (fun (x5:iS)=> ((eq_ref b) (x4 x5))) as proof of (((eq b) (x4 x5)) (x' x5))
% Found (fun (x5:iS)=> ((eq_ref b) (x4 x5))) as proof of (forall (x:iS), (((eq b) (x4 x)) (x' x)))
% Found (functional_extensionality_dep0000 (fun (x5:iS)=> ((eq_ref b) (x4 x5)))) as proof of (((eq (iS->b)) x4) x')
% Found ((functional_extensionality_dep000 x') (fun (x5:iS)=> ((eq_ref b) (x4 x5)))) as proof of (((eq (iS->b)) x4) x')
% Found (((functional_extensionality_dep00 x4) x') (fun (x5:iS)=> ((eq_ref b) (x4 x5)))) as proof of (((eq (iS->b)) x4) x')
% Found ((((functional_extensionality_dep0 (fun (x7:iS)=> b)) x4) x') (fun (x5:iS)=> ((eq_ref b) (x4 x5)))) as proof of (((eq (iS->b)) x4) x')
% Found (((((functional_extensionality_dep iS) (fun (x7:iS)=> b)) x4) x') (fun (x5:iS)=> ((eq_ref b) (x4 x5)))) as proof of (((eq (iS->b)) x4) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((((functional_extensionality_dep iS) (fun (x7:iS)=> b)) x4) x') (fun (x5:iS)=> ((eq_ref b) (x4 x5))))) as proof of (((eq (iS->b)) x4) x')
% Found eta_expansion000:=(eta_expansion00 ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))):(((eq ((iS->b)->Prop)) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) (fun (x:(iS->b))=> (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x)))
% Found (eta_expansion00 ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) as proof of (((eq ((iS->b)->Prop)) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) b0)
% Found ((eta_expansion0 Prop) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) as proof of (((eq ((iS->b)->Prop)) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) b0)
% Found (((eta_expansion (iS->b)) Prop) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) as proof of (((eq ((iS->b)->Prop)) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) b0)
% Found (((eta_expansion (iS->b)) Prop) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) as proof of (((eq ((iS->b)->Prop)) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) b0)
% Found (((eta_expansion (iS->b)) Prop) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) as proof of (((eq ((iS->b)->Prop)) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) b0)
% Found eq_ref000:=(eq_ref00 P):((P x8)->(P x8))
% Found (eq_ref00 P) as proof of ((P x8)->(P x'))
% Found ((eq_ref0 x8) P) as proof of ((P x8)->(P x'))
% Found (((eq_ref (iS->b)) x8) P) as proof of ((P x8)->(P x'))
% Found (((eq_ref (iS->b)) x8) P) as proof of ((P x8)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x8) P)) as proof of ((P x8)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x8) P)) as proof of (((eq (iS->b)) x8) x')
% Found x9:(P x8)
% Instantiate: x8:=x':(iS->b)
% Found (fun (x9:(P x8))=> x9) as proof of (P x')
% Found (fun (P:((iS->b)->Prop)) (x9:(P x8))=> x9) as proof of ((P x8)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop)) (x9:(P x8))=> x9) as proof of (((eq (iS->b)) x8) x')
% Found eq_ref00:=(eq_ref0 (x0 c0)):(((eq b) (x0 c0)) (x0 c0))
% Found (eq_ref0 (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found eq_ref00:=(eq_ref0 (x2 c0)):(((eq b) (x2 c0)) (x2 c0))
% Found (eq_ref0 (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found eq_ref00:=(eq_ref0 (f x6)):(((eq Prop) (f x6)) (f x6))
% Found (eq_ref0 (f x6)) as proof of (((eq Prop) (f x6)) ((and ((and (((eq b) (x6 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x6 ((cP Xx) Xy))) ((cP2 (x6 Xx)) (x6 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x6) Xg)))))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) ((and ((and (((eq b) (x6 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x6 ((cP Xx) Xy))) ((cP2 (x6 Xx)) (x6 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x6) Xg)))))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) ((and ((and (((eq b) (x6 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x6 ((cP Xx) Xy))) ((cP2 (x6 Xx)) (x6 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x6) Xg)))))
% Found (fun (x6:(iS->b))=> ((eq_ref Prop) (f x6))) as proof of (((eq Prop) (f x6)) ((and ((and (((eq b) (x6 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x6 ((cP Xx) Xy))) ((cP2 (x6 Xx)) (x6 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x6) Xg)))))
% Found (fun (x6:(iS->b))=> ((eq_ref Prop) (f x6))) as proof of (forall (x:(iS->b)), (((eq Prop) (f x)) ((and ((and (((eq b) (x c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x ((cP Xx) Xy))) ((cP2 (x Xx)) (x Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x) Xg))))))
% Found eq_ref00:=(eq_ref0 (f x6)):(((eq Prop) (f x6)) (f x6))
% Found (eq_ref0 (f x6)) as proof of (((eq Prop) (f x6)) ((and ((and (((eq b) (x6 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x6 ((cP Xx) Xy))) ((cP2 (x6 Xx)) (x6 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x6) Xg)))))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) ((and ((and (((eq b) (x6 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x6 ((cP Xx) Xy))) ((cP2 (x6 Xx)) (x6 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x6) Xg)))))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) ((and ((and (((eq b) (x6 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x6 ((cP Xx) Xy))) ((cP2 (x6 Xx)) (x6 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x6) Xg)))))
% Found (fun (x6:(iS->b))=> ((eq_ref Prop) (f x6))) as proof of (((eq Prop) (f x6)) ((and ((and (((eq b) (x6 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x6 ((cP Xx) Xy))) ((cP2 (x6 Xx)) (x6 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x6) Xg)))))
% Found (fun (x6:(iS->b))=> ((eq_ref Prop) (f x6))) as proof of (forall (x:(iS->b)), (((eq Prop) (f x)) ((and ((and (((eq b) (x c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x ((cP Xx) Xy))) ((cP2 (x Xx)) (x Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x) Xg))))))
% Found eq_ref00:=(eq_ref0 (x0 c0)):(((eq b) (x0 c0)) (x0 c0))
% Found (eq_ref0 (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found eq_ref00:=(eq_ref0 x6):(((eq (iS->b)) x6) x6)
% Found (eq_ref0 x6) as proof of (((eq (iS->b)) x6) Xg)
% Found ((eq_ref (iS->b)) x6) as proof of (((eq (iS->b)) x6) Xg)
% Found ((eq_ref (iS->b)) x6) as proof of (((eq (iS->b)) x6) Xg)
% Found (fun (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x6)) as proof of (((eq (iS->b)) x6) Xg)
% Found (fun (x7:(((eq b) (Xg c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x6)) as proof of ((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->(((eq (iS->b)) x6) Xg))
% Found (fun (x7:(((eq b) (Xg c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x6)) as proof of ((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->(((eq (iS->b)) x6) Xg)))
% Found (and_rect30 (fun (x7:(((eq b) (Xg c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x6))) as proof of (((eq (iS->b)) x6) Xg)
% Found ((and_rect3 (((eq (iS->b)) x6) Xg)) (fun (x7:(((eq b) (Xg c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x6))) as proof of (((eq (iS->b)) x6) Xg)
% Found (((fun (P:Type) (x7:((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->P)))=> (((((and_rect (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))) P) x7) x00)) (((eq (iS->b)) x6) Xg)) (fun (x7:(((eq b) (Xg c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x6))) as proof of (((eq (iS->b)) x6) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((fun (P:Type) (x7:((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->P)))=> (((((and_rect (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))) P) x7) x00)) (((eq (iS->b)) x6) Xg)) (fun (x7:(((eq b) (Xg c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x6)))) as proof of (((eq (iS->b)) x6) Xg)
% Found eq_ref00:=(eq_ref0 (x2 c0)):(((eq b) (x2 c0)) (x2 c0))
% Found (eq_ref0 (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found eq_ref00:=(eq_ref0 (x4 c0)):(((eq b) (x4 c0)) (x4 c0))
% Found (eq_ref0 (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found ((eq_ref b) (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found ((eq_ref b) (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found ((eq_ref b) (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xg):(((eq (iS->b)) Xg) (fun (x:iS)=> (Xg x)))
% Found (eta_expansion_dep00 Xg) as proof of (((eq (iS->b)) Xg) x6)
% Found ((eta_expansion_dep0 (fun (x8:iS)=> b)) Xg) as proof of (((eq (iS->b)) Xg) x6)
% Found (((eta_expansion_dep iS) (fun (x8:iS)=> b)) Xg) as proof of (((eq (iS->b)) Xg) x6)
% Found (((eta_expansion_dep iS) (fun (x8:iS)=> b)) Xg) as proof of (((eq (iS->b)) Xg) x6)
% Found (((eta_expansion_dep iS) (fun (x8:iS)=> b)) Xg) as proof of (((eq (iS->b)) Xg) x6)
% Found (eq_sym000 (((eta_expansion_dep iS) (fun (x8:iS)=> b)) Xg)) as proof of (((eq (iS->b)) x6) Xg)
% Found ((eq_sym00 x6) (((eta_expansion_dep iS) (fun (x8:iS)=> b)) Xg)) as proof of (((eq (iS->b)) x6) Xg)
% Found (((eq_sym0 Xg) x6) (((eta_expansion_dep iS) (fun (x8:iS)=> b)) Xg)) as proof of (((eq (iS->b)) x6) Xg)
% Found ((((eq_sym (iS->b)) Xg) x6) (((eta_expansion_dep iS) (fun (x8:iS)=> b)) Xg)) as proof of (((eq (iS->b)) x6) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((((eq_sym (iS->b)) Xg) x6) (((eta_expansion_dep iS) (fun (x8:iS)=> b)) Xg))) as proof of (((eq (iS->b)) x6) Xg)
% Found eq_ref00:=(eq_ref0 (x0 x5)):(((eq b) (x0 x5)) (x0 x5))
% Found (eq_ref0 (x0 x5)) as proof of (((eq b) (x0 x5)) (x' x5))
% Found ((eq_ref b) (x0 x5)) as proof of (((eq b) (x0 x5)) (x' x5))
% Found ((eq_ref b) (x0 x5)) as proof of (((eq b) (x0 x5)) (x' x5))
% Found (fun (x5:iS)=> ((eq_ref b) (x0 x5))) as proof of (((eq b) (x0 x5)) (x' x5))
% Found (fun (x5:iS)=> ((eq_ref b) (x0 x5))) as proof of (forall (x:iS), (((eq b) (x0 x)) (x' x)))
% Found (functional_extensionality_dep0000 (fun (x5:iS)=> ((eq_ref b) (x0 x5)))) as proof of (((eq (iS->b)) x0) x')
% Found ((functional_extensionality_dep000 x') (fun (x5:iS)=> ((eq_ref b) (x0 x5)))) as proof of (((eq (iS->b)) x0) x')
% Found (((functional_extensionality_dep00 x0) x') (fun (x5:iS)=> ((eq_ref b) (x0 x5)))) as proof of (((eq (iS->b)) x0) x')
% Found ((((functional_extensionality_dep0 (fun (x7:iS)=> b)) x0) x') (fun (x5:iS)=> ((eq_ref b) (x0 x5)))) as proof of (((eq (iS->b)) x0) x')
% Found (((((functional_extensionality_dep iS) (fun (x7:iS)=> b)) x0) x') (fun (x5:iS)=> ((eq_ref b) (x0 x5)))) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((((functional_extensionality_dep iS) (fun (x7:iS)=> b)) x0) x') (fun (x5:iS)=> ((eq_ref b) (x0 x5))))) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 (x0 x5)):(((eq b) (x0 x5)) (x0 x5))
% Found (eq_ref0 (x0 x5)) as proof of (((eq b) (x0 x5)) (x' x5))
% Found ((eq_ref b) (x0 x5)) as proof of (((eq b) (x0 x5)) (x' x5))
% Found ((eq_ref b) (x0 x5)) as proof of (((eq b) (x0 x5)) (x' x5))
% Found (fun (x5:iS)=> ((eq_ref b) (x0 x5))) as proof of (((eq b) (x0 x5)) (x' x5))
% Found (fun (x5:iS)=> ((eq_ref b) (x0 x5))) as proof of (forall (x:iS), (((eq b) (x0 x)) (x' x)))
% Found (functional_extensionality0000 (fun (x5:iS)=> ((eq_ref b) (x0 x5)))) as proof of (((eq (iS->b)) x0) x')
% Found ((functional_extensionality000 x') (fun (x5:iS)=> ((eq_ref b) (x0 x5)))) as proof of (((eq (iS->b)) x0) x')
% Found (((functional_extensionality00 x0) x') (fun (x5:iS)=> ((eq_ref b) (x0 x5)))) as proof of (((eq (iS->b)) x0) x')
% Found ((((functional_extensionality0 b) x0) x') (fun (x5:iS)=> ((eq_ref b) (x0 x5)))) as proof of (((eq (iS->b)) x0) x')
% Found (((((functional_extensionality iS) b) x0) x') (fun (x5:iS)=> ((eq_ref b) (x0 x5)))) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((((functional_extensionality iS) b) x0) x') (fun (x5:iS)=> ((eq_ref b) (x0 x5))))) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 (x2 x5)):(((eq b) (x2 x5)) (x2 x5))
% Found (eq_ref0 (x2 x5)) as proof of (((eq b) (x2 x5)) (x' x5))
% Found ((eq_ref b) (x2 x5)) as proof of (((eq b) (x2 x5)) (x' x5))
% Found ((eq_ref b) (x2 x5)) as proof of (((eq b) (x2 x5)) (x' x5))
% Found (fun (x5:iS)=> ((eq_ref b) (x2 x5))) as proof of (((eq b) (x2 x5)) (x' x5))
% Found (fun (x5:iS)=> ((eq_ref b) (x2 x5))) as proof of (forall (x:iS), (((eq b) (x2 x)) (x' x)))
% Found (functional_extensionality0000 (fun (x5:iS)=> ((eq_ref b) (x2 x5)))) as proof of (((eq (iS->b)) x2) x')
% Found ((functional_extensionality000 x') (fun (x5:iS)=> ((eq_ref b) (x2 x5)))) as proof of (((eq (iS->b)) x2) x')
% Found (((functional_extensionality00 x2) x') (fun (x5:iS)=> ((eq_ref b) (x2 x5)))) as proof of (((eq (iS->b)) x2) x')
% Found ((((functional_extensionality0 b) x2) x') (fun (x5:iS)=> ((eq_ref b) (x2 x5)))) as proof of (((eq (iS->b)) x2) x')
% Found (((((functional_extensionality iS) b) x2) x') (fun (x5:iS)=> ((eq_ref b) (x2 x5)))) as proof of (((eq (iS->b)) x2) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((((functional_extensionality iS) b) x2) x') (fun (x5:iS)=> ((eq_ref b) (x2 x5))))) as proof of (((eq (iS->b)) x2) x')
% Found eq_ref00:=(eq_ref0 (x2 x5)):(((eq b) (x2 x5)) (x2 x5))
% Found (eq_ref0 (x2 x5)) as proof of (((eq b) (x2 x5)) (x' x5))
% Found ((eq_ref b) (x2 x5)) as proof of (((eq b) (x2 x5)) (x' x5))
% Found ((eq_ref b) (x2 x5)) as proof of (((eq b) (x2 x5)) (x' x5))
% Found (fun (x5:iS)=> ((eq_ref b) (x2 x5))) as proof of (((eq b) (x2 x5)) (x' x5))
% Found (fun (x5:iS)=> ((eq_ref b) (x2 x5))) as proof of (forall (x:iS), (((eq b) (x2 x)) (x' x)))
% Found (functional_extensionality_dep0000 (fun (x5:iS)=> ((eq_ref b) (x2 x5)))) as proof of (((eq (iS->b)) x2) x')
% Found ((functional_extensionality_dep000 x') (fun (x5:iS)=> ((eq_ref b) (x2 x5)))) as proof of (((eq (iS->b)) x2) x')
% Found (((functional_extensionality_dep00 x2) x') (fun (x5:iS)=> ((eq_ref b) (x2 x5)))) as proof of (((eq (iS->b)) x2) x')
% Found ((((functional_extensionality_dep0 (fun (x7:iS)=> b)) x2) x') (fun (x5:iS)=> ((eq_ref b) (x2 x5)))) as proof of (((eq (iS->b)) x2) x')
% Found (((((functional_extensionality_dep iS) (fun (x7:iS)=> b)) x2) x') (fun (x5:iS)=> ((eq_ref b) (x2 x5)))) as proof of (((eq (iS->b)) x2) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((((functional_extensionality_dep iS) (fun (x7:iS)=> b)) x2) x') (fun (x5:iS)=> ((eq_ref b) (x2 x5))))) as proof of (((eq (iS->b)) x2) x')
% Found x9:(P x0)
% Instantiate: x0:=x':(iS->b)
% Found (fun (x9:(P x0))=> x9) as proof of (P x')
% Found (fun (P:((iS->b)->Prop)) (x9:(P x0))=> x9) as proof of ((P x0)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop)) (x9:(P x0))=> x9) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of ((P x0)->(P x'))
% Found ((eq_ref0 x0) P) as proof of ((P x0)->(P x'))
% Found (((eq_ref (iS->b)) x0) P) as proof of ((P x0)->(P x'))
% Found (((eq_ref (iS->b)) x0) P) as proof of ((P x0)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x0) P)) as proof of (((eq (iS->b)) x0) x')
% Found x9:(P x2)
% Instantiate: x2:=x':(iS->b)
% Found (fun (x9:(P x2))=> x9) as proof of (P x')
% Found (fun (P:((iS->b)->Prop)) (x9:(P x2))=> x9) as proof of ((P x2)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop)) (x9:(P x2))=> x9) as proof of (((eq (iS->b)) x2) x')
% Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% Found (eq_ref00 P) as proof of ((P x2)->(P x'))
% Found ((eq_ref0 x2) P) as proof of ((P x2)->(P x'))
% Found (((eq_ref (iS->b)) x2) P) as proof of ((P x2)->(P x'))
% Found (((eq_ref (iS->b)) x2) P) as proof of ((P x2)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x2) P)) as proof of (((eq (iS->b)) x2) x')
% Found x9:(P x4)
% Instantiate: x4:=x':(iS->b)
% Found (fun (x9:(P x4))=> x9) as proof of (P x')
% Found (fun (P:((iS->b)->Prop)) (x9:(P x4))=> x9) as proof of ((P x4)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop)) (x9:(P x4))=> x9) as proof of (((eq (iS->b)) x4) x')
% Found eq_ref000:=(eq_ref00 P):((P x4)->(P x4))
% Found (eq_ref00 P) as proof of ((P x4)->(P x'))
% Found ((eq_ref0 x4) P) as proof of ((P x4)->(P x'))
% Found (((eq_ref (iS->b)) x4) P) as proof of ((P x4)->(P x'))
% Found (((eq_ref (iS->b)) x4) P) as proof of ((P x4)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x4) P)) as proof of ((P x4)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x4) P)) as proof of (((eq (iS->b)) x4) x')
% Found x9:(P x6)
% Instantiate: x6:=x':(iS->b)
% Found (fun (x9:(P x6))=> x9) as proof of (P x')
% Found (fun (P:((iS->b)->Prop)) (x9:(P x6))=> x9) as proof of ((P x6)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop)) (x9:(P x6))=> x9) as proof of (((eq (iS->b)) x6) x')
% Found eq_ref000:=(eq_ref00 P):((P x6)->(P x6))
% Found (eq_ref00 P) as proof of ((P x6)->(P x'))
% Found ((eq_ref0 x6) P) as proof of ((P x6)->(P x'))
% Found (((eq_ref (iS->b)) x6) P) as proof of ((P x6)->(P x'))
% Found (((eq_ref (iS->b)) x6) P) as proof of ((P x6)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x6) P)) as proof of ((P x6)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x6) P)) as proof of (((eq (iS->b)) x6) x')
% Found eq_ref00:=(eq_ref0 x0):(((eq (iS->b)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (iS->b)) x0) Xg)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) Xg)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x0)) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x7:(((eq b) (Xg c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x0)) as proof of ((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->(((eq (iS->b)) x0) Xg))
% Found (fun (x7:(((eq b) (Xg c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x0)) as proof of ((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->(((eq (iS->b)) x0) Xg)))
% Found (and_rect30 (fun (x7:(((eq b) (Xg c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) Xg)
% Found ((and_rect3 (((eq (iS->b)) x0) Xg)) (fun (x7:(((eq b) (Xg c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) Xg)
% Found (((fun (P:Type) (x7:((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->P)))=> (((((and_rect (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))) P) x7) x00)) (((eq (iS->b)) x0) Xg)) (fun (x7:(((eq b) (Xg c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((fun (P:Type) (x7:((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->P)))=> (((((and_rect (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))) P) x7) x00)) (((eq (iS->b)) x0) Xg)) (fun (x7:(((eq b) (Xg c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x0)))) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 x2):(((eq (iS->b)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (iS->b)) x2) Xg)
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) Xg)
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) Xg)
% Found (fun (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x2)) as proof of (((eq (iS->b)) x2) Xg)
% Found (fun (x7:(((eq b) (Xg c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x2)) as proof of ((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->(((eq (iS->b)) x2) Xg))
% Found (fun (x7:(((eq b) (Xg c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x2)) as proof of ((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->(((eq (iS->b)) x2) Xg)))
% Found (and_rect30 (fun (x7:(((eq b) (Xg c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) Xg)
% Found ((and_rect3 (((eq (iS->b)) x2) Xg)) (fun (x7:(((eq b) (Xg c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) Xg)
% Found (((fun (P:Type) (x7:((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->P)))=> (((((and_rect (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))) P) x7) x00)) (((eq (iS->b)) x2) Xg)) (fun (x7:(((eq b) (Xg c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((fun (P:Type) (x7:((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->P)))=> (((((and_rect (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))) P) x7) x00)) (((eq (iS->b)) x2) Xg)) (fun (x7:(((eq b) (Xg c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x2)))) as proof of (((eq (iS->b)) x2) Xg)
% Found eq_ref00:=(eq_ref0 x4):(((eq (iS->b)) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq (iS->b)) x4) Xg)
% Found ((eq_ref (iS->b)) x4) as proof of (((eq (iS->b)) x4) Xg)
% Found ((eq_ref (iS->b)) x4) as proof of (((eq (iS->b)) x4) Xg)
% Found (fun (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x4)) as proof of (((eq (iS->b)) x4) Xg)
% Found (fun (x7:(((eq b) (Xg c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x4)) as proof of ((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->(((eq (iS->b)) x4) Xg))
% Found (fun (x7:(((eq b) (Xg c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x4)) as proof of ((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->(((eq (iS->b)) x4) Xg)))
% Found (and_rect30 (fun (x7:(((eq b) (Xg c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x4))) as proof of (((eq (iS->b)) x4) Xg)
% Found ((and_rect3 (((eq (iS->b)) x4) Xg)) (fun (x7:(((eq b) (Xg c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x4))) as proof of (((eq (iS->b)) x4) Xg)
% Found (((fun (P:Type) (x7:((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->P)))=> (((((and_rect (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))) P) x7) x00)) (((eq (iS->b)) x4) Xg)) (fun (x7:(((eq b) (Xg c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x4))) as proof of (((eq (iS->b)) x4) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((fun (P:Type) (x7:((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->P)))=> (((((and_rect (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))) P) x7) x00)) (((eq (iS->b)) x4) Xg)) (fun (x7:(((eq b) (Xg c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x4)))) as proof of (((eq (iS->b)) x4) Xg)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xg):(((eq (iS->b)) Xg) (fun (x:iS)=> (Xg x)))
% Found (eta_expansion_dep00 Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found ((eta_expansion_dep0 (fun (x8:iS)=> b)) Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found (((eta_expansion_dep iS) (fun (x8:iS)=> b)) Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found (((eta_expansion_dep iS) (fun (x8:iS)=> b)) Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found (((eta_expansion_dep iS) (fun (x8:iS)=> b)) Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found (eq_sym000 (((eta_expansion_dep iS) (fun (x8:iS)=> b)) Xg)) as proof of (((eq (iS->b)) x0) Xg)
% Found ((eq_sym00 x0) (((eta_expansion_dep iS) (fun (x8:iS)=> b)) Xg)) as proof of (((eq (iS->b)) x0) Xg)
% Found (((eq_sym0 Xg) x0) (((eta_expansion_dep iS) (fun (x8:iS)=> b)) Xg)) as proof of (((eq (iS->b)) x0) Xg)
% Found ((((eq_sym (iS->b)) Xg) x0) (((eta_expansion_dep iS) (fun (x8:iS)=> b)) Xg)) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((((eq_sym (iS->b)) Xg) x0) (((eta_expansion_dep iS) (fun (x8:iS)=> b)) Xg))) as proof of (((eq (iS->b)) x0) Xg)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xg):(((eq (iS->b)) Xg) (fun (x:iS)=> (Xg x)))
% Found (eta_expansion_dep00 Xg) as proof of (((eq (iS->b)) Xg) x2)
% Found ((eta_expansion_dep0 (fun (x8:iS)=> b)) Xg) as proof of (((eq (iS->b)) Xg) x2)
% Found (((eta_expansion_dep iS) (fun (x8:iS)=> b)) Xg) as proof of (((eq (iS->b)) Xg) x2)
% Found (((eta_expansion_dep iS) (fun (x8:iS)=> b)) Xg) as proof of (((eq (iS->b)) Xg) x2)
% Found (((eta_expansion_dep iS) (fun (x8:iS)=> b)) Xg) as proof of (((eq (iS->b)) Xg) x2)
% Found (eq_sym000 (((eta_expansion_dep iS) (fun (x8:iS)=> b)) Xg)) as proof of (((eq (iS->b)) x2) Xg)
% Found ((eq_sym00 x2) (((eta_expansion_dep iS) (fun (x8:iS)=> b)) Xg)) as proof of (((eq (iS->b)) x2) Xg)
% Found (((eq_sym0 Xg) x2) (((eta_expansion_dep iS) (fun (x8:iS)=> b)) Xg)) as proof of (((eq (iS->b)) x2) Xg)
% Found ((((eq_sym (iS->b)) Xg) x2) (((eta_expansion_dep iS) (fun (x8:iS)=> b)) Xg)) as proof of (((eq (iS->b)) x2) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((((eq_sym (iS->b)) Xg) x2) (((eta_expansion_dep iS) (fun (x8:iS)=> b)) Xg))) as proof of (((eq (iS->b)) x2) Xg)
% Found eq_ref00:=(eq_ref0 Xg):(((eq (iS->b)) Xg) Xg)
% Found (eq_ref0 Xg) as proof of (((eq (iS->b)) Xg) x4)
% Found ((eq_ref (iS->b)) Xg) as proof of (((eq (iS->b)) Xg) x4)
% Found ((eq_ref (iS->b)) Xg) as proof of (((eq (iS->b)) Xg) x4)
% Found ((eq_ref (iS->b)) Xg) as proof of (((eq (iS->b)) Xg) x4)
% Found (eq_sym000 ((eq_ref (iS->b)) Xg)) as proof of (((eq (iS->b)) x4) Xg)
% Found ((eq_sym00 x4) ((eq_ref (iS->b)) Xg)) as proof of (((eq (iS->b)) x4) Xg)
% Found (((eq_sym0 Xg) x4) ((eq_ref (iS->b)) Xg)) as proof of (((eq (iS->b)) x4) Xg)
% Found ((((eq_sym (iS->b)) Xg) x4) ((eq_ref (iS->b)) Xg)) as proof of (((eq (iS->b)) x4) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((((eq_sym (iS->b)) Xg) x4) ((eq_ref (iS->b)) Xg))) as proof of (((eq (iS->b)) x4) Xg)
% Found eq_ref00:=(eq_ref0 (f x6)):(((eq Prop) (f x6)) (f x6))
% Found (eq_ref0 (f x6)) as proof of (((eq Prop) (f x6)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x6))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x6))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x6))
% Found (fun (x6:(iS->b))=> ((eq_ref Prop) (f x6))) as proof of (((eq Prop) (f x6)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x6))
% Found (fun (x6:(iS->b))=> ((eq_ref Prop) (f x6))) as proof of (forall (x:(iS->b)), (((eq Prop) (f x)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x)))
% Found eq_ref00:=(eq_ref0 (f x6)):(((eq Prop) (f x6)) (f x6))
% Found (eq_ref0 (f x6)) as proof of (((eq Prop) (f x6)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x6))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x6))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x6))
% Found (fun (x6:(iS->b))=> ((eq_ref Prop) (f x6))) as proof of (((eq Prop) (f x6)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x6))
% Found (fun (x6:(iS->b))=> ((eq_ref Prop) (f x6))) as proof of (forall (x:(iS->b)), (((eq Prop) (f x)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x)))
% Found eq_ref00:=(eq_ref0 x6):(((eq (iS->b)) x6) x6)
% Found (eq_ref0 x6) as proof of (((eq (iS->b)) x6) x')
% Found ((eq_ref (iS->b)) x6) as proof of (((eq (iS->b)) x6) x')
% Found ((eq_ref (iS->b)) x6) as proof of (((eq (iS->b)) x6) x')
% Found (fun (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x6)) as proof of (((eq (iS->b)) x6) x')
% Found (fun (x7:(((eq b) (x' c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x6)) as proof of ((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->(((eq (iS->b)) x6) x'))
% Found (fun (x7:(((eq b) (x' c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x6)) as proof of ((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->(((eq (iS->b)) x6) x')))
% Found (and_rect30 (fun (x7:(((eq b) (x' c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x6))) as proof of (((eq (iS->b)) x6) x')
% Found ((and_rect3 (((eq (iS->b)) x6) x')) (fun (x7:(((eq b) (x' c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x6))) as proof of (((eq (iS->b)) x6) x')
% Found (((fun (P:Type) (x7:((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->P)))=> (((((and_rect (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))) P) x7) x00)) (((eq (iS->b)) x6) x')) (fun (x7:(((eq b) (x' c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x6))) as proof of (((eq (iS->b)) x6) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((fun (P:Type) (x7:((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->P)))=> (((((and_rect (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))) P) x7) x00)) (((eq (iS->b)) x6) x')) (fun (x7:(((eq b) (x' c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x6)))) as proof of (((eq (iS->b)) x6) x')
% Found eq_ref00:=(eq_ref0 x0):(((eq (iS->b)) x0) x0)
% Found (eq_ref0 x0) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P Xg)))
% Found ((eq_ref (iS->b)) x0) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P Xg)))
% Found ((eq_ref (iS->b)) x0) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P Xg)))
% Found ((eq_ref (iS->b)) x0) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P Xg)))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((eq_ref (iS->b)) x0)) as proof of (((eq (iS->b)) x0) Xg)
% Found eta_expansion_dep000:=(eta_expansion_dep00 x'):(((eq (iS->b)) x') (fun (x:iS)=> (x' x)))
% Found (eta_expansion_dep00 x') as proof of (((eq (iS->b)) x') x6)
% Found ((eta_expansion_dep0 (fun (x8:iS)=> b)) x') as proof of (((eq (iS->b)) x') x6)
% Found (((eta_expansion_dep iS) (fun (x8:iS)=> b)) x') as proof of (((eq (iS->b)) x') x6)
% Found (((eta_expansion_dep iS) (fun (x8:iS)=> b)) x') as proof of (((eq (iS->b)) x') x6)
% Found (((eta_expansion_dep iS) (fun (x8:iS)=> b)) x') as proof of (((eq (iS->b)) x') x6)
% Found (eq_sym000 (((eta_expansion_dep iS) (fun (x8:iS)=> b)) x')) as proof of (((eq (iS->b)) x6) x')
% Found ((eq_sym00 x6) (((eta_expansion_dep iS) (fun (x8:iS)=> b)) x')) as proof of (((eq (iS->b)) x6) x')
% Found (((eq_sym0 x') x6) (((eta_expansion_dep iS) (fun (x8:iS)=> b)) x')) as proof of (((eq (iS->b)) x6) x')
% Found ((((eq_sym (iS->b)) x') x6) (((eta_expansion_dep iS) (fun (x8:iS)=> b)) x')) as proof of (((eq (iS->b)) x6) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((((eq_sym (iS->b)) x') x6) (((eta_expansion_dep iS) (fun (x8:iS)=> b)) x'))) as proof of (((eq (iS->b)) x6) x')
% Found eq_ref00:=(eq_ref0 (x0 c0)):(((eq b) (x0 c0)) (x0 c0))
% Found (eq_ref0 (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found eq_ref00:=(eq_ref0 x0):(((eq (iS->b)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (iS->b)) x0) Xg)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) Xg)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x8:(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)))))=> ((eq_ref (iS->b)) x0)) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x0)) as proof of ((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))))->(((eq (iS->b)) x0) Xg))
% Found (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x0)) as proof of ((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))))->(((eq (iS->b)) x0) Xg)))
% Found (and_rect30 (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) Xg)
% Found ((and_rect3 (((eq (iS->b)) x0) Xg)) (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) Xg)
% Found (((fun (P:Type) (x7:((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))))->P)))=> (((((and_rect (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))))) P) x7) x5)) (((eq (iS->b)) x0) Xg)) (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((fun (P:Type) (x7:((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))))->P)))=> (((((and_rect (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))))) P) x7) x5)) (((eq (iS->b)) x0) Xg)) (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x0)))) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 x2):(((eq (iS->b)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (iS->b)) x2) Xg)
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) Xg)
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) Xg)
% Found (fun (x8:(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)))))=> ((eq_ref (iS->b)) x2)) as proof of (((eq (iS->b)) x2) Xg)
% Found (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x2)) as proof of ((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))))->(((eq (iS->b)) x2) Xg))
% Found (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x2)) as proof of ((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))))->(((eq (iS->b)) x2) Xg)))
% Found (and_rect30 (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) Xg)
% Found ((and_rect3 (((eq (iS->b)) x2) Xg)) (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) Xg)
% Found (((fun (P:Type) (x7:((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))))->P)))=> (((((and_rect (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))))) P) x7) x5)) (((eq (iS->b)) x2) Xg)) (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((fun (P:Type) (x7:((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))))->P)))=> (((((and_rect (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))))) P) x7) x5)) (((eq (iS->b)) x2) Xg)) (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x2)))) as proof of (((eq (iS->b)) x2) Xg)
% Found eq_ref00:=(eq_ref0 x4):(((eq (iS->b)) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq (iS->b)) x4) Xg)
% Found ((eq_ref (iS->b)) x4) as proof of (((eq (iS->b)) x4) Xg)
% Found ((eq_ref (iS->b)) x4) as proof of (((eq (iS->b)) x4) Xg)
% Found (fun (x8:(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)))))=> ((eq_ref (iS->b)) x4)) as proof of (((eq (iS->b)) x4) Xg)
% Found (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x4)) as proof of ((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))))->(((eq (iS->b)) x4) Xg))
% Found (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x4)) as proof of ((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))))->(((eq (iS->b)) x4) Xg)))
% Found (and_rect30 (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x4))) as proof of (((eq (iS->b)) x4) Xg)
% Found ((and_rect3 (((eq (iS->b)) x4) Xg)) (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x4))) as proof of (((eq (iS->b)) x4) Xg)
% Found (((fun (P:Type) (x7:((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))))->P)))=> (((((and_rect (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))))) P) x7) x5)) (((eq (iS->b)) x4) Xg)) (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x4))) as proof of (((eq (iS->b)) x4) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((fun (P:Type) (x7:((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))))->P)))=> (((((and_rect (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))))) P) x7) x5)) (((eq (iS->b)) x4) Xg)) (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x4)))) as proof of (((eq (iS->b)) x4) Xg)
% Found eq_ref00:=(eq_ref0 x6):(((eq (iS->b)) x6) x6)
% Found (eq_ref0 x6) as proof of (((eq (iS->b)) x6) Xg)
% Found ((eq_ref (iS->b)) x6) as proof of (((eq (iS->b)) x6) Xg)
% Found ((eq_ref (iS->b)) x6) as proof of (((eq (iS->b)) x6) Xg)
% Found (fun (x8:(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)))))=> ((eq_ref (iS->b)) x6)) as proof of (((eq (iS->b)) x6) Xg)
% Found (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x6)) as proof of ((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))))->(((eq (iS->b)) x6) Xg))
% Found (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x6)) as proof of ((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))))->(((eq (iS->b)) x6) Xg)))
% Found (and_rect30 (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x6))) as proof of (((eq (iS->b)) x6) Xg)
% Found ((and_rect3 (((eq (iS->b)) x6) Xg)) (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x6))) as proof of (((eq (iS->b)) x6) Xg)
% Found (((fun (P:Type) (x7:((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))))->P)))=> (((((and_rect (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))))) P) x7) x4)) (((eq (iS->b)) x6) Xg)) (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x6))) as proof of (((eq (iS->b)) x6) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((fun (P:Type) (x7:((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))))->P)))=> (((((and_rect (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))))) P) x7) x4)) (((eq (iS->b)) x6) Xg)) (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x6)))) as proof of (((eq (iS->b)) x6) Xg)
% Found eq_ref00:=(eq_ref0 x0):(((eq (iS->b)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (iS->b)) x0) x')
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) x')
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x0)) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x7:(((eq b) (x' c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x0)) as proof of ((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->(((eq (iS->b)) x0) x'))
% Found (fun (x7:(((eq b) (x' c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x0)) as proof of ((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->(((eq (iS->b)) x0) x')))
% Found (and_rect30 (fun (x7:(((eq b) (x' c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) x')
% Found ((and_rect3 (((eq (iS->b)) x0) x')) (fun (x7:(((eq b) (x' c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) x')
% Found (((fun (P:Type) (x7:((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->P)))=> (((((and_rect (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))) P) x7) x00)) (((eq (iS->b)) x0) x')) (fun (x7:(((eq b) (x' c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((fun (P:Type) (x7:((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->P)))=> (((((and_rect (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))) P) x7) x00)) (((eq (iS->b)) x0) x')) (fun (x7:(((eq b) (x' c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x0)))) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 x2):(((eq (iS->b)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (iS->b)) x2) x')
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) x')
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) x')
% Found (fun (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x2)) as proof of (((eq (iS->b)) x2) x')
% Found (fun (x7:(((eq b) (x' c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x2)) as proof of ((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->(((eq (iS->b)) x2) x'))
% Found (fun (x7:(((eq b) (x' c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x2)) as proof of ((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->(((eq (iS->b)) x2) x')))
% Found (and_rect30 (fun (x7:(((eq b) (x' c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) x')
% Found ((and_rect3 (((eq (iS->b)) x2) x')) (fun (x7:(((eq b) (x' c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) x')
% Found (((fun (P:Type) (x7:((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->P)))=> (((((and_rect (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))) P) x7) x00)) (((eq (iS->b)) x2) x')) (fun (x7:(((eq b) (x' c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((fun (P:Type) (x7:((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->P)))=> (((((and_rect (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))) P) x7) x00)) (((eq (iS->b)) x2) x')) (fun (x7:(((eq b) (x' c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x2)))) as proof of (((eq (iS->b)) x2) x')
% Found eq_ref00:=(eq_ref0 x4):(((eq (iS->b)) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq (iS->b)) x4) x')
% Found ((eq_ref (iS->b)) x4) as proof of (((eq (iS->b)) x4) x')
% Found ((eq_ref (iS->b)) x4) as proof of (((eq (iS->b)) x4) x')
% Found (fun (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x4)) as proof of (((eq (iS->b)) x4) x')
% Found (fun (x7:(((eq b) (x' c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x4)) as proof of ((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->(((eq (iS->b)) x4) x'))
% Found (fun (x7:(((eq b) (x' c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x4)) as proof of ((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->(((eq (iS->b)) x4) x')))
% Found (and_rect30 (fun (x7:(((eq b) (x' c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x4))) as proof of (((eq (iS->b)) x4) x')
% Found ((and_rect3 (((eq (iS->b)) x4) x')) (fun (x7:(((eq b) (x' c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x4))) as proof of (((eq (iS->b)) x4) x')
% Found (((fun (P:Type) (x7:((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->P)))=> (((((and_rect (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))) P) x7) x00)) (((eq (iS->b)) x4) x')) (fun (x7:(((eq b) (x' c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x4))) as proof of (((eq (iS->b)) x4) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((fun (P:Type) (x7:((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->P)))=> (((((and_rect (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))) P) x7) x00)) (((eq (iS->b)) x4) x')) (fun (x7:(((eq b) (x' c0)) c02)) (x8:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x4)))) as proof of (((eq (iS->b)) x4) x')
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))):(((eq ((iS->b)->Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) (fun (x:(iS->b))=> ((and ((and (((eq b) (x c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x ((cP Xx) Xy))) ((cP2 (x Xx)) (x Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x) Xg))))))
% Found (eta_expansion_dep00 (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) as proof of (((eq ((iS->b)->Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) b0)
% Found ((eta_expansion_dep0 (fun (x9:(iS->b))=> Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) as proof of (((eq ((iS->b)->Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) b0)
% Found (((eta_expansion_dep (iS->b)) (fun (x9:(iS->b))=> Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) as proof of (((eq ((iS->b)->Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) b0)
% Found (((eta_expansion_dep (iS->b)) (fun (x9:(iS->b))=> Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) as proof of (((eq ((iS->b)->Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) b0)
% Found (((eta_expansion_dep (iS->b)) (fun (x9:(iS->b))=> Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) as proof of (((eq ((iS->b)->Prop)) (fun (Xf:(iS->b))=> ((and ((and (((eq b) (Xf c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xf ((cP Xx) Xy))) ((cP2 (Xf Xx)) (Xf Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) Xf) Xg)))))) b0)
% 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 (iS->b)) x0) P) as proof of ((P x0)->(P Xg))
% Found (((eq_ref (iS->b)) x0) P) as proof of ((P x0)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x0) P)) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P Xg)))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x0) P)) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (iS->b)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (iS->b)) x0) P) as proof of (P0 x0)
% Found eta_expansion000:=(eta_expansion00 x'):(((eq (iS->b)) x') (fun (x:iS)=> (x' x)))
% Found (eta_expansion00 x') as proof of (((eq (iS->b)) x') x0)
% Found ((eta_expansion0 b) x') as proof of (((eq (iS->b)) x') x0)
% Found (((eta_expansion iS) b) x') as proof of (((eq (iS->b)) x') x0)
% Found (((eta_expansion iS) b) x') as proof of (((eq (iS->b)) x') x0)
% Found (((eta_expansion iS) b) x') as proof of (((eq (iS->b)) x') x0)
% Found (eq_sym000 (((eta_expansion iS) b) x')) as proof of (((eq (iS->b)) x0) x')
% Found ((eq_sym00 x0) (((eta_expansion iS) b) x')) as proof of (((eq (iS->b)) x0) x')
% Found (((eq_sym0 x') x0) (((eta_expansion iS) b) x')) as proof of (((eq (iS->b)) x0) x')
% Found ((((eq_sym (iS->b)) x') x0) (((eta_expansion iS) b) x')) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((((eq_sym (iS->b)) x') x0) (((eta_expansion iS) b) x'))) as proof of (((eq (iS->b)) x0) x')
% Found x1:(P x0)
% Instantiate: x0:=Xg:(iS->b)
% Found (fun (x1:(P x0))=> x1) as proof of (P Xg)
% Found (fun (P:((iS->b)->Prop)) (x1:(P x0))=> x1) as proof of ((P x0)->(P Xg))
% Found (fun (P:((iS->b)->Prop)) (x1:(P x0))=> x1) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P Xg)))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop)) (x1:(P x0))=> x1) as proof of (((eq (iS->b)) x0) Xg)
% Found eta_expansion_dep000:=(eta_expansion_dep00 x'):(((eq (iS->b)) x') (fun (x:iS)=> (x' x)))
% Found (eta_expansion_dep00 x') as proof of (((eq (iS->b)) x') x2)
% Found ((eta_expansion_dep0 (fun (x8:iS)=> b)) x') as proof of (((eq (iS->b)) x') x2)
% Found (((eta_expansion_dep iS) (fun (x8:iS)=> b)) x') as proof of (((eq (iS->b)) x') x2)
% Found (((eta_expansion_dep iS) (fun (x8:iS)=> b)) x') as proof of (((eq (iS->b)) x') x2)
% Found (((eta_expansion_dep iS) (fun (x8:iS)=> b)) x') as proof of (((eq (iS->b)) x') x2)
% Found (eq_sym000 (((eta_expansion_dep iS) (fun (x8:iS)=> b)) x')) as proof of (((eq (iS->b)) x2) x')
% Found ((eq_sym00 x2) (((eta_expansion_dep iS) (fun (x8:iS)=> b)) x')) as proof of (((eq (iS->b)) x2) x')
% Found (((eq_sym0 x') x2) (((eta_expansion_dep iS) (fun (x8:iS)=> b)) x')) as proof of (((eq (iS->b)) x2) x')
% Found ((((eq_sym (iS->b)) x') x2) (((eta_expansion_dep iS) (fun (x8:iS)=> b)) x')) as proof of (((eq (iS->b)) x2) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((((eq_sym (iS->b)) x') x2) (((eta_expansion_dep iS) (fun (x8:iS)=> b)) x'))) as proof of (((eq (iS->b)) x2) x')
% Found eta_expansion000:=(eta_expansion00 x'):(((eq (iS->b)) x') (fun (x:iS)=> (x' x)))
% Found (eta_expansion00 x') as proof of (((eq (iS->b)) x') x4)
% Found ((eta_expansion0 b) x') as proof of (((eq (iS->b)) x') x4)
% Found (((eta_expansion iS) b) x') as proof of (((eq (iS->b)) x') x4)
% Found (((eta_expansion iS) b) x') as proof of (((eq (iS->b)) x') x4)
% Found (((eta_expansion iS) b) x') as proof of (((eq (iS->b)) x') x4)
% Found (eq_sym000 (((eta_expansion iS) b) x')) as proof of (((eq (iS->b)) x4) x')
% Found ((eq_sym00 x4) (((eta_expansion iS) b) x')) as proof of (((eq (iS->b)) x4) x')
% Found (((eq_sym0 x') x4) (((eta_expansion iS) b) x')) as proof of (((eq (iS->b)) x4) x')
% Found ((((eq_sym (iS->b)) x') x4) (((eta_expansion iS) b) x')) as proof of (((eq (iS->b)) x4) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((((eq_sym (iS->b)) x') x4) (((eta_expansion iS) b) x'))) as proof of (((eq (iS->b)) x4) x')
% Found eq_ref00:=(eq_ref0 Xg):(((eq (iS->b)) Xg) Xg)
% Found (eq_ref0 Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found ((eq_ref (iS->b)) Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found ((eq_ref (iS->b)) Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found ((eq_ref (iS->b)) Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found (eq_sym0000 ((eq_ref (iS->b)) Xg)) as proof of ((P x0)->(P Xg))
% Found (eq_sym0000 ((eq_ref (iS->b)) Xg)) as proof of ((P x0)->(P Xg))
% Found ((fun (x1:(((eq (iS->b)) Xg) x0))=> ((eq_sym000 x1) P)) ((eq_ref (iS->b)) Xg)) as proof of ((P x0)->(P Xg))
% Found ((fun (x1:(((eq (iS->b)) Xg) x0))=> (((eq_sym00 x0) x1) P)) ((eq_ref (iS->b)) Xg)) as proof of ((P x0)->(P Xg))
% Found ((fun (x1:(((eq (iS->b)) Xg) x0))=> ((((eq_sym0 Xg) x0) x1) P)) ((eq_ref (iS->b)) Xg)) as proof of ((P x0)->(P Xg))
% Found ((fun (x1:(((eq (iS->b)) Xg) x0))=> (((((eq_sym (iS->b)) Xg) x0) x1) P)) ((eq_ref (iS->b)) Xg)) as proof of ((P x0)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> ((fun (x1:(((eq (iS->b)) Xg) x0))=> (((((eq_sym (iS->b)) Xg) x0) x1) P)) ((eq_ref (iS->b)) Xg))) as proof of ((P x0)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> ((fun (x1:(((eq (iS->b)) Xg) x0))=> (((((eq_sym (iS->b)) Xg) x0) x1) P)) ((eq_ref (iS->b)) Xg))) as proof of (((eq (iS->b)) x0) Xg)
% Found eta_expansion000:=(eta_expansion00 Xg):(((eq (iS->b)) Xg) (fun (x:iS)=> (Xg x)))
% Found (eta_expansion00 Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found ((eta_expansion0 b) Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found (((eta_expansion iS) b) Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found (((eta_expansion iS) b) Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found (((eta_expansion iS) b) Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found ((eq_sym0000 (((eta_expansion iS) b) Xg)) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found ((eq_sym0000 (((eta_expansion iS) b) Xg)) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (((fun (x1:(((eq (iS->b)) Xg) x0))=> ((eq_sym000 x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) (((eta_expansion iS) b) Xg)) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (((fun (x1:(((eq (iS->b)) Xg) x0))=> (((eq_sym00 x0) x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) (((eta_expansion iS) b) Xg)) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (((fun (x1:(((eq (iS->b)) Xg) x0))=> ((((eq_sym0 Xg) x0) x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) (((eta_expansion iS) b) Xg)) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (((fun (x1:(((eq (iS->b)) Xg) x0))=> (((((eq_sym (iS->b)) Xg) x0) x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) (((eta_expansion iS) b) Xg)) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> (((fun (x1:(((eq (iS->b)) Xg) x0))=> (((((eq_sym (iS->b)) Xg) x0) x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) (((eta_expansion iS) b) Xg)) (((eq_ref (iS->b)) x0) P))) as proof of ((P x0)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> (((fun (x1:(((eq (iS->b)) Xg) x0))=> (((((eq_sym (iS->b)) Xg) x0) x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) (((eta_expansion iS) b) Xg)) (((eq_ref (iS->b)) x0) P))) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 (x0 c0)):(((eq b) (x0 c0)) (x0 c0))
% Found (eq_ref0 (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found eq_ref00:=(eq_ref0 x0):(((eq (iS->b)) x0) x0)
% Found (eq_ref0 x0) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P x')))
% Found ((eq_ref (iS->b)) x0) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P x')))
% Found ((eq_ref (iS->b)) x0) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P x')))
% Found ((eq_ref (iS->b)) x0) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P x')))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((eq_ref (iS->b)) x0)) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 (x2 c0)):(((eq b) (x2 c0)) (x2 c0))
% Found (eq_ref0 (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found eq_ref00:=(eq_ref0 (x8 c0)):(((eq b) (x8 c0)) (x8 c0))
% Found (eq_ref0 (x8 c0)) as proof of (((eq b) (x8 c0)) c02)
% Found ((eq_ref b) (x8 c0)) as proof of (((eq b) (x8 c0)) c02)
% Found ((eq_ref b) (x8 c0)) as proof of (((eq b) (x8 c0)) c02)
% Found ((eq_ref b) (x8 c0)) as proof of (((eq b) (x8 c0)) c02)
% Found eq_ref00:=(eq_ref0 (x4 c0)):(((eq b) (x4 c0)) (x4 c0))
% Found (eq_ref0 (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found ((eq_ref b) (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found ((eq_ref b) (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found ((eq_ref b) (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found eq_ref00:=(eq_ref0 (x6 x7)):(((eq b) (x6 x7)) (x6 x7))
% Found (eq_ref0 (x6 x7)) as proof of (((eq b) (x6 x7)) (Xg x7))
% Found ((eq_ref b) (x6 x7)) as proof of (((eq b) (x6 x7)) (Xg x7))
% Found ((eq_ref b) (x6 x7)) as proof of (((eq b) (x6 x7)) (Xg x7))
% Found (fun (x7:iS)=> ((eq_ref b) (x6 x7))) as proof of (((eq b) (x6 x7)) (Xg x7))
% Found (fun (x7:iS)=> ((eq_ref b) (x6 x7))) as proof of (forall (x:iS), (((eq b) (x6 x)) (Xg x)))
% Found (functional_extensionality_dep0000 (fun (x7:iS)=> ((eq_ref b) (x6 x7)))) as proof of (((eq (iS->b)) x6) Xg)
% Found ((functional_extensionality_dep000 Xg) (fun (x7:iS)=> ((eq_ref b) (x6 x7)))) as proof of (((eq (iS->b)) x6) Xg)
% Found (((functional_extensionality_dep00 x6) Xg) (fun (x7:iS)=> ((eq_ref b) (x6 x7)))) as proof of (((eq (iS->b)) x6) Xg)
% Found ((((functional_extensionality_dep0 (fun (x9:iS)=> b)) x6) Xg) (fun (x7:iS)=> ((eq_ref b) (x6 x7)))) as proof of (((eq (iS->b)) x6) Xg)
% Found (((((functional_extensionality_dep iS) (fun (x9:iS)=> b)) x6) Xg) (fun (x7:iS)=> ((eq_ref b) (x6 x7)))) as proof of (((eq (iS->b)) x6) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((((functional_extensionality_dep iS) (fun (x9:iS)=> b)) x6) Xg) (fun (x7:iS)=> ((eq_ref b) (x6 x7))))) as proof of (((eq (iS->b)) x6) Xg)
% Found eq_ref00:=(eq_ref0 (x6 x7)):(((eq b) (x6 x7)) (x6 x7))
% Found (eq_ref0 (x6 x7)) as proof of (((eq b) (x6 x7)) (Xg x7))
% Found ((eq_ref b) (x6 x7)) as proof of (((eq b) (x6 x7)) (Xg x7))
% Found ((eq_ref b) (x6 x7)) as proof of (((eq b) (x6 x7)) (Xg x7))
% Found (fun (x7:iS)=> ((eq_ref b) (x6 x7))) as proof of (((eq b) (x6 x7)) (Xg x7))
% Found (fun (x7:iS)=> ((eq_ref b) (x6 x7))) as proof of (forall (x:iS), (((eq b) (x6 x)) (Xg x)))
% Found (functional_extensionality0000 (fun (x7:iS)=> ((eq_ref b) (x6 x7)))) as proof of (((eq (iS->b)) x6) Xg)
% Found ((functional_extensionality000 Xg) (fun (x7:iS)=> ((eq_ref b) (x6 x7)))) as proof of (((eq (iS->b)) x6) Xg)
% Found (((functional_extensionality00 x6) Xg) (fun (x7:iS)=> ((eq_ref b) (x6 x7)))) as proof of (((eq (iS->b)) x6) Xg)
% Found ((((functional_extensionality0 b) x6) Xg) (fun (x7:iS)=> ((eq_ref b) (x6 x7)))) as proof of (((eq (iS->b)) x6) Xg)
% Found (((((functional_extensionality iS) b) x6) Xg) (fun (x7:iS)=> ((eq_ref b) (x6 x7)))) as proof of (((eq (iS->b)) x6) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((((functional_extensionality iS) b) x6) Xg) (fun (x7:iS)=> ((eq_ref b) (x6 x7))))) as proof of (((eq (iS->b)) x6) Xg)
% Found eq_ref00:=(eq_ref0 (x0 c0)):(((eq b) (x0 c0)) (x0 c0))
% Found (eq_ref0 (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found eq_ref00:=(eq_ref0 (x0 c0)):(((eq b) (x0 c0)) (x0 c0))
% Found (eq_ref0 (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found eq_ref00:=(eq_ref0 (x2 c0)):(((eq b) (x2 c0)) (x2 c0))
% Found (eq_ref0 (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found eq_ref00:=(eq_ref0 (x4 c0)):(((eq b) (x4 c0)) (x4 c0))
% Found (eq_ref0 (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found ((eq_ref b) (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found ((eq_ref b) (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found ((eq_ref b) (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found eq_ref00:=(eq_ref0 (x6 c0)):(((eq b) (x6 c0)) (x6 c0))
% Found (eq_ref0 (x6 c0)) as proof of (((eq b) (x6 c0)) c02)
% Found ((eq_ref b) (x6 c0)) as proof of (((eq b) (x6 c0)) c02)
% Found ((eq_ref b) (x6 c0)) as proof of (((eq b) (x6 c0)) c02)
% Found ((eq_ref b) (x6 c0)) as proof of (((eq b) (x6 c0)) c02)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of ((P x0)->(P x'))
% Found ((eq_ref0 x0) P) as proof of ((P x0)->(P x'))
% Found (((eq_ref (iS->b)) x0) P) as proof of ((P x0)->(P x'))
% Found (((eq_ref (iS->b)) x0) P) as proof of ((P x0)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x0) P)) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P x')))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x0) P)) as proof of (((eq (iS->b)) x0) x')
% Found x1:(P x0)
% Instantiate: x0:=x':(iS->b)
% Found (fun (x1:(P x0))=> x1) as proof of (P x')
% Found (fun (P:((iS->b)->Prop)) (x1:(P x0))=> x1) as proof of ((P x0)->(P x'))
% Found (fun (P:((iS->b)->Prop)) (x1:(P x0))=> x1) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P x')))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop)) (x1:(P x0))=> x1) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 (x0 x7)):(((eq b) (x0 x7)) (x0 x7))
% Found (eq_ref0 (x0 x7)) as proof of (((eq b) (x0 x7)) (Xg x7))
% Found ((eq_ref b) (x0 x7)) as proof of (((eq b) (x0 x7)) (Xg x7))
% Found ((eq_ref b) (x0 x7)) as proof of (((eq b) (x0 x7)) (Xg x7))
% Found (fun (x7:iS)=> ((eq_ref b) (x0 x7))) as proof of (((eq b) (x0 x7)) (Xg x7))
% Found (fun (x7:iS)=> ((eq_ref b) (x0 x7))) as proof of (forall (x:iS), (((eq b) (x0 x)) (Xg x)))
% Found (functional_extensionality0000 (fun (x7:iS)=> ((eq_ref b) (x0 x7)))) as proof of (((eq (iS->b)) x0) Xg)
% Found ((functional_extensionality000 Xg) (fun (x7:iS)=> ((eq_ref b) (x0 x7)))) as proof of (((eq (iS->b)) x0) Xg)
% Found (((functional_extensionality00 x0) Xg) (fun (x7:iS)=> ((eq_ref b) (x0 x7)))) as proof of (((eq (iS->b)) x0) Xg)
% Found ((((functional_extensionality0 b) x0) Xg) (fun (x7:iS)=> ((eq_ref b) (x0 x7)))) as proof of (((eq (iS->b)) x0) Xg)
% Found (((((functional_extensionality iS) b) x0) Xg) (fun (x7:iS)=> ((eq_ref b) (x0 x7)))) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((((functional_extensionality iS) b) x0) Xg) (fun (x7:iS)=> ((eq_ref b) (x0 x7))))) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 (x0 x7)):(((eq b) (x0 x7)) (x0 x7))
% Found (eq_ref0 (x0 x7)) as proof of (((eq b) (x0 x7)) (Xg x7))
% Found ((eq_ref b) (x0 x7)) as proof of (((eq b) (x0 x7)) (Xg x7))
% Found ((eq_ref b) (x0 x7)) as proof of (((eq b) (x0 x7)) (Xg x7))
% Found (fun (x7:iS)=> ((eq_ref b) (x0 x7))) as proof of (((eq b) (x0 x7)) (Xg x7))
% Found (fun (x7:iS)=> ((eq_ref b) (x0 x7))) as proof of (forall (x:iS), (((eq b) (x0 x)) (Xg x)))
% Found (functional_extensionality_dep0000 (fun (x7:iS)=> ((eq_ref b) (x0 x7)))) as proof of (((eq (iS->b)) x0) Xg)
% Found ((functional_extensionality_dep000 Xg) (fun (x7:iS)=> ((eq_ref b) (x0 x7)))) as proof of (((eq (iS->b)) x0) Xg)
% Found (((functional_extensionality_dep00 x0) Xg) (fun (x7:iS)=> ((eq_ref b) (x0 x7)))) as proof of (((eq (iS->b)) x0) Xg)
% Found ((((functional_extensionality_dep0 (fun (x9:iS)=> b)) x0) Xg) (fun (x7:iS)=> ((eq_ref b) (x0 x7)))) as proof of (((eq (iS->b)) x0) Xg)
% Found (((((functional_extensionality_dep iS) (fun (x9:iS)=> b)) x0) Xg) (fun (x7:iS)=> ((eq_ref b) (x0 x7)))) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((((functional_extensionality_dep iS) (fun (x9:iS)=> b)) x0) Xg) (fun (x7:iS)=> ((eq_ref b) (x0 x7))))) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 x0):(((eq (iS->b)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (iS->b)) x0) x')
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) x')
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x8:(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)))))=> ((eq_ref (iS->b)) x0)) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x0)) as proof of ((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))))->(((eq (iS->b)) x0) x'))
% Found (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x0)) as proof of ((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))))->(((eq (iS->b)) x0) x')))
% Found (and_rect30 (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) x')
% Found ((and_rect3 (((eq (iS->b)) x0) x')) (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) x')
% Found (((fun (P:Type) (x7:((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))))->P)))=> (((((and_rect (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))))) P) x7) x5)) (((eq (iS->b)) x0) x')) (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((fun (P:Type) (x7:((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))))->P)))=> (((((and_rect (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))))) P) x7) x5)) (((eq (iS->b)) x0) x')) (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x0)))) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 x2):(((eq (iS->b)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (iS->b)) x2) x')
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) x')
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) x')
% Found (fun (x8:(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)))))=> ((eq_ref (iS->b)) x2)) as proof of (((eq (iS->b)) x2) x')
% Found (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x2)) as proof of ((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))))->(((eq (iS->b)) x2) x'))
% Found (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x2)) as proof of ((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))))->(((eq (iS->b)) x2) x')))
% Found (and_rect30 (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) x')
% Found ((and_rect3 (((eq (iS->b)) x2) x')) (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) x')
% Found (((fun (P:Type) (x7:((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))))->P)))=> (((((and_rect (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))))) P) x7) x5)) (((eq (iS->b)) x2) x')) (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((fun (P:Type) (x7:((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))))->P)))=> (((((and_rect (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))))) P) x7) x5)) (((eq (iS->b)) x2) x')) (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x2)))) as proof of (((eq (iS->b)) x2) x')
% Found eq_ref00:=(eq_ref0 (x2 x7)):(((eq b) (x2 x7)) (x2 x7))
% Found (eq_ref0 (x2 x7)) as proof of (((eq b) (x2 x7)) (Xg x7))
% Found ((eq_ref b) (x2 x7)) as proof of (((eq b) (x2 x7)) (Xg x7))
% Found ((eq_ref b) (x2 x7)) as proof of (((eq b) (x2 x7)) (Xg x7))
% Found (fun (x7:iS)=> ((eq_ref b) (x2 x7))) as proof of (((eq b) (x2 x7)) (Xg x7))
% Found (fun (x7:iS)=> ((eq_ref b) (x2 x7))) as proof of (forall (x:iS), (((eq b) (x2 x)) (Xg x)))
% Found (functional_extensionality_dep0000 (fun (x7:iS)=> ((eq_ref b) (x2 x7)))) as proof of (((eq (iS->b)) x2) Xg)
% Found ((functional_extensionality_dep000 Xg) (fun (x7:iS)=> ((eq_ref b) (x2 x7)))) as proof of (((eq (iS->b)) x2) Xg)
% Found (((functional_extensionality_dep00 x2) Xg) (fun (x7:iS)=> ((eq_ref b) (x2 x7)))) as proof of (((eq (iS->b)) x2) Xg)
% Found ((((functional_extensionality_dep0 (fun (x9:iS)=> b)) x2) Xg) (fun (x7:iS)=> ((eq_ref b) (x2 x7)))) as proof of (((eq (iS->b)) x2) Xg)
% Found (((((functional_extensionality_dep iS) (fun (x9:iS)=> b)) x2) Xg) (fun (x7:iS)=> ((eq_ref b) (x2 x7)))) as proof of (((eq (iS->b)) x2) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((((functional_extensionality_dep iS) (fun (x9:iS)=> b)) x2) Xg) (fun (x7:iS)=> ((eq_ref b) (x2 x7))))) as proof of (((eq (iS->b)) x2) Xg)
% Found eq_ref00:=(eq_ref0 (x2 x7)):(((eq b) (x2 x7)) (x2 x7))
% Found (eq_ref0 (x2 x7)) as proof of (((eq b) (x2 x7)) (Xg x7))
% Found ((eq_ref b) (x2 x7)) as proof of (((eq b) (x2 x7)) (Xg x7))
% Found ((eq_ref b) (x2 x7)) as proof of (((eq b) (x2 x7)) (Xg x7))
% Found (fun (x7:iS)=> ((eq_ref b) (x2 x7))) as proof of (((eq b) (x2 x7)) (Xg x7))
% Found (fun (x7:iS)=> ((eq_ref b) (x2 x7))) as proof of (forall (x:iS), (((eq b) (x2 x)) (Xg x)))
% Found (functional_extensionality0000 (fun (x7:iS)=> ((eq_ref b) (x2 x7)))) as proof of (((eq (iS->b)) x2) Xg)
% Found ((functional_extensionality000 Xg) (fun (x7:iS)=> ((eq_ref b) (x2 x7)))) as proof of (((eq (iS->b)) x2) Xg)
% Found (((functional_extensionality00 x2) Xg) (fun (x7:iS)=> ((eq_ref b) (x2 x7)))) as proof of (((eq (iS->b)) x2) Xg)
% Found ((((functional_extensionality0 b) x2) Xg) (fun (x7:iS)=> ((eq_ref b) (x2 x7)))) as proof of (((eq (iS->b)) x2) Xg)
% Found (((((functional_extensionality iS) b) x2) Xg) (fun (x7:iS)=> ((eq_ref b) (x2 x7)))) as proof of (((eq (iS->b)) x2) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((((functional_extensionality iS) b) x2) Xg) (fun (x7:iS)=> ((eq_ref b) (x2 x7))))) as proof of (((eq (iS->b)) x2) Xg)
% Found eq_ref00:=(eq_ref0 x4):(((eq (iS->b)) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq (iS->b)) x4) x')
% Found ((eq_ref (iS->b)) x4) as proof of (((eq (iS->b)) x4) x')
% Found ((eq_ref (iS->b)) x4) as proof of (((eq (iS->b)) x4) x')
% Found (fun (x8:(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)))))=> ((eq_ref (iS->b)) x4)) as proof of (((eq (iS->b)) x4) x')
% Found (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x4)) as proof of ((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))))->(((eq (iS->b)) x4) x'))
% Found (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x4)) as proof of ((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))))->(((eq (iS->b)) x4) x')))
% Found (and_rect30 (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x4))) as proof of (((eq (iS->b)) x4) x')
% Found ((and_rect3 (((eq (iS->b)) x4) x')) (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x4))) as proof of (((eq (iS->b)) x4) x')
% Found (((fun (P:Type) (x7:((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))))->P)))=> (((((and_rect (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))))) P) x7) x5)) (((eq (iS->b)) x4) x')) (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x4))) as proof of (((eq (iS->b)) x4) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((fun (P:Type) (x7:((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))))->P)))=> (((((and_rect (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))))) P) x7) x5)) (((eq (iS->b)) x4) x')) (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x4)))) as proof of (((eq (iS->b)) x4) x')
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (iS->b)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (iS->b)) x0) P) as proof of (P0 x0)
% Found eq_ref00:=(eq_ref0 x6):(((eq (iS->b)) x6) x6)
% Found (eq_ref0 x6) as proof of (((eq (iS->b)) x6) x')
% Found ((eq_ref (iS->b)) x6) as proof of (((eq (iS->b)) x6) x')
% Found ((eq_ref (iS->b)) x6) as proof of (((eq (iS->b)) x6) x')
% Found (fun (x8:(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)))))=> ((eq_ref (iS->b)) x6)) as proof of (((eq (iS->b)) x6) x')
% Found (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x6)) as proof of ((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))))->(((eq (iS->b)) x6) x'))
% Found (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x6)) as proof of ((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))))->(((eq (iS->b)) x6) x')))
% Found (and_rect30 (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x6))) as proof of (((eq (iS->b)) x6) x')
% Found ((and_rect3 (((eq (iS->b)) x6) x')) (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x6))) as proof of (((eq (iS->b)) x6) x')
% Found (((fun (P:Type) (x7:((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))))->P)))=> (((((and_rect (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))))) P) x7) x4)) (((eq (iS->b)) x6) x')) (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x6))) as proof of (((eq (iS->b)) x6) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((fun (P:Type) (x7:((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))))->P)))=> (((((and_rect (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))))) P) x7) x4)) (((eq (iS->b)) x6) x')) (fun (x7:(forall (Xx:iS) (Xy:iS), (not (((eq iS) ((cP Xx) Xy)) c0)))) (x8:(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)))))=> ((eq_ref (iS->b)) x6)))) as proof of (((eq (iS->b)) x6) x')
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (iS->b)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (iS->b)) x0) P) as proof of (P0 x0)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (iS->b)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (iS->b)) x0) P) as proof of (P0 x0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (iS->b)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (iS->b)) b0) Xg)
% Found ((eq_ref (iS->b)) b0) as proof of (((eq (iS->b)) b0) Xg)
% Found ((eq_ref (iS->b)) b0) as proof of (((eq (iS->b)) b0) Xg)
% Found ((eq_ref (iS->b)) b0) as proof of (((eq (iS->b)) b0) Xg)
% Found eq_ref00:=(eq_ref0 x0):(((eq (iS->b)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (iS->b)) x0) b0)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) b0)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) b0)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) b0)
% Found ((eq_trans0000 ((eq_ref (iS->b)) x0)) ((eq_ref (iS->b)) b0)) as proof of (((eq (iS->b)) x0) Xg)
% Found (((eq_trans000 Xg) ((eq_ref (iS->b)) x0)) ((eq_ref (iS->b)) Xg)) as proof of (((eq (iS->b)) x0) Xg)
% Found ((((fun (b0:(iS->b))=> ((eq_trans00 b0) Xg)) Xg) ((eq_ref (iS->b)) x0)) ((eq_ref (iS->b)) Xg)) as proof of (((eq (iS->b)) x0) Xg)
% Found ((((fun (b0:(iS->b))=> (((eq_trans0 x0) b0) Xg)) Xg) ((eq_ref (iS->b)) x0)) ((eq_ref (iS->b)) Xg)) as proof of (((eq (iS->b)) x0) Xg)
% Found ((((fun (b0:(iS->b))=> ((((eq_trans (iS->b)) x0) b0) Xg)) Xg) ((eq_ref (iS->b)) x0)) ((eq_ref (iS->b)) Xg)) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((((fun (b0:(iS->b))=> ((((eq_trans (iS->b)) x0) b0) Xg)) Xg) ((eq_ref (iS->b)) x0)) ((eq_ref (iS->b)) Xg))) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 (x4 x7)):(((eq b) (x4 x7)) (x4 x7))
% Found (eq_ref0 (x4 x7)) as proof of (((eq b) (x4 x7)) (Xg x7))
% Found ((eq_ref b) (x4 x7)) as proof of (((eq b) (x4 x7)) (Xg x7))
% Found ((eq_ref b) (x4 x7)) as proof of (((eq b) (x4 x7)) (Xg x7))
% Found (fun (x7:iS)=> ((eq_ref b) (x4 x7))) as proof of (((eq b) (x4 x7)) (Xg x7))
% Found (fun (x7:iS)=> ((eq_ref b) (x4 x7))) as proof of (forall (x:iS), (((eq b) (x4 x)) (Xg x)))
% Found (functional_extensionality_dep0000 (fun (x7:iS)=> ((eq_ref b) (x4 x7)))) as proof of (((eq (iS->b)) x4) Xg)
% Found ((functional_extensionality_dep000 Xg) (fun (x7:iS)=> ((eq_ref b) (x4 x7)))) as proof of (((eq (iS->b)) x4) Xg)
% Found (((functional_extensionality_dep00 x4) Xg) (fun (x7:iS)=> ((eq_ref b) (x4 x7)))) as proof of (((eq (iS->b)) x4) Xg)
% Found ((((functional_extensionality_dep0 (fun (x9:iS)=> b)) x4) Xg) (fun (x7:iS)=> ((eq_ref b) (x4 x7)))) as proof of (((eq (iS->b)) x4) Xg)
% Found (((((functional_extensionality_dep iS) (fun (x9:iS)=> b)) x4) Xg) (fun (x7:iS)=> ((eq_ref b) (x4 x7)))) as proof of (((eq (iS->b)) x4) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((((functional_extensionality_dep iS) (fun (x9:iS)=> b)) x4) Xg) (fun (x7:iS)=> ((eq_ref b) (x4 x7))))) as proof of (((eq (iS->b)) x4) Xg)
% Found eq_ref00:=(eq_ref0 (x4 x7)):(((eq b) (x4 x7)) (x4 x7))
% Found (eq_ref0 (x4 x7)) as proof of (((eq b) (x4 x7)) (Xg x7))
% Found ((eq_ref b) (x4 x7)) as proof of (((eq b) (x4 x7)) (Xg x7))
% Found ((eq_ref b) (x4 x7)) as proof of (((eq b) (x4 x7)) (Xg x7))
% Found (fun (x7:iS)=> ((eq_ref b) (x4 x7))) as proof of (((eq b) (x4 x7)) (Xg x7))
% Found (fun (x7:iS)=> ((eq_ref b) (x4 x7))) as proof of (forall (x:iS), (((eq b) (x4 x)) (Xg x)))
% Found (functional_extensionality0000 (fun (x7:iS)=> ((eq_ref b) (x4 x7)))) as proof of (((eq (iS->b)) x4) Xg)
% Found ((functional_extensionality000 Xg) (fun (x7:iS)=> ((eq_ref b) (x4 x7)))) as proof of (((eq (iS->b)) x4) Xg)
% Found (((functional_extensionality00 x4) Xg) (fun (x7:iS)=> ((eq_ref b) (x4 x7)))) as proof of (((eq (iS->b)) x4) Xg)
% Found ((((functional_extensionality0 b) x4) Xg) (fun (x7:iS)=> ((eq_ref b) (x4 x7)))) as proof of (((eq (iS->b)) x4) Xg)
% Found (((((functional_extensionality iS) b) x4) Xg) (fun (x7:iS)=> ((eq_ref b) (x4 x7)))) as proof of (((eq (iS->b)) x4) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((((functional_extensionality iS) b) x4) Xg) (fun (x7:iS)=> ((eq_ref b) (x4 x7))))) as proof of (((eq (iS->b)) x4) Xg)
% Found eta_expansion000:=(eta_expansion00 x'):(((eq (iS->b)) x') (fun (x:iS)=> (x' x)))
% Found (eta_expansion00 x') as proof of (((eq (iS->b)) x') x0)
% Found ((eta_expansion0 b) x') as proof of (((eq (iS->b)) x') x0)
% Found (((eta_expansion iS) b) x') as proof of (((eq (iS->b)) x') x0)
% Found (((eta_expansion iS) b) x') as proof of (((eq (iS->b)) x') x0)
% Found (((eta_expansion iS) b) x') as proof of (((eq (iS->b)) x') x0)
% Found (eq_sym0000 (((eta_expansion iS) b) x')) as proof of ((P x0)->(P x'))
% Found (eq_sym0000 (((eta_expansion iS) b) x')) as proof of ((P x0)->(P x'))
% Found ((fun (x1:(((eq (iS->b)) x') x0))=> ((eq_sym000 x1) P)) (((eta_expansion iS) b) x')) as proof of ((P x0)->(P x'))
% Found ((fun (x1:(((eq (iS->b)) x') x0))=> (((eq_sym00 x0) x1) P)) (((eta_expansion iS) b) x')) as proof of ((P x0)->(P x'))
% Found ((fun (x1:(((eq (iS->b)) x') x0))=> ((((eq_sym0 x') x0) x1) P)) (((eta_expansion iS) b) x')) as proof of ((P x0)->(P x'))
% Found ((fun (x1:(((eq (iS->b)) x') x0))=> (((((eq_sym (iS->b)) x') x0) x1) P)) (((eta_expansion iS) b) x')) as proof of ((P x0)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> ((fun (x1:(((eq (iS->b)) x') x0))=> (((((eq_sym (iS->b)) x') x0) x1) P)) (((eta_expansion iS) b) x'))) as proof of ((P x0)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop))=> ((fun (x1:(((eq (iS->b)) x') x0))=> (((((eq_sym (iS->b)) x') x0) x1) P)) (((eta_expansion iS) b) x'))) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 x'):(((eq (iS->b)) x') x')
% Found (eq_ref0 x') as proof of (((eq (iS->b)) x') x0)
% Found ((eq_ref (iS->b)) x') as proof of (((eq (iS->b)) x') x0)
% Found ((eq_ref (iS->b)) x') as proof of (((eq (iS->b)) x') x0)
% Found ((eq_ref (iS->b)) x') as proof of (((eq (iS->b)) x') x0)
% Found ((eq_sym0000 ((eq_ref (iS->b)) x')) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found ((eq_sym0000 ((eq_ref (iS->b)) x')) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found (((fun (x1:(((eq (iS->b)) x') x0))=> ((eq_sym000 x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) ((eq_ref (iS->b)) x')) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found (((fun (x1:(((eq (iS->b)) x') x0))=> (((eq_sym00 x0) x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) ((eq_ref (iS->b)) x')) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found (((fun (x1:(((eq (iS->b)) x') x0))=> ((((eq_sym0 x') x0) x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) ((eq_ref (iS->b)) x')) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found (((fun (x1:(((eq (iS->b)) x') x0))=> (((((eq_sym (iS->b)) x') x0) x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) ((eq_ref (iS->b)) x')) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> (((fun (x1:(((eq (iS->b)) x') x0))=> (((((eq_sym (iS->b)) x') x0) x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) ((eq_ref (iS->b)) x')) (((eq_ref (iS->b)) x0) P))) as proof of ((P x0)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop))=> (((fun (x1:(((eq (iS->b)) x') x0))=> (((((eq_sym (iS->b)) x') x0) x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) ((eq_ref (iS->b)) x')) (((eq_ref (iS->b)) x0) P))) as proof of (((eq (iS->b)) x0) x')
% Found eta_expansion000:=(eta_expansion00 ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))):(((eq ((iS->b)->Prop)) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) (fun (x:(iS->b))=> (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x)))
% Found (eta_expansion00 ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) as proof of (((eq ((iS->b)->Prop)) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) b0)
% Found ((eta_expansion0 Prop) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) as proof of (((eq ((iS->b)->Prop)) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) b0)
% Found (((eta_expansion (iS->b)) Prop) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) as proof of (((eq ((iS->b)->Prop)) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) b0)
% Found (((eta_expansion (iS->b)) Prop) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) as proof of (((eq ((iS->b)->Prop)) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) b0)
% Found (((eta_expansion (iS->b)) Prop) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) as proof of (((eq ((iS->b)->Prop)) ((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy)))))))) b0)
% Found eq_ref00:=(eq_ref0 (x0 c0)):(((eq b) (x0 c0)) (x0 c0))
% Found (eq_ref0 (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found eq_ref00:=(eq_ref0 (x2 c0)):(((eq b) (x2 c0)) (x2 c0))
% Found (eq_ref0 (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found eq_ref00:=(eq_ref0 (x4 c0)):(((eq b) (x4 c0)) (x4 c0))
% Found (eq_ref0 (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found ((eq_ref b) (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found ((eq_ref b) (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found ((eq_ref b) (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found eq_ref00:=(eq_ref0 x2):(((eq (iS->b)) x2) x2)
% Found (eq_ref0 x2) as proof of (forall (P:((iS->b)->Prop)), ((P x2)->(P Xg)))
% Found ((eq_ref (iS->b)) x2) as proof of (forall (P:((iS->b)->Prop)), ((P x2)->(P Xg)))
% Found ((eq_ref (iS->b)) x2) as proof of (forall (P:((iS->b)->Prop)), ((P x2)->(P Xg)))
% Found ((eq_ref (iS->b)) x2) as proof of (forall (P:((iS->b)->Prop)), ((P x2)->(P Xg)))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((eq_ref (iS->b)) x2)) as proof of (((eq (iS->b)) x2) Xg)
% Found eq_ref00:=(eq_ref0 (x8 c0)):(((eq b) (x8 c0)) (x8 c0))
% Found (eq_ref0 (x8 c0)) as proof of (((eq b) (x8 c0)) c02)
% Found ((eq_ref b) (x8 c0)) as proof of (((eq b) (x8 c0)) c02)
% Found ((eq_ref b) (x8 c0)) as proof of (((eq b) (x8 c0)) c02)
% Found ((eq_ref b) (x8 c0)) as proof of (((eq b) (x8 c0)) c02)
% Found eq_ref00:=(eq_ref0 (x6 x7)):(((eq b) (x6 x7)) (x6 x7))
% Found (eq_ref0 (x6 x7)) as proof of (((eq b) (x6 x7)) (x' x7))
% Found ((eq_ref b) (x6 x7)) as proof of (((eq b) (x6 x7)) (x' x7))
% Found ((eq_ref b) (x6 x7)) as proof of (((eq b) (x6 x7)) (x' x7))
% Found (fun (x7:iS)=> ((eq_ref b) (x6 x7))) as proof of (((eq b) (x6 x7)) (x' x7))
% Found (fun (x7:iS)=> ((eq_ref b) (x6 x7))) as proof of (forall (x:iS), (((eq b) (x6 x)) (x' x)))
% Found (functional_extensionality_dep0000 (fun (x7:iS)=> ((eq_ref b) (x6 x7)))) as proof of (((eq (iS->b)) x6) x')
% Found ((functional_extensionality_dep000 x') (fun (x7:iS)=> ((eq_ref b) (x6 x7)))) as proof of (((eq (iS->b)) x6) x')
% Found (((functional_extensionality_dep00 x6) x') (fun (x7:iS)=> ((eq_ref b) (x6 x7)))) as proof of (((eq (iS->b)) x6) x')
% Found ((((functional_extensionality_dep0 (fun (x9:iS)=> b)) x6) x') (fun (x7:iS)=> ((eq_ref b) (x6 x7)))) as proof of (((eq (iS->b)) x6) x')
% Found (((((functional_extensionality_dep iS) (fun (x9:iS)=> b)) x6) x') (fun (x7:iS)=> ((eq_ref b) (x6 x7)))) as proof of (((eq (iS->b)) x6) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((((functional_extensionality_dep iS) (fun (x9:iS)=> b)) x6) x') (fun (x7:iS)=> ((eq_ref b) (x6 x7))))) as proof of (((eq (iS->b)) x6) x')
% Found eq_ref00:=(eq_ref0 (x6 x7)):(((eq b) (x6 x7)) (x6 x7))
% Found (eq_ref0 (x6 x7)) as proof of (((eq b) (x6 x7)) (x' x7))
% Found ((eq_ref b) (x6 x7)) as proof of (((eq b) (x6 x7)) (x' x7))
% Found ((eq_ref b) (x6 x7)) as proof of (((eq b) (x6 x7)) (x' x7))
% Found (fun (x7:iS)=> ((eq_ref b) (x6 x7))) as proof of (((eq b) (x6 x7)) (x' x7))
% Found (fun (x7:iS)=> ((eq_ref b) (x6 x7))) as proof of (forall (x:iS), (((eq b) (x6 x)) (x' x)))
% Found (functional_extensionality0000 (fun (x7:iS)=> ((eq_ref b) (x6 x7)))) as proof of (((eq (iS->b)) x6) x')
% Found ((functional_extensionality000 x') (fun (x7:iS)=> ((eq_ref b) (x6 x7)))) as proof of (((eq (iS->b)) x6) x')
% Found (((functional_extensionality00 x6) x') (fun (x7:iS)=> ((eq_ref b) (x6 x7)))) as proof of (((eq (iS->b)) x6) x')
% Found ((((functional_extensionality0 b) x6) x') (fun (x7:iS)=> ((eq_ref b) (x6 x7)))) as proof of (((eq (iS->b)) x6) x')
% Found (((((functional_extensionality iS) b) x6) x') (fun (x7:iS)=> ((eq_ref b) (x6 x7)))) as proof of (((eq (iS->b)) x6) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((((functional_extensionality iS) b) x6) x') (fun (x7:iS)=> ((eq_ref b) (x6 x7))))) as proof of (((eq (iS->b)) x6) x')
% Found eq_ref00:=(eq_ref0 (f x8)):(((eq Prop) (f x8)) (f x8))
% Found (eq_ref0 (f x8)) as proof of (((eq Prop) (f x8)) ((and ((and (((eq b) (x8 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x8 ((cP Xx) Xy))) ((cP2 (x8 Xx)) (x8 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x8) Xg)))))
% Found ((eq_ref Prop) (f x8)) as proof of (((eq Prop) (f x8)) ((and ((and (((eq b) (x8 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x8 ((cP Xx) Xy))) ((cP2 (x8 Xx)) (x8 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x8) Xg)))))
% Found ((eq_ref Prop) (f x8)) as proof of (((eq Prop) (f x8)) ((and ((and (((eq b) (x8 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x8 ((cP Xx) Xy))) ((cP2 (x8 Xx)) (x8 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x8) Xg)))))
% Found (fun (x8:(iS->b))=> ((eq_ref Prop) (f x8))) as proof of (((eq Prop) (f x8)) ((and ((and (((eq b) (x8 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x8 ((cP Xx) Xy))) ((cP2 (x8 Xx)) (x8 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x8) Xg)))))
% Found (fun (x8:(iS->b))=> ((eq_ref Prop) (f x8))) as proof of (forall (x:(iS->b)), (((eq Prop) (f x)) ((and ((and (((eq b) (x c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x ((cP Xx) Xy))) ((cP2 (x Xx)) (x Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x) Xg))))))
% Found eq_ref00:=(eq_ref0 (f x8)):(((eq Prop) (f x8)) (f x8))
% Found (eq_ref0 (f x8)) as proof of (((eq Prop) (f x8)) ((and ((and (((eq b) (x8 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x8 ((cP Xx) Xy))) ((cP2 (x8 Xx)) (x8 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x8) Xg)))))
% Found ((eq_ref Prop) (f x8)) as proof of (((eq Prop) (f x8)) ((and ((and (((eq b) (x8 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x8 ((cP Xx) Xy))) ((cP2 (x8 Xx)) (x8 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x8) Xg)))))
% Found ((eq_ref Prop) (f x8)) as proof of (((eq Prop) (f x8)) ((and ((and (((eq b) (x8 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x8 ((cP Xx) Xy))) ((cP2 (x8 Xx)) (x8 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x8) Xg)))))
% Found (fun (x8:(iS->b))=> ((eq_ref Prop) (f x8))) as proof of (((eq Prop) (f x8)) ((and ((and (((eq b) (x8 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x8 ((cP Xx) Xy))) ((cP2 (x8 Xx)) (x8 Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x8) Xg)))))
% Found (fun (x8:(iS->b))=> ((eq_ref Prop) (f x8))) as proof of (forall (x:(iS->b)), (((eq Prop) (f x)) ((and ((and (((eq b) (x c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x ((cP Xx) Xy))) ((cP2 (x Xx)) (x Xy)))))) (forall (Xg:(iS->b)), (((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))->(((eq (iS->b)) x) Xg))))))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq b) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq b) (x0 x1)) (Xg x1))
% Found ((eq_ref b) (x0 x1)) as proof of (((eq b) (x0 x1)) (Xg x1))
% Found ((eq_ref b) (x0 x1)) as proof of (((eq b) (x0 x1)) (Xg x1))
% Found (fun (x1:iS)=> ((eq_ref b) (x0 x1))) as proof of (((eq b) (x0 x1)) (Xg x1))
% Found (fun (x1:iS)=> ((eq_ref b) (x0 x1))) as proof of (forall (x:iS), (((eq b) (x0 x)) (Xg x)))
% Found ((functional_extensionality00000 (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found ((functional_extensionality00000 (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((functional_extensionality0000 x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> (((functional_extensionality000 Xg) x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((((functional_extensionality00 x0) Xg) x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> (((((functional_extensionality0 b) x0) Xg) x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((((((functional_extensionality iS) b) x0) Xg) x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> (((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((((((functional_extensionality iS) b) x0) Xg) x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) (((eq_ref (iS->b)) x0) P))) as proof of ((P x0)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> (((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((((((functional_extensionality iS) b) x0) Xg) x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) (((eq_ref (iS->b)) x0) P))) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq b) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq b) (x0 x1)) (Xg x1))
% Found ((eq_ref b) (x0 x1)) as proof of (((eq b) (x0 x1)) (Xg x1))
% Found ((eq_ref b) (x0 x1)) as proof of (((eq b) (x0 x1)) (Xg x1))
% Found (fun (x1:iS)=> ((eq_ref b) (x0 x1))) as proof of (((eq b) (x0 x1)) (Xg x1))
% Found (fun (x1:iS)=> ((eq_ref b) (x0 x1))) as proof of (forall (x:iS), (((eq b) (x0 x)) (Xg x)))
% Found (functional_extensionality00000 (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of ((P x0)->(P Xg))
% Found (functional_extensionality00000 (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of ((P x0)->(P Xg))
% Found ((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((functional_extensionality0000 x1) P)) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of ((P x0)->(P Xg))
% Found ((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> (((functional_extensionality000 Xg) x1) P)) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of ((P x0)->(P Xg))
% Found ((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((((functional_extensionality00 x0) Xg) x1) P)) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of ((P x0)->(P Xg))
% Found ((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> (((((functional_extensionality0 b) x0) Xg) x1) P)) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of ((P x0)->(P Xg))
% Found ((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((((((functional_extensionality iS) b) x0) Xg) x1) P)) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of ((P x0)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> ((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((((((functional_extensionality iS) b) x0) Xg) x1) P)) (fun (x1:iS)=> ((eq_ref b) (x0 x1))))) as proof of ((P x0)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> ((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((((((functional_extensionality iS) b) x0) Xg) x1) P)) (fun (x1:iS)=> ((eq_ref b) (x0 x1))))) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq b) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq b) (x0 x1)) (Xg x1))
% Found ((eq_ref b) (x0 x1)) as proof of (((eq b) (x0 x1)) (Xg x1))
% Found ((eq_ref b) (x0 x1)) as proof of (((eq b) (x0 x1)) (Xg x1))
% Found (fun (x1:iS)=> ((eq_ref b) (x0 x1))) as proof of (((eq b) (x0 x1)) (Xg x1))
% Found (fun (x1:iS)=> ((eq_ref b) (x0 x1))) as proof of (forall (x:iS), (((eq b) (x0 x)) (Xg x)))
% Found ((functional_extensionality_dep00000 (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found ((functional_extensionality_dep00000 (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((functional_extensionality_dep0000 x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> (((functional_extensionality_dep000 Xg) x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((((functional_extensionality_dep00 x0) Xg) x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> (((((functional_extensionality_dep0 (fun (x3:iS)=> b)) x0) Xg) x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((((((functional_extensionality_dep iS) (fun (x3:iS)=> b)) x0) Xg) x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> (((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((((((functional_extensionality_dep iS) (fun (x3:iS)=> b)) x0) Xg) x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) (((eq_ref (iS->b)) x0) P))) as proof of ((P x0)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> (((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((((((functional_extensionality_dep iS) (fun (x3:iS)=> b)) x0) Xg) x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) (((eq_ref (iS->b)) x0) P))) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq b) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq b) (x0 x1)) (Xg x1))
% Found ((eq_ref b) (x0 x1)) as proof of (((eq b) (x0 x1)) (Xg x1))
% Found ((eq_ref b) (x0 x1)) as proof of (((eq b) (x0 x1)) (Xg x1))
% Found (fun (x1:iS)=> ((eq_ref b) (x0 x1))) as proof of (((eq b) (x0 x1)) (Xg x1))
% Found (fun (x1:iS)=> ((eq_ref b) (x0 x1))) as proof of (forall (x:iS), (((eq b) (x0 x)) (Xg x)))
% Found (functional_extensionality_dep00000 (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of ((P x0)->(P Xg))
% Found (functional_extensionality_dep00000 (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of ((P x0)->(P Xg))
% Found ((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((functional_extensionality_dep0000 x1) P)) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of ((P x0)->(P Xg))
% Found ((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> (((functional_extensionality_dep000 Xg) x1) P)) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of ((P x0)->(P Xg))
% Found ((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((((functional_extensionality_dep00 x0) Xg) x1) P)) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of ((P x0)->(P Xg))
% Found ((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> (((((functional_extensionality_dep0 (fun (x3:iS)=> b)) x0) Xg) x1) P)) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of ((P x0)->(P Xg))
% Found ((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((((((functional_extensionality_dep iS) (fun (x3:iS)=> b)) x0) Xg) x1) P)) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of ((P x0)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> ((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((((((functional_extensionality_dep iS) (fun (x3:iS)=> b)) x0) Xg) x1) P)) (fun (x1:iS)=> ((eq_ref b) (x0 x1))))) as proof of ((P x0)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> ((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((((((functional_extensionality_dep iS) (fun (x3:iS)=> b)) x0) Xg) x1) P)) (fun (x1:iS)=> ((eq_ref b) (x0 x1))))) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 b0):(((eq (iS->b)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (iS->b)) b0) x')
% Found ((eq_ref (iS->b)) b0) as proof of (((eq (iS->b)) b0) x')
% Found ((eq_ref (iS->b)) b0) as proof of (((eq (iS->b)) b0) x')
% Found ((eq_ref (iS->b)) b0) as proof of (((eq (iS->b)) b0) x')
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (iS->b)) x0) (fun (x:iS)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (iS->b)) x0) b0)
% Found ((eta_expansion_dep0 (fun (x2:iS)=> b)) x0) as proof of (((eq (iS->b)) x0) b0)
% Found (((eta_expansion_dep iS) (fun (x2:iS)=> b)) x0) as proof of (((eq (iS->b)) x0) b0)
% Found (((eta_expansion_dep iS) (fun (x2:iS)=> b)) x0) as proof of (((eq (iS->b)) x0) b0)
% Found (((eta_expansion_dep iS) (fun (x2:iS)=> b)) x0) as proof of (((eq (iS->b)) x0) b0)
% Found eq_ref00:=(eq_ref0 x0):(((eq (iS->b)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (iS->b)) x0) b0)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) b0)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) b0)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) b0)
% Found ((eq_trans0000 ((eq_ref (iS->b)) x0)) ((eq_ref (iS->b)) b0)) as proof of (((eq (iS->b)) x0) x')
% Found (((eq_trans000 x') ((eq_ref (iS->b)) x0)) ((eq_ref (iS->b)) x')) as proof of (((eq (iS->b)) x0) x')
% Found ((((fun (b0:(iS->b))=> ((eq_trans00 b0) x')) x') ((eq_ref (iS->b)) x0)) ((eq_ref (iS->b)) x')) as proof of (((eq (iS->b)) x0) x')
% Found ((((fun (b0:(iS->b))=> (((eq_trans0 x0) b0) x')) x') ((eq_ref (iS->b)) x0)) ((eq_ref (iS->b)) x')) as proof of (((eq (iS->b)) x0) x')
% Found ((((fun (b0:(iS->b))=> ((((eq_trans (iS->b)) x0) b0) x')) x') ((eq_ref (iS->b)) x0)) ((eq_ref (iS->b)) x')) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((((fun (b0:(iS->b))=> ((((eq_trans (iS->b)) x0) b0) x')) x') ((eq_ref (iS->b)) x0)) ((eq_ref (iS->b)) x'))) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (iS->b)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (iS->b)) x0) P) as proof of (P0 x0)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (iS->b)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (iS->b)) x0) P) as proof of (P0 x0)
% Found eq_ref00:=(eq_ref0 x0):(((eq (iS->b)) x0) x0)
% Found (eq_ref0 x0) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P Xg)))
% Found ((eq_ref (iS->b)) x0) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P Xg)))
% Found ((eq_ref (iS->b)) x0) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P Xg)))
% Found ((eq_ref (iS->b)) x0) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P Xg)))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((eq_ref (iS->b)) x0)) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 (x0 c0)):(((eq b) (x0 c0)) (x0 c0))
% Found (eq_ref0 (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found eq_ref00:=(eq_ref0 (x2 c0)):(((eq b) (x2 c0)) (x2 c0))
% Found (eq_ref0 (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found eq_ref00:=(eq_ref0 (x4 c0)):(((eq b) (x4 c0)) (x4 c0))
% Found (eq_ref0 (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found ((eq_ref b) (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found ((eq_ref b) (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found ((eq_ref b) (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found eq_ref00:=(eq_ref0 (x6 c0)):(((eq b) (x6 c0)) (x6 c0))
% Found (eq_ref0 (x6 c0)) as proof of (((eq b) (x6 c0)) c02)
% Found ((eq_ref b) (x6 c0)) as proof of (((eq b) (x6 c0)) c02)
% Found ((eq_ref b) (x6 c0)) as proof of (((eq b) (x6 c0)) c02)
% Found ((eq_ref b) (x6 c0)) as proof of (((eq b) (x6 c0)) c02)
% Found eq_ref00:=(eq_ref0 (x0 x7)):(((eq b) (x0 x7)) (x0 x7))
% Found (eq_ref0 (x0 x7)) as proof of (((eq b) (x0 x7)) (x' x7))
% Found ((eq_ref b) (x0 x7)) as proof of (((eq b) (x0 x7)) (x' x7))
% Found ((eq_ref b) (x0 x7)) as proof of (((eq b) (x0 x7)) (x' x7))
% Found (fun (x7:iS)=> ((eq_ref b) (x0 x7))) as proof of (((eq b) (x0 x7)) (x' x7))
% Found (fun (x7:iS)=> ((eq_ref b) (x0 x7))) as proof of (forall (x:iS), (((eq b) (x0 x)) (x' x)))
% Found (functional_extensionality_dep0000 (fun (x7:iS)=> ((eq_ref b) (x0 x7)))) as proof of (((eq (iS->b)) x0) x')
% Found ((functional_extensionality_dep000 x') (fun (x7:iS)=> ((eq_ref b) (x0 x7)))) as proof of (((eq (iS->b)) x0) x')
% Found (((functional_extensionality_dep00 x0) x') (fun (x7:iS)=> ((eq_ref b) (x0 x7)))) as proof of (((eq (iS->b)) x0) x')
% Found ((((functional_extensionality_dep0 (fun (x9:iS)=> b)) x0) x') (fun (x7:iS)=> ((eq_ref b) (x0 x7)))) as proof of (((eq (iS->b)) x0) x')
% Found (((((functional_extensionality_dep iS) (fun (x9:iS)=> b)) x0) x') (fun (x7:iS)=> ((eq_ref b) (x0 x7)))) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((((functional_extensionality_dep iS) (fun (x9:iS)=> b)) x0) x') (fun (x7:iS)=> ((eq_ref b) (x0 x7))))) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 (x0 x7)):(((eq b) (x0 x7)) (x0 x7))
% Found (eq_ref0 (x0 x7)) as proof of (((eq b) (x0 x7)) (x' x7))
% Found ((eq_ref b) (x0 x7)) as proof of (((eq b) (x0 x7)) (x' x7))
% Found ((eq_ref b) (x0 x7)) as proof of (((eq b) (x0 x7)) (x' x7))
% Found (fun (x7:iS)=> ((eq_ref b) (x0 x7))) as proof of (((eq b) (x0 x7)) (x' x7))
% Found (fun (x7:iS)=> ((eq_ref b) (x0 x7))) as proof of (forall (x:iS), (((eq b) (x0 x)) (x' x)))
% Found (functional_extensionality0000 (fun (x7:iS)=> ((eq_ref b) (x0 x7)))) as proof of (((eq (iS->b)) x0) x')
% Found ((functional_extensionality000 x') (fun (x7:iS)=> ((eq_ref b) (x0 x7)))) as proof of (((eq (iS->b)) x0) x')
% Found (((functional_extensionality00 x0) x') (fun (x7:iS)=> ((eq_ref b) (x0 x7)))) as proof of (((eq (iS->b)) x0) x')
% Found ((((functional_extensionality0 b) x0) x') (fun (x7:iS)=> ((eq_ref b) (x0 x7)))) as proof of (((eq (iS->b)) x0) x')
% Found (((((functional_extensionality iS) b) x0) x') (fun (x7:iS)=> ((eq_ref b) (x0 x7)))) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((((functional_extensionality iS) b) x0) x') (fun (x7:iS)=> ((eq_ref b) (x0 x7))))) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 (x2 x7)):(((eq b) (x2 x7)) (x2 x7))
% Found (eq_ref0 (x2 x7)) as proof of (((eq b) (x2 x7)) (x' x7))
% Found ((eq_ref b) (x2 x7)) as proof of (((eq b) (x2 x7)) (x' x7))
% Found ((eq_ref b) (x2 x7)) as proof of (((eq b) (x2 x7)) (x' x7))
% Found (fun (x7:iS)=> ((eq_ref b) (x2 x7))) as proof of (((eq b) (x2 x7)) (x' x7))
% Found (fun (x7:iS)=> ((eq_ref b) (x2 x7))) as proof of (forall (x:iS), (((eq b) (x2 x)) (x' x)))
% Found (functional_extensionality_dep0000 (fun (x7:iS)=> ((eq_ref b) (x2 x7)))) as proof of (((eq (iS->b)) x2) x')
% Found ((functional_extensionality_dep000 x') (fun (x7:iS)=> ((eq_ref b) (x2 x7)))) as proof of (((eq (iS->b)) x2) x')
% Found (((functional_extensionality_dep00 x2) x') (fun (x7:iS)=> ((eq_ref b) (x2 x7)))) as proof of (((eq (iS->b)) x2) x')
% Found ((((functional_extensionality_dep0 (fun (x9:iS)=> b)) x2) x') (fun (x7:iS)=> ((eq_ref b) (x2 x7)))) as proof of (((eq (iS->b)) x2) x')
% Found (((((functional_extensionality_dep iS) (fun (x9:iS)=> b)) x2) x') (fun (x7:iS)=> ((eq_ref b) (x2 x7)))) as proof of (((eq (iS->b)) x2) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((((functional_extensionality_dep iS) (fun (x9:iS)=> b)) x2) x') (fun (x7:iS)=> ((eq_ref b) (x2 x7))))) as proof of (((eq (iS->b)) x2) x')
% Found eq_ref00:=(eq_ref0 (x2 x7)):(((eq b) (x2 x7)) (x2 x7))
% Found (eq_ref0 (x2 x7)) as proof of (((eq b) (x2 x7)) (x' x7))
% Found ((eq_ref b) (x2 x7)) as proof of (((eq b) (x2 x7)) (x' x7))
% Found ((eq_ref b) (x2 x7)) as proof of (((eq b) (x2 x7)) (x' x7))
% Found (fun (x7:iS)=> ((eq_ref b) (x2 x7))) as proof of (((eq b) (x2 x7)) (x' x7))
% Found (fun (x7:iS)=> ((eq_ref b) (x2 x7))) as proof of (forall (x:iS), (((eq b) (x2 x)) (x' x)))
% Found (functional_extensionality0000 (fun (x7:iS)=> ((eq_ref b) (x2 x7)))) as proof of (((eq (iS->b)) x2) x')
% Found ((functional_extensionality000 x') (fun (x7:iS)=> ((eq_ref b) (x2 x7)))) as proof of (((eq (iS->b)) x2) x')
% Found (((functional_extensionality00 x2) x') (fun (x7:iS)=> ((eq_ref b) (x2 x7)))) as proof of (((eq (iS->b)) x2) x')
% Found ((((functional_extensionality0 b) x2) x') (fun (x7:iS)=> ((eq_ref b) (x2 x7)))) as proof of (((eq (iS->b)) x2) x')
% Found (((((functional_extensionality iS) b) x2) x') (fun (x7:iS)=> ((eq_ref b) (x2 x7)))) as proof of (((eq (iS->b)) x2) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((((functional_extensionality iS) b) x2) x') (fun (x7:iS)=> ((eq_ref b) (x2 x7))))) as proof of (((eq (iS->b)) x2) x')
% 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 (iS->b)) x2) P) as proof of ((P x2)->(P Xg))
% Found (((eq_ref (iS->b)) x2) P) as proof of ((P x2)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x2) P)) as proof of (forall (P:((iS->b)->Prop)), ((P x2)->(P Xg)))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x2) P)) as proof of (((eq (iS->b)) x2) Xg)
% Found x3:(P x2)
% Instantiate: x2:=Xg:(iS->b)
% Found (fun (x3:(P x2))=> x3) as proof of (P Xg)
% Found (fun (P:((iS->b)->Prop)) (x3:(P x2))=> x3) as proof of ((P x2)->(P Xg))
% Found (fun (P:((iS->b)->Prop)) (x3:(P x2))=> x3) as proof of (forall (P:((iS->b)->Prop)), ((P x2)->(P Xg)))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop)) (x3:(P x2))=> x3) as proof of (((eq (iS->b)) x2) Xg)
% Found eq_ref00:=(eq_ref0 (x4 x7)):(((eq b) (x4 x7)) (x4 x7))
% Found (eq_ref0 (x4 x7)) as proof of (((eq b) (x4 x7)) (x' x7))
% Found ((eq_ref b) (x4 x7)) as proof of (((eq b) (x4 x7)) (x' x7))
% Found ((eq_ref b) (x4 x7)) as proof of (((eq b) (x4 x7)) (x' x7))
% Found (fun (x7:iS)=> ((eq_ref b) (x4 x7))) as proof of (((eq b) (x4 x7)) (x' x7))
% Found (fun (x7:iS)=> ((eq_ref b) (x4 x7))) as proof of (forall (x:iS), (((eq b) (x4 x)) (x' x)))
% Found (functional_extensionality0000 (fun (x7:iS)=> ((eq_ref b) (x4 x7)))) as proof of (((eq (iS->b)) x4) x')
% Found ((functional_extensionality000 x') (fun (x7:iS)=> ((eq_ref b) (x4 x7)))) as proof of (((eq (iS->b)) x4) x')
% Found (((functional_extensionality00 x4) x') (fun (x7:iS)=> ((eq_ref b) (x4 x7)))) as proof of (((eq (iS->b)) x4) x')
% Found ((((functional_extensionality0 b) x4) x') (fun (x7:iS)=> ((eq_ref b) (x4 x7)))) as proof of (((eq (iS->b)) x4) x')
% Found (((((functional_extensionality iS) b) x4) x') (fun (x7:iS)=> ((eq_ref b) (x4 x7)))) as proof of (((eq (iS->b)) x4) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((((functional_extensionality iS) b) x4) x') (fun (x7:iS)=> ((eq_ref b) (x4 x7))))) as proof of (((eq (iS->b)) x4) x')
% Found eq_ref00:=(eq_ref0 (x4 x7)):(((eq b) (x4 x7)) (x4 x7))
% Found (eq_ref0 (x4 x7)) as proof of (((eq b) (x4 x7)) (x' x7))
% Found ((eq_ref b) (x4 x7)) as proof of (((eq b) (x4 x7)) (x' x7))
% Found ((eq_ref b) (x4 x7)) as proof of (((eq b) (x4 x7)) (x' x7))
% Found (fun (x7:iS)=> ((eq_ref b) (x4 x7))) as proof of (((eq b) (x4 x7)) (x' x7))
% Found (fun (x7:iS)=> ((eq_ref b) (x4 x7))) as proof of (forall (x:iS), (((eq b) (x4 x)) (x' x)))
% Found (functional_extensionality_dep0000 (fun (x7:iS)=> ((eq_ref b) (x4 x7)))) as proof of (((eq (iS->b)) x4) x')
% Found ((functional_extensionality_dep000 x') (fun (x7:iS)=> ((eq_ref b) (x4 x7)))) as proof of (((eq (iS->b)) x4) x')
% Found (((functional_extensionality_dep00 x4) x') (fun (x7:iS)=> ((eq_ref b) (x4 x7)))) as proof of (((eq (iS->b)) x4) x')
% Found ((((functional_extensionality_dep0 (fun (x9:iS)=> b)) x4) x') (fun (x7:iS)=> ((eq_ref b) (x4 x7)))) as proof of (((eq (iS->b)) x4) x')
% Found (((((functional_extensionality_dep iS) (fun (x9:iS)=> b)) x4) x') (fun (x7:iS)=> ((eq_ref b) (x4 x7)))) as proof of (((eq (iS->b)) x4) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((((functional_extensionality_dep iS) (fun (x9:iS)=> b)) x4) x') (fun (x7:iS)=> ((eq_ref b) (x4 x7))))) as proof of (((eq (iS->b)) x4) x')
% Found eq_ref00:=(eq_ref0 x2):(((eq (iS->b)) x2) x2)
% Found (eq_ref0 x2) as proof of (forall (P:((iS->b)->Prop)), ((P x2)->(P x')))
% Found ((eq_ref (iS->b)) x2) as proof of (forall (P:((iS->b)->Prop)), ((P x2)->(P x')))
% Found ((eq_ref (iS->b)) x2) as proof of (forall (P:((iS->b)->Prop)), ((P x2)->(P x')))
% Found ((eq_ref (iS->b)) x2) as proof of (forall (P:((iS->b)->Prop)), ((P x2)->(P x')))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((eq_ref (iS->b)) x2)) as proof of (((eq (iS->b)) x2) x')
% Found eq_ref00:=(eq_ref0 x8):(((eq (iS->b)) x8) x8)
% Found (eq_ref0 x8) as proof of (((eq (iS->b)) x8) Xg)
% Found ((eq_ref (iS->b)) x8) as proof of (((eq (iS->b)) x8) Xg)
% Found ((eq_ref (iS->b)) x8) as proof of (((eq (iS->b)) x8) Xg)
% Found (fun (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x8)) as proof of (((eq (iS->b)) x8) Xg)
% Found (fun (x9:(((eq b) (Xg c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x8)) as proof of ((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->(((eq (iS->b)) x8) Xg))
% Found (fun (x9:(((eq b) (Xg c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x8)) as proof of ((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->(((eq (iS->b)) x8) Xg)))
% Found (and_rect40 (fun (x9:(((eq b) (Xg c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x8))) as proof of (((eq (iS->b)) x8) Xg)
% Found ((and_rect4 (((eq (iS->b)) x8) Xg)) (fun (x9:(((eq b) (Xg c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x8))) as proof of (((eq (iS->b)) x8) Xg)
% Found (((fun (P:Type) (x9:((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->P)))=> (((((and_rect (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))) P) x9) x00)) (((eq (iS->b)) x8) Xg)) (fun (x9:(((eq b) (Xg c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x8))) as proof of (((eq (iS->b)) x8) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((fun (P:Type) (x9:((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->P)))=> (((((and_rect (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))) P) x9) x00)) (((eq (iS->b)) x8) Xg)) (fun (x9:(((eq b) (Xg c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x8)))) as proof of (((eq (iS->b)) x8) Xg)
% Found eta_expansion000:=(eta_expansion00 Xg):(((eq (iS->b)) Xg) (fun (x:iS)=> (Xg x)))
% Found (eta_expansion00 Xg) as proof of (((eq (iS->b)) Xg) x8)
% Found ((eta_expansion0 b) Xg) as proof of (((eq (iS->b)) Xg) x8)
% Found (((eta_expansion iS) b) Xg) as proof of (((eq (iS->b)) Xg) x8)
% Found (((eta_expansion iS) b) Xg) as proof of (((eq (iS->b)) Xg) x8)
% Found (((eta_expansion iS) b) Xg) as proof of (((eq (iS->b)) Xg) x8)
% Found (eq_sym000 (((eta_expansion iS) b) Xg)) as proof of (((eq (iS->b)) x8) Xg)
% Found ((eq_sym00 x8) (((eta_expansion iS) b) Xg)) as proof of (((eq (iS->b)) x8) Xg)
% Found (((eq_sym0 Xg) x8) (((eta_expansion iS) b) Xg)) as proof of (((eq (iS->b)) x8) Xg)
% Found ((((eq_sym (iS->b)) Xg) x8) (((eta_expansion iS) b) Xg)) as proof of (((eq (iS->b)) x8) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((((eq_sym (iS->b)) Xg) x8) (((eta_expansion iS) b) Xg))) as proof of (((eq (iS->b)) x8) Xg)
% Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% Found (eq_ref00 P) as proof of (P0 x2)
% Found ((eq_ref0 x2) P) as proof of (P0 x2)
% Found (((eq_ref (iS->b)) x2) P) as proof of (P0 x2)
% Found (((eq_ref (iS->b)) x2) P) as proof of (P0 x2)
% Found eq_ref00:=(eq_ref0 (f x8)):(((eq Prop) (f x8)) (f x8))
% Found (eq_ref0 (f x8)) as proof of (((eq Prop) (f x8)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x8))
% Found ((eq_ref Prop) (f x8)) as proof of (((eq Prop) (f x8)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x8))
% Found ((eq_ref Prop) (f x8)) as proof of (((eq Prop) (f x8)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x8))
% Found (fun (x8:(iS->b))=> ((eq_ref Prop) (f x8))) as proof of (((eq Prop) (f x8)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x8))
% Found (fun (x8:(iS->b))=> ((eq_ref Prop) (f x8))) as proof of (forall (x:(iS->b)), (((eq Prop) (f x)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x)))
% Found eq_ref00:=(eq_ref0 (f x8)):(((eq Prop) (f x8)) (f x8))
% Found (eq_ref0 (f x8)) as proof of (((eq Prop) (f x8)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x8))
% Found ((eq_ref Prop) (f x8)) as proof of (((eq Prop) (f x8)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x8))
% Found ((eq_ref Prop) (f x8)) as proof of (((eq Prop) (f x8)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x8))
% Found (fun (x8:(iS->b))=> ((eq_ref Prop) (f x8))) as proof of (((eq Prop) (f x8)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x8))
% Found (fun (x8:(iS->b))=> ((eq_ref Prop) (f x8))) as proof of (forall (x:(iS->b)), (((eq Prop) (f x)) (((unique (iS->b)) (fun (x10:(iS->b))=> ((and (((eq b) (x10 c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x10 ((cP Xx) Xy))) ((cP2 (x10 Xx)) (x10 Xy))))))) x)))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq b) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq b) (x0 x1)) (x' x1))
% Found ((eq_ref b) (x0 x1)) as proof of (((eq b) (x0 x1)) (x' x1))
% Found ((eq_ref b) (x0 x1)) as proof of (((eq b) (x0 x1)) (x' x1))
% Found (fun (x1:iS)=> ((eq_ref b) (x0 x1))) as proof of (((eq b) (x0 x1)) (x' x1))
% Found (fun (x1:iS)=> ((eq_ref b) (x0 x1))) as proof of (forall (x:iS), (((eq b) (x0 x)) (x' x)))
% Found ((functional_extensionality00000 (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found ((functional_extensionality00000 (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found (((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> ((functional_extensionality0000 x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found (((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> (((functional_extensionality000 x') x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found (((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> ((((functional_extensionality00 x0) x') x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found (((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> (((((functional_extensionality0 b) x0) x') x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found (((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> ((((((functional_extensionality iS) b) x0) x') x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> (((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> ((((((functional_extensionality iS) b) x0) x') x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) (((eq_ref (iS->b)) x0) P))) as proof of ((P x0)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop))=> (((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> ((((((functional_extensionality iS) b) x0) x') x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) (((eq_ref (iS->b)) x0) P))) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq b) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq b) (x0 x1)) (x' x1))
% Found ((eq_ref b) (x0 x1)) as proof of (((eq b) (x0 x1)) (x' x1))
% Found ((eq_ref b) (x0 x1)) as proof of (((eq b) (x0 x1)) (x' x1))
% Found (fun (x1:iS)=> ((eq_ref b) (x0 x1))) as proof of (((eq b) (x0 x1)) (x' x1))
% Found (fun (x1:iS)=> ((eq_ref b) (x0 x1))) as proof of (forall (x:iS), (((eq b) (x0 x)) (x' x)))
% Found (functional_extensionality00000 (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of ((P x0)->(P x'))
% Found (functional_extensionality00000 (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of ((P x0)->(P x'))
% Found ((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> ((functional_extensionality0000 x1) P)) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of ((P x0)->(P x'))
% Found ((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> (((functional_extensionality000 x') x1) P)) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of ((P x0)->(P x'))
% Found ((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> ((((functional_extensionality00 x0) x') x1) P)) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of ((P x0)->(P x'))
% Found ((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> (((((functional_extensionality0 b) x0) x') x1) P)) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of ((P x0)->(P x'))
% Found ((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> ((((((functional_extensionality iS) b) x0) x') x1) P)) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of ((P x0)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> ((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> ((((((functional_extensionality iS) b) x0) x') x1) P)) (fun (x1:iS)=> ((eq_ref b) (x0 x1))))) as proof of ((P x0)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop))=> ((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> ((((((functional_extensionality iS) b) x0) x') x1) P)) (fun (x1:iS)=> ((eq_ref b) (x0 x1))))) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq b) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq b) (x0 x1)) (x' x1))
% Found ((eq_ref b) (x0 x1)) as proof of (((eq b) (x0 x1)) (x' x1))
% Found ((eq_ref b) (x0 x1)) as proof of (((eq b) (x0 x1)) (x' x1))
% Found (fun (x1:iS)=> ((eq_ref b) (x0 x1))) as proof of (((eq b) (x0 x1)) (x' x1))
% Found (fun (x1:iS)=> ((eq_ref b) (x0 x1))) as proof of (forall (x:iS), (((eq b) (x0 x)) (x' x)))
% Found ((functional_extensionality_dep00000 (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found ((functional_extensionality_dep00000 (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found (((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> ((functional_extensionality_dep0000 x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found (((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> (((functional_extensionality_dep000 x') x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found (((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> ((((functional_extensionality_dep00 x0) x') x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found (((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> (((((functional_extensionality_dep0 (fun (x3:iS)=> b)) x0) x') x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found (((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> ((((((functional_extensionality_dep iS) (fun (x3:iS)=> b)) x0) x') x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> (((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> ((((((functional_extensionality_dep iS) (fun (x3:iS)=> b)) x0) x') x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) (((eq_ref (iS->b)) x0) P))) as proof of ((P x0)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop))=> (((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> ((((((functional_extensionality_dep iS) (fun (x3:iS)=> b)) x0) x') x1) (fun (x3:(iS->b))=> ((P x0)->(P x3))))) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) (((eq_ref (iS->b)) x0) P))) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq b) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq b) (x0 x1)) (x' x1))
% Found ((eq_ref b) (x0 x1)) as proof of (((eq b) (x0 x1)) (x' x1))
% Found ((eq_ref b) (x0 x1)) as proof of (((eq b) (x0 x1)) (x' x1))
% Found (fun (x1:iS)=> ((eq_ref b) (x0 x1))) as proof of (((eq b) (x0 x1)) (x' x1))
% Found (fun (x1:iS)=> ((eq_ref b) (x0 x1))) as proof of (forall (x:iS), (((eq b) (x0 x)) (x' x)))
% Found (functional_extensionality_dep00000 (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of ((P x0)->(P x'))
% Found (functional_extensionality_dep00000 (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of ((P x0)->(P x'))
% Found ((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> ((functional_extensionality_dep0000 x1) P)) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of ((P x0)->(P x'))
% Found ((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> (((functional_extensionality_dep000 x') x1) P)) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of ((P x0)->(P x'))
% Found ((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> ((((functional_extensionality_dep00 x0) x') x1) P)) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of ((P x0)->(P x'))
% Found ((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> (((((functional_extensionality_dep0 (fun (x3:iS)=> b)) x0) x') x1) P)) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of ((P x0)->(P x'))
% Found ((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> ((((((functional_extensionality_dep iS) (fun (x3:iS)=> b)) x0) x') x1) P)) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of ((P x0)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> ((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> ((((((functional_extensionality_dep iS) (fun (x3:iS)=> b)) x0) x') x1) P)) (fun (x1:iS)=> ((eq_ref b) (x0 x1))))) as proof of ((P x0)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop))=> ((fun (x1:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> ((((((functional_extensionality_dep iS) (fun (x3:iS)=> b)) x0) x') x1) P)) (fun (x1:iS)=> ((eq_ref b) (x0 x1))))) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 (x0 c0)):(((eq b) (x0 c0)) (x0 c0))
% Found (eq_ref0 (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found eq_ref00:=(eq_ref0 Xg):(((eq (iS->b)) Xg) Xg)
% Found (eq_ref0 Xg) as proof of (((eq (iS->b)) Xg) x2)
% Found ((eq_ref (iS->b)) Xg) as proof of (((eq (iS->b)) Xg) x2)
% Found ((eq_ref (iS->b)) Xg) as proof of (((eq (iS->b)) Xg) x2)
% Found ((eq_ref (iS->b)) Xg) as proof of (((eq (iS->b)) Xg) x2)
% Found ((eq_sym0000 ((eq_ref (iS->b)) Xg)) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P Xg))
% Found ((eq_sym0000 ((eq_ref (iS->b)) Xg)) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P Xg))
% Found (((fun (x3:(((eq (iS->b)) Xg) x2))=> ((eq_sym000 x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) ((eq_ref (iS->b)) Xg)) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P Xg))
% Found (((fun (x3:(((eq (iS->b)) Xg) x2))=> (((eq_sym00 x2) x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) ((eq_ref (iS->b)) Xg)) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P Xg))
% Found (((fun (x3:(((eq (iS->b)) Xg) x2))=> ((((eq_sym0 Xg) x2) x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) ((eq_ref (iS->b)) Xg)) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P Xg))
% Found (((fun (x3:(((eq (iS->b)) Xg) x2))=> (((((eq_sym (iS->b)) Xg) x2) x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) ((eq_ref (iS->b)) Xg)) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> (((fun (x3:(((eq (iS->b)) Xg) x2))=> (((((eq_sym (iS->b)) Xg) x2) x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) ((eq_ref (iS->b)) Xg)) (((eq_ref (iS->b)) x2) P))) as proof of ((P x2)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> (((fun (x3:(((eq (iS->b)) Xg) x2))=> (((((eq_sym (iS->b)) Xg) x2) x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) ((eq_ref (iS->b)) Xg)) (((eq_ref (iS->b)) x2) P))) as proof of (((eq (iS->b)) x2) Xg)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xg):(((eq (iS->b)) Xg) (fun (x:iS)=> (Xg x)))
% Found (eta_expansion_dep00 Xg) as proof of (((eq (iS->b)) Xg) x2)
% Found ((eta_expansion_dep0 (fun (x4:iS)=> b)) Xg) as proof of (((eq (iS->b)) Xg) x2)
% Found (((eta_expansion_dep iS) (fun (x4:iS)=> b)) Xg) as proof of (((eq (iS->b)) Xg) x2)
% Found (((eta_expansion_dep iS) (fun (x4:iS)=> b)) Xg) as proof of (((eq (iS->b)) Xg) x2)
% Found (((eta_expansion_dep iS) (fun (x4:iS)=> b)) Xg) as proof of (((eq (iS->b)) Xg) x2)
% Found (eq_sym0000 (((eta_expansion_dep iS) (fun (x4:iS)=> b)) Xg)) as proof of ((P x2)->(P Xg))
% Found (eq_sym0000 (((eta_expansion_dep iS) (fun (x4:iS)=> b)) Xg)) as proof of ((P x2)->(P Xg))
% Found ((fun (x3:(((eq (iS->b)) Xg) x2))=> ((eq_sym000 x3) P)) (((eta_expansion_dep iS) (fun (x4:iS)=> b)) Xg)) as proof of ((P x2)->(P Xg))
% Found ((fun (x3:(((eq (iS->b)) Xg) x2))=> (((eq_sym00 x2) x3) P)) (((eta_expansion_dep iS) (fun (x4:iS)=> b)) Xg)) as proof of ((P x2)->(P Xg))
% Found ((fun (x3:(((eq (iS->b)) Xg) x2))=> ((((eq_sym0 Xg) x2) x3) P)) (((eta_expansion_dep iS) (fun (x4:iS)=> b)) Xg)) as proof of ((P x2)->(P Xg))
% Found ((fun (x3:(((eq (iS->b)) Xg) x2))=> (((((eq_sym (iS->b)) Xg) x2) x3) P)) (((eta_expansion_dep iS) (fun (x4:iS)=> b)) Xg)) as proof of ((P x2)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> ((fun (x3:(((eq (iS->b)) Xg) x2))=> (((((eq_sym (iS->b)) Xg) x2) x3) P)) (((eta_expansion_dep iS) (fun (x4:iS)=> b)) Xg))) as proof of ((P x2)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> ((fun (x3:(((eq (iS->b)) Xg) x2))=> (((((eq_sym (iS->b)) Xg) x2) x3) P)) (((eta_expansion_dep iS) (fun (x4:iS)=> b)) Xg))) as proof of (((eq (iS->b)) x2) Xg)
% Found eq_ref00:=(eq_ref0 (x2 c0)):(((eq b) (x2 c0)) (x2 c0))
% Found (eq_ref0 (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% 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 (iS->b)) x0) P) as proof of ((P x0)->(P Xg))
% Found (((eq_ref (iS->b)) x0) P) as proof of ((P x0)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x0) P)) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P Xg)))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x0) P)) as proof of (((eq (iS->b)) x0) Xg)
% Found x3:(P x0)
% Instantiate: x0:=Xg:(iS->b)
% Found (fun (x3:(P x0))=> x3) as proof of (P Xg)
% Found (fun (P:((iS->b)->Prop)) (x3:(P x0))=> x3) as proof of ((P x0)->(P Xg))
% Found (fun (P:((iS->b)->Prop)) (x3:(P x0))=> x3) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P Xg)))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop)) (x3:(P x0))=> x3) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 x0):(((eq (iS->b)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (iS->b)) x0) Xg)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) Xg)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x0)) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x9:(((eq b) (Xg c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x0)) as proof of ((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->(((eq (iS->b)) x0) Xg))
% Found (fun (x9:(((eq b) (Xg c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x0)) as proof of ((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->(((eq (iS->b)) x0) Xg)))
% Found (and_rect40 (fun (x9:(((eq b) (Xg c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) Xg)
% Found ((and_rect4 (((eq (iS->b)) x0) Xg)) (fun (x9:(((eq b) (Xg c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) Xg)
% Found (((fun (P:Type) (x9:((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->P)))=> (((((and_rect (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))) P) x9) x00)) (((eq (iS->b)) x0) Xg)) (fun (x9:(((eq b) (Xg c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((fun (P:Type) (x9:((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->P)))=> (((((and_rect (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))) P) x9) x00)) (((eq (iS->b)) x0) Xg)) (fun (x9:(((eq b) (Xg c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x0)))) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 x2):(((eq (iS->b)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (iS->b)) x2) Xg)
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) Xg)
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) Xg)
% Found (fun (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x2)) as proof of (((eq (iS->b)) x2) Xg)
% Found (fun (x9:(((eq b) (Xg c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x2)) as proof of ((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->(((eq (iS->b)) x2) Xg))
% Found (fun (x9:(((eq b) (Xg c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x2)) as proof of ((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->(((eq (iS->b)) x2) Xg)))
% Found (and_rect40 (fun (x9:(((eq b) (Xg c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) Xg)
% Found ((and_rect4 (((eq (iS->b)) x2) Xg)) (fun (x9:(((eq b) (Xg c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) Xg)
% Found (((fun (P:Type) (x9:((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->P)))=> (((((and_rect (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))) P) x9) x00)) (((eq (iS->b)) x2) Xg)) (fun (x9:(((eq b) (Xg c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((fun (P:Type) (x9:((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->P)))=> (((((and_rect (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))) P) x9) x00)) (((eq (iS->b)) x2) Xg)) (fun (x9:(((eq b) (Xg c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x2)))) as proof of (((eq (iS->b)) x2) Xg)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xg):(((eq (iS->b)) Xg) (fun (x:iS)=> (Xg x)))
% Found (eta_expansion_dep00 Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found ((eta_expansion_dep0 (fun (x10:iS)=> b)) Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found (((eta_expansion_dep iS) (fun (x10:iS)=> b)) Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found (((eta_expansion_dep iS) (fun (x10:iS)=> b)) Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found (((eta_expansion_dep iS) (fun (x10:iS)=> b)) Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found (eq_sym000 (((eta_expansion_dep iS) (fun (x10:iS)=> b)) Xg)) as proof of (((eq (iS->b)) x0) Xg)
% Found ((eq_sym00 x0) (((eta_expansion_dep iS) (fun (x10:iS)=> b)) Xg)) as proof of (((eq (iS->b)) x0) Xg)
% Found (((eq_sym0 Xg) x0) (((eta_expansion_dep iS) (fun (x10:iS)=> b)) Xg)) as proof of (((eq (iS->b)) x0) Xg)
% Found ((((eq_sym (iS->b)) Xg) x0) (((eta_expansion_dep iS) (fun (x10:iS)=> b)) Xg)) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((((eq_sym (iS->b)) Xg) x0) (((eta_expansion_dep iS) (fun (x10:iS)=> b)) Xg))) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 x0):(((eq (iS->b)) x0) x0)
% Found (eq_ref0 x0) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P x')))
% Found ((eq_ref (iS->b)) x0) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P x')))
% Found ((eq_ref (iS->b)) x0) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P x')))
% Found ((eq_ref (iS->b)) x0) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P x')))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((eq_ref (iS->b)) x0)) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 x4):(((eq (iS->b)) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq (iS->b)) x4) Xg)
% Found ((eq_ref (iS->b)) x4) as proof of (((eq (iS->b)) x4) Xg)
% Found ((eq_ref (iS->b)) x4) as proof of (((eq (iS->b)) x4) Xg)
% Found (fun (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x4)) as proof of (((eq (iS->b)) x4) Xg)
% Found (fun (x9:(((eq b) (Xg c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x4)) as proof of ((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->(((eq (iS->b)) x4) Xg))
% Found (fun (x9:(((eq b) (Xg c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x4)) as proof of ((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->(((eq (iS->b)) x4) Xg)))
% Found (and_rect40 (fun (x9:(((eq b) (Xg c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x4))) as proof of (((eq (iS->b)) x4) Xg)
% Found ((and_rect4 (((eq (iS->b)) x4) Xg)) (fun (x9:(((eq b) (Xg c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x4))) as proof of (((eq (iS->b)) x4) Xg)
% Found (((fun (P:Type) (x9:((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->P)))=> (((((and_rect (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))) P) x9) x00)) (((eq (iS->b)) x4) Xg)) (fun (x9:(((eq b) (Xg c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x4))) as proof of (((eq (iS->b)) x4) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((fun (P:Type) (x9:((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->P)))=> (((((and_rect (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))) P) x9) x00)) (((eq (iS->b)) x4) Xg)) (fun (x9:(((eq b) (Xg c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x4)))) as proof of (((eq (iS->b)) x4) Xg)
% Found eta_expansion000:=(eta_expansion00 Xg):(((eq (iS->b)) Xg) (fun (x:iS)=> (Xg x)))
% Found (eta_expansion00 Xg) as proof of (((eq (iS->b)) Xg) x2)
% Found ((eta_expansion0 b) Xg) as proof of (((eq (iS->b)) Xg) x2)
% Found (((eta_expansion iS) b) Xg) as proof of (((eq (iS->b)) Xg) x2)
% Found (((eta_expansion iS) b) Xg) as proof of (((eq (iS->b)) Xg) x2)
% Found (((eta_expansion iS) b) Xg) as proof of (((eq (iS->b)) Xg) x2)
% Found (eq_sym000 (((eta_expansion iS) b) Xg)) as proof of (((eq (iS->b)) x2) Xg)
% Found ((eq_sym00 x2) (((eta_expansion iS) b) Xg)) as proof of (((eq (iS->b)) x2) Xg)
% Found (((eq_sym0 Xg) x2) (((eta_expansion iS) b) Xg)) as proof of (((eq (iS->b)) x2) Xg)
% Found ((((eq_sym (iS->b)) Xg) x2) (((eta_expansion iS) b) Xg)) as proof of (((eq (iS->b)) x2) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((((eq_sym (iS->b)) Xg) x2) (((eta_expansion iS) b) Xg))) as proof of (((eq (iS->b)) x2) Xg)
% Found eq_ref00:=(eq_ref0 x6):(((eq (iS->b)) x6) x6)
% Found (eq_ref0 x6) as proof of (((eq (iS->b)) x6) Xg)
% Found ((eq_ref (iS->b)) x6) as proof of (((eq (iS->b)) x6) Xg)
% Found ((eq_ref (iS->b)) x6) as proof of (((eq (iS->b)) x6) Xg)
% Found (fun (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x6)) as proof of (((eq (iS->b)) x6) Xg)
% Found (fun (x9:(((eq b) (Xg c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x6)) as proof of ((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->(((eq (iS->b)) x6) Xg))
% Found (fun (x9:(((eq b) (Xg c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x6)) as proof of ((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->(((eq (iS->b)) x6) Xg)))
% Found (and_rect40 (fun (x9:(((eq b) (Xg c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x6))) as proof of (((eq (iS->b)) x6) Xg)
% Found ((and_rect4 (((eq (iS->b)) x6) Xg)) (fun (x9:(((eq b) (Xg c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x6))) as proof of (((eq (iS->b)) x6) Xg)
% Found (((fun (P:Type) (x9:((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->P)))=> (((((and_rect (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))) P) x9) x00)) (((eq (iS->b)) x6) Xg)) (fun (x9:(((eq b) (Xg c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x6))) as proof of (((eq (iS->b)) x6) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((fun (P:Type) (x9:((((eq b) (Xg c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))->P)))=> (((((and_rect (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))) P) x9) x00)) (((eq (iS->b)) x6) Xg)) (fun (x9:(((eq b) (Xg c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))=> ((eq_ref (iS->b)) x6)))) as proof of (((eq (iS->b)) x6) Xg)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xg):(((eq (iS->b)) Xg) (fun (x:iS)=> (Xg x)))
% Found (eta_expansion_dep00 Xg) as proof of (((eq (iS->b)) Xg) x4)
% Found ((eta_expansion_dep0 (fun (x10:iS)=> b)) Xg) as proof of (((eq (iS->b)) Xg) x4)
% Found (((eta_expansion_dep iS) (fun (x10:iS)=> b)) Xg) as proof of (((eq (iS->b)) Xg) x4)
% Found (((eta_expansion_dep iS) (fun (x10:iS)=> b)) Xg) as proof of (((eq (iS->b)) Xg) x4)
% Found (((eta_expansion_dep iS) (fun (x10:iS)=> b)) Xg) as proof of (((eq (iS->b)) Xg) x4)
% Found (eq_sym000 (((eta_expansion_dep iS) (fun (x10:iS)=> b)) Xg)) as proof of (((eq (iS->b)) x4) Xg)
% Found ((eq_sym00 x4) (((eta_expansion_dep iS) (fun (x10:iS)=> b)) Xg)) as proof of (((eq (iS->b)) x4) Xg)
% Found (((eq_sym0 Xg) x4) (((eta_expansion_dep iS) (fun (x10:iS)=> b)) Xg)) as proof of (((eq (iS->b)) x4) Xg)
% Found ((((eq_sym (iS->b)) Xg) x4) (((eta_expansion_dep iS) (fun (x10:iS)=> b)) Xg)) as proof of (((eq (iS->b)) x4) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((((eq_sym (iS->b)) Xg) x4) (((eta_expansion_dep iS) (fun (x10:iS)=> b)) Xg))) as proof of (((eq (iS->b)) x4) Xg)
% Found eta_expansion000:=(eta_expansion00 Xg):(((eq (iS->b)) Xg) (fun (x:iS)=> (Xg x)))
% Found (eta_expansion00 Xg) as proof of (((eq (iS->b)) Xg) x6)
% Found ((eta_expansion0 b) Xg) as proof of (((eq (iS->b)) Xg) x6)
% Found (((eta_expansion iS) b) Xg) as proof of (((eq (iS->b)) Xg) x6)
% Found (((eta_expansion iS) b) Xg) as proof of (((eq (iS->b)) Xg) x6)
% Found (((eta_expansion iS) b) Xg) as proof of (((eq (iS->b)) Xg) x6)
% Found (eq_sym000 (((eta_expansion iS) b) Xg)) as proof of (((eq (iS->b)) x6) Xg)
% Found ((eq_sym00 x6) (((eta_expansion iS) b) Xg)) as proof of (((eq (iS->b)) x6) Xg)
% Found (((eq_sym0 Xg) x6) (((eta_expansion iS) b) Xg)) as proof of (((eq (iS->b)) x6) Xg)
% Found ((((eq_sym (iS->b)) Xg) x6) (((eta_expansion iS) b) Xg)) as proof of (((eq (iS->b)) x6) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((((eq_sym (iS->b)) Xg) x6) (((eta_expansion iS) b) Xg))) as proof of (((eq (iS->b)) x6) Xg)
% Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% Found (eq_ref00 P) as proof of ((P x2)->(P x'))
% Found ((eq_ref0 x2) P) as proof of ((P x2)->(P x'))
% Found (((eq_ref (iS->b)) x2) P) as proof of ((P x2)->(P x'))
% Found (((eq_ref (iS->b)) x2) P) as proof of ((P x2)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x2) P)) as proof of (forall (P:((iS->b)->Prop)), ((P x2)->(P x')))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x2) P)) as proof of (((eq (iS->b)) x2) x')
% Found x3:(P x2)
% Instantiate: x2:=x':(iS->b)
% Found (fun (x3:(P x2))=> x3) as proof of (P x')
% Found (fun (P:((iS->b)->Prop)) (x3:(P x2))=> x3) as proof of ((P x2)->(P x'))
% Found (fun (P:((iS->b)->Prop)) (x3:(P x2))=> x3) as proof of (forall (P:((iS->b)->Prop)), ((P x2)->(P x')))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop)) (x3:(P x2))=> x3) as proof of (((eq (iS->b)) x2) x')
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (iS->b)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (iS->b)) x0) P) as proof of (P0 x0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (iS->b)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (iS->b)) b0) Xg)
% Found ((eq_ref (iS->b)) b0) as proof of (((eq (iS->b)) b0) Xg)
% Found ((eq_ref (iS->b)) b0) as proof of (((eq (iS->b)) b0) Xg)
% Found ((eq_ref (iS->b)) b0) as proof of (((eq (iS->b)) b0) Xg)
% Found eta_expansion000:=(eta_expansion00 x2):(((eq (iS->b)) x2) (fun (x:iS)=> (x2 x)))
% Found (eta_expansion00 x2) as proof of (((eq (iS->b)) x2) b0)
% Found ((eta_expansion0 b) x2) as proof of (((eq (iS->b)) x2) b0)
% Found (((eta_expansion iS) b) x2) as proof of (((eq (iS->b)) x2) b0)
% Found (((eta_expansion iS) b) x2) as proof of (((eq (iS->b)) x2) b0)
% Found (((eta_expansion iS) b) x2) as proof of (((eq (iS->b)) x2) b0)
% Found eq_ref00:=(eq_ref0 x2):(((eq (iS->b)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (iS->b)) x2) b0)
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) b0)
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) b0)
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) b0)
% Found ((eq_trans0000 ((eq_ref (iS->b)) x2)) ((eq_ref (iS->b)) b0)) as proof of (((eq (iS->b)) x2) Xg)
% Found (((eq_trans000 Xg) ((eq_ref (iS->b)) x2)) ((eq_ref (iS->b)) Xg)) as proof of (((eq (iS->b)) x2) Xg)
% Found ((((fun (b0:(iS->b))=> ((eq_trans00 b0) Xg)) Xg) ((eq_ref (iS->b)) x2)) ((eq_ref (iS->b)) Xg)) as proof of (((eq (iS->b)) x2) Xg)
% Found ((((fun (b0:(iS->b))=> (((eq_trans0 x2) b0) Xg)) Xg) ((eq_ref (iS->b)) x2)) ((eq_ref (iS->b)) Xg)) as proof of (((eq (iS->b)) x2) Xg)
% Found ((((fun (b0:(iS->b))=> ((((eq_trans (iS->b)) x2) b0) Xg)) Xg) ((eq_ref (iS->b)) x2)) ((eq_ref (iS->b)) Xg)) as proof of (((eq (iS->b)) x2) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((((fun (b0:(iS->b))=> ((((eq_trans (iS->b)) x2) b0) Xg)) Xg) ((eq_ref (iS->b)) x2)) ((eq_ref (iS->b)) Xg))) as proof of (((eq (iS->b)) x2) Xg)
% Found eta_expansion000:=(eta_expansion00 Xg):(((eq (iS->b)) Xg) (fun (x:iS)=> (Xg x)))
% Found (eta_expansion00 Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found ((eta_expansion0 b) Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found (((eta_expansion iS) b) Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found (((eta_expansion iS) b) Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found (((eta_expansion iS) b) Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found ((eq_sym0000 (((eta_expansion iS) b) Xg)) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found ((eq_sym0000 (((eta_expansion iS) b) Xg)) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (((fun (x3:(((eq (iS->b)) Xg) x0))=> ((eq_sym000 x3) (fun (x5:(iS->b))=> ((P x0)->(P x5))))) (((eta_expansion iS) b) Xg)) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (((fun (x3:(((eq (iS->b)) Xg) x0))=> (((eq_sym00 x0) x3) (fun (x5:(iS->b))=> ((P x0)->(P x5))))) (((eta_expansion iS) b) Xg)) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (((fun (x3:(((eq (iS->b)) Xg) x0))=> ((((eq_sym0 Xg) x0) x3) (fun (x5:(iS->b))=> ((P x0)->(P x5))))) (((eta_expansion iS) b) Xg)) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (((fun (x3:(((eq (iS->b)) Xg) x0))=> (((((eq_sym (iS->b)) Xg) x0) x3) (fun (x5:(iS->b))=> ((P x0)->(P x5))))) (((eta_expansion iS) b) Xg)) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> (((fun (x3:(((eq (iS->b)) Xg) x0))=> (((((eq_sym (iS->b)) Xg) x0) x3) (fun (x5:(iS->b))=> ((P x0)->(P x5))))) (((eta_expansion iS) b) Xg)) (((eq_ref (iS->b)) x0) P))) as proof of ((P x0)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> (((fun (x3:(((eq (iS->b)) Xg) x0))=> (((((eq_sym (iS->b)) Xg) x0) x3) (fun (x5:(iS->b))=> ((P x0)->(P x5))))) (((eta_expansion iS) b) Xg)) (((eq_ref (iS->b)) x0) P))) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 Xg):(((eq (iS->b)) Xg) Xg)
% Found (eq_ref0 Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found ((eq_ref (iS->b)) Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found ((eq_ref (iS->b)) Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found ((eq_ref (iS->b)) Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found (eq_sym0000 ((eq_ref (iS->b)) Xg)) as proof of ((P x0)->(P Xg))
% Found (eq_sym0000 ((eq_ref (iS->b)) Xg)) as proof of ((P x0)->(P Xg))
% Found ((fun (x3:(((eq (iS->b)) Xg) x0))=> ((eq_sym000 x3) P)) ((eq_ref (iS->b)) Xg)) as proof of ((P x0)->(P Xg))
% Found ((fun (x3:(((eq (iS->b)) Xg) x0))=> (((eq_sym00 x0) x3) P)) ((eq_ref (iS->b)) Xg)) as proof of ((P x0)->(P Xg))
% Found ((fun (x3:(((eq (iS->b)) Xg) x0))=> ((((eq_sym0 Xg) x0) x3) P)) ((eq_ref (iS->b)) Xg)) as proof of ((P x0)->(P Xg))
% Found ((fun (x3:(((eq (iS->b)) Xg) x0))=> (((((eq_sym (iS->b)) Xg) x0) x3) P)) ((eq_ref (iS->b)) Xg)) as proof of ((P x0)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> ((fun (x3:(((eq (iS->b)) Xg) x0))=> (((((eq_sym (iS->b)) Xg) x0) x3) P)) ((eq_ref (iS->b)) Xg))) as proof of ((P x0)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> ((fun (x3:(((eq (iS->b)) Xg) x0))=> (((((eq_sym (iS->b)) Xg) x0) x3) P)) ((eq_ref (iS->b)) Xg))) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 x8):(((eq (iS->b)) x8) x8)
% Found (eq_ref0 x8) as proof of (((eq (iS->b)) x8) x')
% Found ((eq_ref (iS->b)) x8) as proof of (((eq (iS->b)) x8) x')
% Found ((eq_ref (iS->b)) x8) as proof of (((eq (iS->b)) x8) x')
% Found (fun (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x8)) as proof of (((eq (iS->b)) x8) x')
% Found (fun (x9:(((eq b) (x' c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x8)) as proof of ((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->(((eq (iS->b)) x8) x'))
% Found (fun (x9:(((eq b) (x' c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x8)) as proof of ((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->(((eq (iS->b)) x8) x')))
% Found (and_rect40 (fun (x9:(((eq b) (x' c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x8))) as proof of (((eq (iS->b)) x8) x')
% Found ((and_rect4 (((eq (iS->b)) x8) x')) (fun (x9:(((eq b) (x' c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x8))) as proof of (((eq (iS->b)) x8) x')
% Found (((fun (P:Type) (x9:((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->P)))=> (((((and_rect (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))) P) x9) x00)) (((eq (iS->b)) x8) x')) (fun (x9:(((eq b) (x' c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x8))) as proof of (((eq (iS->b)) x8) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((fun (P:Type) (x9:((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->P)))=> (((((and_rect (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))) P) x9) x00)) (((eq (iS->b)) x8) x')) (fun (x9:(((eq b) (x' c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x8)))) as proof of (((eq (iS->b)) x8) x')
% Found eq_ref00:=(eq_ref0 x'):(((eq (iS->b)) x') x')
% Found (eq_ref0 x') as proof of (((eq (iS->b)) x') x8)
% Found ((eq_ref (iS->b)) x') as proof of (((eq (iS->b)) x') x8)
% Found ((eq_ref (iS->b)) x') as proof of (((eq (iS->b)) x') x8)
% Found ((eq_ref (iS->b)) x') as proof of (((eq (iS->b)) x') x8)
% Found (eq_sym000 ((eq_ref (iS->b)) x')) as proof of (((eq (iS->b)) x8) x')
% Found ((eq_sym00 x8) ((eq_ref (iS->b)) x')) as proof of (((eq (iS->b)) x8) x')
% Found (((eq_sym0 x') x8) ((eq_ref (iS->b)) x')) as proof of (((eq (iS->b)) x8) x')
% Found ((((eq_sym (iS->b)) x') x8) ((eq_ref (iS->b)) x')) as proof of (((eq (iS->b)) x8) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((((eq_sym (iS->b)) x') x8) ((eq_ref (iS->b)) x'))) as proof of (((eq (iS->b)) x8) x')
% Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% Found (eq_ref00 P) as proof of (P0 x2)
% Found ((eq_ref0 x2) P) as proof of (P0 x2)
% Found (((eq_ref (iS->b)) x2) P) as proof of (P0 x2)
% Found (((eq_ref (iS->b)) x2) P) as proof of (P0 x2)
% Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% Found (eq_ref00 P) as proof of (P0 x2)
% Found ((eq_ref0 x2) P) as proof of (P0 x2)
% Found (((eq_ref (iS->b)) x2) P) as proof of (P0 x2)
% Found (((eq_ref (iS->b)) x2) P) as proof of (P0 x2)
% Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% Found (eq_ref00 P) as proof of (P0 x2)
% Found ((eq_ref0 x2) P) as proof of (P0 x2)
% Found (((eq_ref (iS->b)) x2) P) as proof of (P0 x2)
% Found (((eq_ref (iS->b)) x2) P) as proof of (P0 x2)
% Found eta_expansion000:=(eta_expansion00 x'):(((eq (iS->b)) x') (fun (x:iS)=> (x' x)))
% Found (eta_expansion00 x') as proof of (((eq (iS->b)) x') x2)
% Found ((eta_expansion0 b) x') as proof of (((eq (iS->b)) x') x2)
% Found (((eta_expansion iS) b) x') as proof of (((eq (iS->b)) x') x2)
% Found (((eta_expansion iS) b) x') as proof of (((eq (iS->b)) x') x2)
% Found (((eta_expansion iS) b) x') as proof of (((eq (iS->b)) x') x2)
% Found ((eq_sym0000 (((eta_expansion iS) b) x')) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P x'))
% Found ((eq_sym0000 (((eta_expansion iS) b) x')) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P x'))
% Found (((fun (x3:(((eq (iS->b)) x') x2))=> ((eq_sym000 x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) (((eta_expansion iS) b) x')) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P x'))
% Found (((fun (x3:(((eq (iS->b)) x') x2))=> (((eq_sym00 x2) x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) (((eta_expansion iS) b) x')) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P x'))
% Found (((fun (x3:(((eq (iS->b)) x') x2))=> ((((eq_sym0 x') x2) x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) (((eta_expansion iS) b) x')) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P x'))
% Found (((fun (x3:(((eq (iS->b)) x') x2))=> (((((eq_sym (iS->b)) x') x2) x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) (((eta_expansion iS) b) x')) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> (((fun (x3:(((eq (iS->b)) x') x2))=> (((((eq_sym (iS->b)) x') x2) x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) (((eta_expansion iS) b) x')) (((eq_ref (iS->b)) x2) P))) as proof of ((P x2)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop))=> (((fun (x3:(((eq (iS->b)) x') x2))=> (((((eq_sym (iS->b)) x') x2) x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) (((eta_expansion iS) b) x')) (((eq_ref (iS->b)) x2) P))) as proof of (((eq (iS->b)) x2) x')
% Found eta_expansion_dep000:=(eta_expansion_dep00 x'):(((eq (iS->b)) x') (fun (x:iS)=> (x' x)))
% Found (eta_expansion_dep00 x') as proof of (((eq (iS->b)) x') x2)
% Found ((eta_expansion_dep0 (fun (x4:iS)=> b)) x') as proof of (((eq (iS->b)) x') x2)
% Found (((eta_expansion_dep iS) (fun (x4:iS)=> b)) x') as proof of (((eq (iS->b)) x') x2)
% Found (((eta_expansion_dep iS) (fun (x4:iS)=> b)) x') as proof of (((eq (iS->b)) x') x2)
% Found (((eta_expansion_dep iS) (fun (x4:iS)=> b)) x') as proof of (((eq (iS->b)) x') x2)
% Found (eq_sym0000 (((eta_expansion_dep iS) (fun (x4:iS)=> b)) x')) as proof of ((P x2)->(P x'))
% Found (eq_sym0000 (((eta_expansion_dep iS) (fun (x4:iS)=> b)) x')) as proof of ((P x2)->(P x'))
% Found ((fun (x3:(((eq (iS->b)) x') x2))=> ((eq_sym000 x3) P)) (((eta_expansion_dep iS) (fun (x4:iS)=> b)) x')) as proof of ((P x2)->(P x'))
% Found ((fun (x3:(((eq (iS->b)) x') x2))=> (((eq_sym00 x2) x3) P)) (((eta_expansion_dep iS) (fun (x4:iS)=> b)) x')) as proof of ((P x2)->(P x'))
% Found ((fun (x3:(((eq (iS->b)) x') x2))=> ((((eq_sym0 x') x2) x3) P)) (((eta_expansion_dep iS) (fun (x4:iS)=> b)) x')) as proof of ((P x2)->(P x'))
% Found ((fun (x3:(((eq (iS->b)) x') x2))=> (((((eq_sym (iS->b)) x') x2) x3) P)) (((eta_expansion_dep iS) (fun (x4:iS)=> b)) x')) as proof of ((P x2)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> ((fun (x3:(((eq (iS->b)) x') x2))=> (((((eq_sym (iS->b)) x') x2) x3) P)) (((eta_expansion_dep iS) (fun (x4:iS)=> b)) x'))) as proof of ((P x2)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop))=> ((fun (x3:(((eq (iS->b)) x') x2))=> (((((eq_sym (iS->b)) x') x2) x3) P)) (((eta_expansion_dep iS) (fun (x4:iS)=> b)) x'))) as proof of (((eq (iS->b)) x2) x')
% Found x3:(P x0)
% Instantiate: x0:=x':(iS->b)
% Found (fun (x3:(P x0))=> x3) as proof of (P x')
% Found (fun (P:((iS->b)->Prop)) (x3:(P x0))=> x3) as proof of ((P x0)->(P x'))
% Found (fun (P:((iS->b)->Prop)) (x3:(P x0))=> x3) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P x')))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop)) (x3:(P x0))=> x3) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of ((P x0)->(P x'))
% Found ((eq_ref0 x0) P) as proof of ((P x0)->(P x'))
% Found (((eq_ref (iS->b)) x0) P) as proof of ((P x0)->(P x'))
% Found (((eq_ref (iS->b)) x0) P) as proof of ((P x0)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x0) P)) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P x')))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x0) P)) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 (x0 c0)):(((eq b) (x0 c0)) (x0 c0))
% Found (eq_ref0 (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found eq_ref00:=(eq_ref0 (x2 c0)):(((eq b) (x2 c0)) (x2 c0))
% Found (eq_ref0 (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found eq_ref00:=(eq_ref0 (x0 c0)):(((eq b) (x0 c0)) (x0 c0))
% Found (eq_ref0 (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found eq_ref00:=(eq_ref0 (x2 c0)):(((eq b) (x2 c0)) (x2 c0))
% Found (eq_ref0 (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found eq_ref00:=(eq_ref0 x4):(((eq (iS->b)) x4) x4)
% Found (eq_ref0 x4) as proof of (forall (P:((iS->b)->Prop)), ((P x4)->(P Xg)))
% Found ((eq_ref (iS->b)) x4) as proof of (forall (P:((iS->b)->Prop)), ((P x4)->(P Xg)))
% Found ((eq_ref (iS->b)) x4) as proof of (forall (P:((iS->b)->Prop)), ((P x4)->(P Xg)))
% Found ((eq_ref (iS->b)) x4) as proof of (forall (P:((iS->b)->Prop)), ((P x4)->(P Xg)))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((eq_ref (iS->b)) x4)) as proof of (((eq (iS->b)) x4) Xg)
% Found eq_ref00:=(eq_ref0 (x4 c0)):(((eq b) (x4 c0)) (x4 c0))
% Found (eq_ref0 (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found ((eq_ref b) (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found ((eq_ref b) (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found ((eq_ref b) (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found eq_ref00:=(eq_ref0 (x6 c0)):(((eq b) (x6 c0)) (x6 c0))
% Found (eq_ref0 (x6 c0)) as proof of (((eq b) (x6 c0)) c02)
% Found ((eq_ref b) (x6 c0)) as proof of (((eq b) (x6 c0)) c02)
% Found ((eq_ref b) (x6 c0)) as proof of (((eq b) (x6 c0)) c02)
% Found ((eq_ref b) (x6 c0)) as proof of (((eq b) (x6 c0)) c02)
% Found eq_ref00:=(eq_ref0 b0):(((eq (iS->b)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (iS->b)) b0) Xg)
% Found ((eq_ref (iS->b)) b0) as proof of (((eq (iS->b)) b0) Xg)
% Found ((eq_ref (iS->b)) b0) as proof of (((eq (iS->b)) b0) Xg)
% Found ((eq_ref (iS->b)) b0) as proof of (((eq (iS->b)) b0) Xg)
% Found eta_expansion000:=(eta_expansion00 x0):(((eq (iS->b)) x0) (fun (x:iS)=> (x0 x)))
% Found (eta_expansion00 x0) as proof of (((eq (iS->b)) x0) b0)
% Found ((eta_expansion0 b) x0) as proof of (((eq (iS->b)) x0) b0)
% Found (((eta_expansion iS) b) x0) as proof of (((eq (iS->b)) x0) b0)
% Found (((eta_expansion iS) b) x0) as proof of (((eq (iS->b)) x0) b0)
% Found (((eta_expansion iS) b) x0) as proof of (((eq (iS->b)) x0) b0)
% Found eq_ref00:=(eq_ref0 x0):(((eq (iS->b)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (iS->b)) x0) b0)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) b0)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) b0)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) b0)
% Found ((eq_trans0000 ((eq_ref (iS->b)) x0)) ((eq_ref (iS->b)) b0)) as proof of (((eq (iS->b)) x0) Xg)
% Found (((eq_trans000 Xg) ((eq_ref (iS->b)) x0)) ((eq_ref (iS->b)) Xg)) as proof of (((eq (iS->b)) x0) Xg)
% Found ((((fun (b0:(iS->b))=> ((eq_trans00 b0) Xg)) Xg) ((eq_ref (iS->b)) x0)) ((eq_ref (iS->b)) Xg)) as proof of (((eq (iS->b)) x0) Xg)
% Found ((((fun (b0:(iS->b))=> (((eq_trans0 x0) b0) Xg)) Xg) ((eq_ref (iS->b)) x0)) ((eq_ref (iS->b)) Xg)) as proof of (((eq (iS->b)) x0) Xg)
% Found ((((fun (b0:(iS->b))=> ((((eq_trans (iS->b)) x0) b0) Xg)) Xg) ((eq_ref (iS->b)) x0)) ((eq_ref (iS->b)) Xg)) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((((fun (b0:(iS->b))=> ((((eq_trans (iS->b)) x0) b0) Xg)) Xg) ((eq_ref (iS->b)) x0)) ((eq_ref (iS->b)) Xg))) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 x0):(((eq (iS->b)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (iS->b)) x0) x')
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) x')
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x0)) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x9:(((eq b) (x' c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x0)) as proof of ((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->(((eq (iS->b)) x0) x'))
% Found (fun (x9:(((eq b) (x' c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x0)) as proof of ((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->(((eq (iS->b)) x0) x')))
% Found (and_rect40 (fun (x9:(((eq b) (x' c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) x')
% Found ((and_rect4 (((eq (iS->b)) x0) x')) (fun (x9:(((eq b) (x' c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) x')
% Found (((fun (P:Type) (x9:((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->P)))=> (((((and_rect (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))) P) x9) x00)) (((eq (iS->b)) x0) x')) (fun (x9:(((eq b) (x' c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x0))) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((fun (P:Type) (x9:((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->P)))=> (((((and_rect (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))) P) x9) x00)) (((eq (iS->b)) x0) x')) (fun (x9:(((eq b) (x' c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x0)))) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 x2):(((eq (iS->b)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (iS->b)) x2) x')
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) x')
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) x')
% Found (fun (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x2)) as proof of (((eq (iS->b)) x2) x')
% Found (fun (x9:(((eq b) (x' c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x2)) as proof of ((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->(((eq (iS->b)) x2) x'))
% Found (fun (x9:(((eq b) (x' c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x2)) as proof of ((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->(((eq (iS->b)) x2) x')))
% Found (and_rect40 (fun (x9:(((eq b) (x' c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) x')
% Found ((and_rect4 (((eq (iS->b)) x2) x')) (fun (x9:(((eq b) (x' c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) x')
% Found (((fun (P:Type) (x9:((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->P)))=> (((((and_rect (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))) P) x9) x00)) (((eq (iS->b)) x2) x')) (fun (x9:(((eq b) (x' c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x2))) as proof of (((eq (iS->b)) x2) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((fun (P:Type) (x9:((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->P)))=> (((((and_rect (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))) P) x9) x00)) (((eq (iS->b)) x2) x')) (fun (x9:(((eq b) (x' c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x2)))) as proof of (((eq (iS->b)) x2) x')
% Found eq_ref00:=(eq_ref0 x'):(((eq (iS->b)) x') x')
% Found (eq_ref0 x') as proof of (((eq (iS->b)) x') x0)
% Found ((eq_ref (iS->b)) x') as proof of (((eq (iS->b)) x') x0)
% Found ((eq_ref (iS->b)) x') as proof of (((eq (iS->b)) x') x0)
% Found ((eq_ref (iS->b)) x') as proof of (((eq (iS->b)) x') x0)
% Found (eq_sym000 ((eq_ref (iS->b)) x')) as proof of (((eq (iS->b)) x0) x')
% Found ((eq_sym00 x0) ((eq_ref (iS->b)) x')) as proof of (((eq (iS->b)) x0) x')
% Found (((eq_sym0 x') x0) ((eq_ref (iS->b)) x')) as proof of (((eq (iS->b)) x0) x')
% Found ((((eq_sym (iS->b)) x') x0) ((eq_ref (iS->b)) x')) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((((eq_sym (iS->b)) x') x0) ((eq_ref (iS->b)) x'))) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 x4):(((eq (iS->b)) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq (iS->b)) x4) x')
% Found ((eq_ref (iS->b)) x4) as proof of (((eq (iS->b)) x4) x')
% Found ((eq_ref (iS->b)) x4) as proof of (((eq (iS->b)) x4) x')
% Found (fun (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x4)) as proof of (((eq (iS->b)) x4) x')
% Found (fun (x9:(((eq b) (x' c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x4)) as proof of ((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->(((eq (iS->b)) x4) x'))
% Found (fun (x9:(((eq b) (x' c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x4)) as proof of ((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->(((eq (iS->b)) x4) x')))
% Found (and_rect40 (fun (x9:(((eq b) (x' c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x4))) as proof of (((eq (iS->b)) x4) x')
% Found ((and_rect4 (((eq (iS->b)) x4) x')) (fun (x9:(((eq b) (x' c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x4))) as proof of (((eq (iS->b)) x4) x')
% Found (((fun (P:Type) (x9:((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->P)))=> (((((and_rect (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))) P) x9) x00)) (((eq (iS->b)) x4) x')) (fun (x9:(((eq b) (x' c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x4))) as proof of (((eq (iS->b)) x4) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((fun (P:Type) (x9:((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->P)))=> (((((and_rect (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))) P) x9) x00)) (((eq (iS->b)) x4) x')) (fun (x9:(((eq b) (x' c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x4)))) as proof of (((eq (iS->b)) x4) x')
% Found eta_expansion_dep000:=(eta_expansion_dep00 x'):(((eq (iS->b)) x') (fun (x:iS)=> (x' x)))
% Found (eta_expansion_dep00 x') as proof of (((eq (iS->b)) x') x2)
% Found ((eta_expansion_dep0 (fun (x10:iS)=> b)) x') as proof of (((eq (iS->b)) x') x2)
% Found (((eta_expansion_dep iS) (fun (x10:iS)=> b)) x') as proof of (((eq (iS->b)) x') x2)
% Found (((eta_expansion_dep iS) (fun (x10:iS)=> b)) x') as proof of (((eq (iS->b)) x') x2)
% Found (((eta_expansion_dep iS) (fun (x10:iS)=> b)) x') as proof of (((eq (iS->b)) x') x2)
% Found (eq_sym000 (((eta_expansion_dep iS) (fun (x10:iS)=> b)) x')) as proof of (((eq (iS->b)) x2) x')
% Found ((eq_sym00 x2) (((eta_expansion_dep iS) (fun (x10:iS)=> b)) x')) as proof of (((eq (iS->b)) x2) x')
% Found (((eq_sym0 x') x2) (((eta_expansion_dep iS) (fun (x10:iS)=> b)) x')) as proof of (((eq (iS->b)) x2) x')
% Found ((((eq_sym (iS->b)) x') x2) (((eta_expansion_dep iS) (fun (x10:iS)=> b)) x')) as proof of (((eq (iS->b)) x2) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((((eq_sym (iS->b)) x') x2) (((eta_expansion_dep iS) (fun (x10:iS)=> b)) x'))) as proof of (((eq (iS->b)) x2) x')
% Found eq_ref00:=(eq_ref0 x6):(((eq (iS->b)) x6) x6)
% Found (eq_ref0 x6) as proof of (((eq (iS->b)) x6) x')
% Found ((eq_ref (iS->b)) x6) as proof of (((eq (iS->b)) x6) x')
% Found ((eq_ref (iS->b)) x6) as proof of (((eq (iS->b)) x6) x')
% Found (fun (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x6)) as proof of (((eq (iS->b)) x6) x')
% Found (fun (x9:(((eq b) (x' c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x6)) as proof of ((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->(((eq (iS->b)) x6) x'))
% Found (fun (x9:(((eq b) (x' c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x6)) as proof of ((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->(((eq (iS->b)) x6) x')))
% Found (and_rect40 (fun (x9:(((eq b) (x' c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x6))) as proof of (((eq (iS->b)) x6) x')
% Found ((and_rect4 (((eq (iS->b)) x6) x')) (fun (x9:(((eq b) (x' c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x6))) as proof of (((eq (iS->b)) x6) x')
% Found (((fun (P:Type) (x9:((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->P)))=> (((((and_rect (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))) P) x9) x00)) (((eq (iS->b)) x6) x')) (fun (x9:(((eq b) (x' c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x6))) as proof of (((eq (iS->b)) x6) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((fun (P:Type) (x9:((((eq b) (x' c0)) c02)->((forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))->P)))=> (((((and_rect (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))) P) x9) x00)) (((eq (iS->b)) x6) x')) (fun (x9:(((eq b) (x' c0)) c02)) (x10:(forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))=> ((eq_ref (iS->b)) x6)))) as proof of (((eq (iS->b)) x6) x')
% Found eta_expansion_dep000:=(eta_expansion_dep00 x'):(((eq (iS->b)) x') (fun (x:iS)=> (x' x)))
% Found (eta_expansion_dep00 x') as proof of (((eq (iS->b)) x') x4)
% Found ((eta_expansion_dep0 (fun (x10:iS)=> b)) x') as proof of (((eq (iS->b)) x') x4)
% Found (((eta_expansion_dep iS) (fun (x10:iS)=> b)) x') as proof of (((eq (iS->b)) x') x4)
% Found (((eta_expansion_dep iS) (fun (x10:iS)=> b)) x') as proof of (((eq (iS->b)) x') x4)
% Found (((eta_expansion_dep iS) (fun (x10:iS)=> b)) x') as proof of (((eq (iS->b)) x') x4)
% Found (eq_sym000 (((eta_expansion_dep iS) (fun (x10:iS)=> b)) x')) as proof of (((eq (iS->b)) x4) x')
% Found ((eq_sym00 x4) (((eta_expansion_dep iS) (fun (x10:iS)=> b)) x')) as proof of (((eq (iS->b)) x4) x')
% Found (((eq_sym0 x') x4) (((eta_expansion_dep iS) (fun (x10:iS)=> b)) x')) as proof of (((eq (iS->b)) x4) x')
% Found ((((eq_sym (iS->b)) x') x4) (((eta_expansion_dep iS) (fun (x10:iS)=> b)) x')) as proof of (((eq (iS->b)) x4) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((((eq_sym (iS->b)) x') x4) (((eta_expansion_dep iS) (fun (x10:iS)=> b)) x'))) as proof of (((eq (iS->b)) x4) x')
% Found eta_expansion000:=(eta_expansion00 x'):(((eq (iS->b)) x') (fun (x:iS)=> (x' x)))
% Found (eta_expansion00 x') as proof of (((eq (iS->b)) x') x6)
% Found ((eta_expansion0 b) x') as proof of (((eq (iS->b)) x') x6)
% Found (((eta_expansion iS) b) x') as proof of (((eq (iS->b)) x') x6)
% Found (((eta_expansion iS) b) x') as proof of (((eq (iS->b)) x') x6)
% Found (((eta_expansion iS) b) x') as proof of (((eq (iS->b)) x') x6)
% Found (eq_sym000 (((eta_expansion iS) b) x')) as proof of (((eq (iS->b)) x6) x')
% Found ((eq_sym00 x6) (((eta_expansion iS) b) x')) as proof of (((eq (iS->b)) x6) x')
% Found (((eq_sym0 x') x6) (((eta_expansion iS) b) x')) as proof of (((eq (iS->b)) x6) x')
% Found ((((eq_sym (iS->b)) x') x6) (((eta_expansion iS) b) x')) as proof of (((eq (iS->b)) x6) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((((eq_sym (iS->b)) x') x6) (((eta_expansion iS) b) x'))) as proof of (((eq (iS->b)) x6) x')
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (iS->b)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (iS->b)) x0) P) as proof of (P0 x0)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (iS->b)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (iS->b)) x0) P) as proof of (P0 x0)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (iS->b)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (iS->b)) x0) P) as proof of (P0 x0)
% Found eq_ref00:=(eq_ref0 Xg):(((eq (iS->b)) Xg) Xg)
% Found (eq_ref0 Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found ((eq_ref (iS->b)) Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found ((eq_ref (iS->b)) Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found ((eq_ref (iS->b)) Xg) as proof of (((eq (iS->b)) Xg) x0)
% Found (eq_sym000 ((eq_ref (iS->b)) Xg)) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P Xg)))
% Found ((eq_sym00 x0) ((eq_ref (iS->b)) Xg)) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P Xg)))
% Found (((eq_sym0 Xg) x0) ((eq_ref (iS->b)) Xg)) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P Xg)))
% Found ((((eq_sym (iS->b)) Xg) x0) ((eq_ref (iS->b)) Xg)) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P Xg)))
% Found ((((eq_sym (iS->b)) Xg) x0) ((eq_ref (iS->b)) Xg)) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P Xg)))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((((eq_sym (iS->b)) Xg) x0) ((eq_ref (iS->b)) Xg))) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref000:=(eq_ref00 P):((P (x0 ((cP Xx) Xy)))->(P (x0 ((cP Xx) Xy))))
% Found (eq_ref00 P) as proof of (P0 (x0 ((cP Xx) Xy)))
% Found ((eq_ref0 (x0 ((cP Xx) Xy))) P) as proof of (P0 (x0 ((cP Xx) Xy)))
% Found (((eq_ref b) (x0 ((cP Xx) Xy))) P) as proof of (P0 (x0 ((cP Xx) Xy)))
% Found (((eq_ref b) (x0 ((cP Xx) Xy))) P) as proof of (P0 (x0 ((cP Xx) Xy)))
% Found eq_ref00:=(eq_ref0 b0):(((eq (iS->b)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (iS->b)) b0) x')
% Found ((eq_ref (iS->b)) b0) as proof of (((eq (iS->b)) b0) x')
% Found ((eq_ref (iS->b)) b0) as proof of (((eq (iS->b)) b0) x')
% Found ((eq_ref (iS->b)) b0) as proof of (((eq (iS->b)) b0) x')
% Found eta_expansion000:=(eta_expansion00 x2):(((eq (iS->b)) x2) (fun (x:iS)=> (x2 x)))
% Found (eta_expansion00 x2) as proof of (((eq (iS->b)) x2) b0)
% Found ((eta_expansion0 b) x2) as proof of (((eq (iS->b)) x2) b0)
% Found (((eta_expansion iS) b) x2) as proof of (((eq (iS->b)) x2) b0)
% Found (((eta_expansion iS) b) x2) as proof of (((eq (iS->b)) x2) b0)
% Found (((eta_expansion iS) b) x2) as proof of (((eq (iS->b)) x2) b0)
% Found eq_ref00:=(eq_ref0 x2):(((eq (iS->b)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (iS->b)) x2) b0)
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) b0)
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) b0)
% Found ((eq_ref (iS->b)) x2) as proof of (((eq (iS->b)) x2) b0)
% Found ((eq_trans0000 ((eq_ref (iS->b)) x2)) ((eq_ref (iS->b)) b0)) as proof of (((eq (iS->b)) x2) x')
% Found (((eq_trans000 x') ((eq_ref (iS->b)) x2)) ((eq_ref (iS->b)) x')) as proof of (((eq (iS->b)) x2) x')
% Found ((((fun (b0:(iS->b))=> ((eq_trans00 b0) x')) x') ((eq_ref (iS->b)) x2)) ((eq_ref (iS->b)) x')) as proof of (((eq (iS->b)) x2) x')
% Found ((((fun (b0:(iS->b))=> (((eq_trans0 x2) b0) x')) x') ((eq_ref (iS->b)) x2)) ((eq_ref (iS->b)) x')) as proof of (((eq (iS->b)) x2) x')
% Found ((((fun (b0:(iS->b))=> ((((eq_trans (iS->b)) x2) b0) x')) x') ((eq_ref (iS->b)) x2)) ((eq_ref (iS->b)) x')) as proof of (((eq (iS->b)) x2) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((((fun (b0:(iS->b))=> ((((eq_trans (iS->b)) x2) b0) x')) x') ((eq_ref (iS->b)) x2)) ((eq_ref (iS->b)) x'))) as proof of (((eq (iS->b)) x2) x')
% Found eq_ref00:=(eq_ref0 x'):(((eq (iS->b)) x') x')
% Found (eq_ref0 x') as proof of (((eq (iS->b)) x') x0)
% Found ((eq_ref (iS->b)) x') as proof of (((eq (iS->b)) x') x0)
% Found ((eq_ref (iS->b)) x') as proof of (((eq (iS->b)) x') x0)
% Found ((eq_ref (iS->b)) x') as proof of (((eq (iS->b)) x') x0)
% Found ((eq_sym0000 ((eq_ref (iS->b)) x')) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found ((eq_sym0000 ((eq_ref (iS->b)) x')) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found (((fun (x3:(((eq (iS->b)) x') x0))=> ((eq_sym000 x3) (fun (x5:(iS->b))=> ((P x0)->(P x5))))) ((eq_ref (iS->b)) x')) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found (((fun (x3:(((eq (iS->b)) x') x0))=> (((eq_sym00 x0) x3) (fun (x5:(iS->b))=> ((P x0)->(P x5))))) ((eq_ref (iS->b)) x')) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found (((fun (x3:(((eq (iS->b)) x') x0))=> ((((eq_sym0 x') x0) x3) (fun (x5:(iS->b))=> ((P x0)->(P x5))))) ((eq_ref (iS->b)) x')) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found (((fun (x3:(((eq (iS->b)) x') x0))=> (((((eq_sym (iS->b)) x') x0) x3) (fun (x5:(iS->b))=> ((P x0)->(P x5))))) ((eq_ref (iS->b)) x')) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> (((fun (x3:(((eq (iS->b)) x') x0))=> (((((eq_sym (iS->b)) x') x0) x3) (fun (x5:(iS->b))=> ((P x0)->(P x5))))) ((eq_ref (iS->b)) x')) (((eq_ref (iS->b)) x0) P))) as proof of ((P x0)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop))=> (((fun (x3:(((eq (iS->b)) x') x0))=> (((((eq_sym (iS->b)) x') x0) x3) (fun (x5:(iS->b))=> ((P x0)->(P x5))))) ((eq_ref (iS->b)) x')) (((eq_ref (iS->b)) x0) P))) as proof of (((eq (iS->b)) x0) x')
% Found eta_expansion_dep000:=(eta_expansion_dep00 x'):(((eq (iS->b)) x') (fun (x:iS)=> (x' x)))
% Found (eta_expansion_dep00 x') as proof of (((eq (iS->b)) x') x0)
% Found ((eta_expansion_dep0 (fun (x4:iS)=> b)) x') as proof of (((eq (iS->b)) x') x0)
% Found (((eta_expansion_dep iS) (fun (x4:iS)=> b)) x') as proof of (((eq (iS->b)) x') x0)
% Found (((eta_expansion_dep iS) (fun (x4:iS)=> b)) x') as proof of (((eq (iS->b)) x') x0)
% Found (((eta_expansion_dep iS) (fun (x4:iS)=> b)) x') as proof of (((eq (iS->b)) x') x0)
% Found (eq_sym0000 (((eta_expansion_dep iS) (fun (x4:iS)=> b)) x')) as proof of ((P x0)->(P x'))
% Found (eq_sym0000 (((eta_expansion_dep iS) (fun (x4:iS)=> b)) x')) as proof of ((P x0)->(P x'))
% Found ((fun (x3:(((eq (iS->b)) x') x0))=> ((eq_sym000 x3) P)) (((eta_expansion_dep iS) (fun (x4:iS)=> b)) x')) as proof of ((P x0)->(P x'))
% Found ((fun (x3:(((eq (iS->b)) x') x0))=> (((eq_sym00 x0) x3) P)) (((eta_expansion_dep iS) (fun (x4:iS)=> b)) x')) as proof of ((P x0)->(P x'))
% Found ((fun (x3:(((eq (iS->b)) x') x0))=> ((((eq_sym0 x') x0) x3) P)) (((eta_expansion_dep iS) (fun (x4:iS)=> b)) x')) as proof of ((P x0)->(P x'))
% Found ((fun (x3:(((eq (iS->b)) x') x0))=> (((((eq_sym (iS->b)) x') x0) x3) P)) (((eta_expansion_dep iS) (fun (x4:iS)=> b)) x')) as proof of ((P x0)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> ((fun (x3:(((eq (iS->b)) x') x0))=> (((((eq_sym (iS->b)) x') x0) x3) P)) (((eta_expansion_dep iS) (fun (x4:iS)=> b)) x'))) as proof of ((P x0)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop))=> ((fun (x3:(((eq (iS->b)) x') x0))=> (((((eq_sym (iS->b)) x') x0) x3) P)) (((eta_expansion_dep iS) (fun (x4:iS)=> b)) x'))) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 (x8 x9)):(((eq b) (x8 x9)) (x8 x9))
% Found (eq_ref0 (x8 x9)) as proof of (((eq b) (x8 x9)) (Xg x9))
% Found ((eq_ref b) (x8 x9)) as proof of (((eq b) (x8 x9)) (Xg x9))
% Found ((eq_ref b) (x8 x9)) as proof of (((eq b) (x8 x9)) (Xg x9))
% Found (fun (x9:iS)=> ((eq_ref b) (x8 x9))) as proof of (((eq b) (x8 x9)) (Xg x9))
% Found (fun (x9:iS)=> ((eq_ref b) (x8 x9))) as proof of (forall (x:iS), (((eq b) (x8 x)) (Xg x)))
% Found (functional_extensionality_dep0000 (fun (x9:iS)=> ((eq_ref b) (x8 x9)))) as proof of (((eq (iS->b)) x8) Xg)
% Found ((functional_extensionality_dep000 Xg) (fun (x9:iS)=> ((eq_ref b) (x8 x9)))) as proof of (((eq (iS->b)) x8) Xg)
% Found (((functional_extensionality_dep00 x8) Xg) (fun (x9:iS)=> ((eq_ref b) (x8 x9)))) as proof of (((eq (iS->b)) x8) Xg)
% Found ((((functional_extensionality_dep0 (fun (x11:iS)=> b)) x8) Xg) (fun (x9:iS)=> ((eq_ref b) (x8 x9)))) as proof of (((eq (iS->b)) x8) Xg)
% Found (((((functional_extensionality_dep iS) (fun (x11:iS)=> b)) x8) Xg) (fun (x9:iS)=> ((eq_ref b) (x8 x9)))) as proof of (((eq (iS->b)) x8) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((((functional_extensionality_dep iS) (fun (x11:iS)=> b)) x8) Xg) (fun (x9:iS)=> ((eq_ref b) (x8 x9))))) as proof of (((eq (iS->b)) x8) Xg)
% Found eq_ref00:=(eq_ref0 (x8 x9)):(((eq b) (x8 x9)) (x8 x9))
% Found (eq_ref0 (x8 x9)) as proof of (((eq b) (x8 x9)) (Xg x9))
% Found ((eq_ref b) (x8 x9)) as proof of (((eq b) (x8 x9)) (Xg x9))
% Found ((eq_ref b) (x8 x9)) as proof of (((eq b) (x8 x9)) (Xg x9))
% Found (fun (x9:iS)=> ((eq_ref b) (x8 x9))) as proof of (((eq b) (x8 x9)) (Xg x9))
% Found (fun (x9:iS)=> ((eq_ref b) (x8 x9))) as proof of (forall (x:iS), (((eq b) (x8 x)) (Xg x)))
% Found (functional_extensionality0000 (fun (x9:iS)=> ((eq_ref b) (x8 x9)))) as proof of (((eq (iS->b)) x8) Xg)
% Found ((functional_extensionality000 Xg) (fun (x9:iS)=> ((eq_ref b) (x8 x9)))) as proof of (((eq (iS->b)) x8) Xg)
% Found (((functional_extensionality00 x8) Xg) (fun (x9:iS)=> ((eq_ref b) (x8 x9)))) as proof of (((eq (iS->b)) x8) Xg)
% Found ((((functional_extensionality0 b) x8) Xg) (fun (x9:iS)=> ((eq_ref b) (x8 x9)))) as proof of (((eq (iS->b)) x8) Xg)
% Found (((((functional_extensionality iS) b) x8) Xg) (fun (x9:iS)=> ((eq_ref b) (x8 x9)))) as proof of (((eq (iS->b)) x8) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((((functional_extensionality iS) b) x8) Xg) (fun (x9:iS)=> ((eq_ref b) (x8 x9))))) as proof of (((eq (iS->b)) x8) Xg)
% Found eq_ref00:=(eq_ref0 (x2 x3)):(((eq b) (x2 x3)) (x2 x3))
% Found (eq_ref0 (x2 x3)) as proof of (((eq b) (x2 x3)) (Xg x3))
% Found ((eq_ref b) (x2 x3)) as proof of (((eq b) (x2 x3)) (Xg x3))
% Found ((eq_ref b) (x2 x3)) as proof of (((eq b) (x2 x3)) (Xg x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x2 x3))) as proof of (((eq b) (x2 x3)) (Xg x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x2 x3))) as proof of (forall (x:iS), (((eq b) (x2 x)) (Xg x)))
% Found ((functional_extensionality00000 (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P Xg))
% Found ((functional_extensionality00000 (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P Xg))
% Found (((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (Xg x))))=> ((functional_extensionality0000 x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P Xg))
% Found (((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (Xg x))))=> (((functional_extensionality000 Xg) x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P Xg))
% Found (((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (Xg x))))=> ((((functional_extensionality00 x2) Xg) x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P Xg))
% Found (((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (Xg x))))=> (((((functional_extensionality0 b) x2) Xg) x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P Xg))
% Found (((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (Xg x))))=> ((((((functional_extensionality iS) b) x2) Xg) x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> (((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (Xg x))))=> ((((((functional_extensionality iS) b) x2) Xg) x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) (((eq_ref (iS->b)) x2) P))) as proof of ((P x2)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> (((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (Xg x))))=> ((((((functional_extensionality iS) b) x2) Xg) x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) (((eq_ref (iS->b)) x2) P))) as proof of (((eq (iS->b)) x2) Xg)
% Found eq_ref00:=(eq_ref0 (x2 x3)):(((eq b) (x2 x3)) (x2 x3))
% Found (eq_ref0 (x2 x3)) as proof of (((eq b) (x2 x3)) (Xg x3))
% Found ((eq_ref b) (x2 x3)) as proof of (((eq b) (x2 x3)) (Xg x3))
% Found ((eq_ref b) (x2 x3)) as proof of (((eq b) (x2 x3)) (Xg x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x2 x3))) as proof of (((eq b) (x2 x3)) (Xg x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x2 x3))) as proof of (forall (x:iS), (((eq b) (x2 x)) (Xg x)))
% Found (functional_extensionality00000 (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of ((P x2)->(P Xg))
% Found (functional_extensionality00000 (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of ((P x2)->(P Xg))
% Found ((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (Xg x))))=> ((functional_extensionality0000 x3) P)) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of ((P x2)->(P Xg))
% Found ((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (Xg x))))=> (((functional_extensionality000 Xg) x3) P)) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of ((P x2)->(P Xg))
% Found ((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (Xg x))))=> ((((functional_extensionality00 x2) Xg) x3) P)) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of ((P x2)->(P Xg))
% Found ((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (Xg x))))=> (((((functional_extensionality0 b) x2) Xg) x3) P)) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of ((P x2)->(P Xg))
% Found ((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (Xg x))))=> ((((((functional_extensionality iS) b) x2) Xg) x3) P)) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of ((P x2)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> ((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (Xg x))))=> ((((((functional_extensionality iS) b) x2) Xg) x3) P)) (fun (x3:iS)=> ((eq_ref b) (x2 x3))))) as proof of ((P x2)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> ((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (Xg x))))=> ((((((functional_extensionality iS) b) x2) Xg) x3) P)) (fun (x3:iS)=> ((eq_ref b) (x2 x3))))) as proof of (((eq (iS->b)) x2) Xg)
% Found eq_ref00:=(eq_ref0 (x2 x3)):(((eq b) (x2 x3)) (x2 x3))
% Found (eq_ref0 (x2 x3)) as proof of (((eq b) (x2 x3)) (Xg x3))
% Found ((eq_ref b) (x2 x3)) as proof of (((eq b) (x2 x3)) (Xg x3))
% Found ((eq_ref b) (x2 x3)) as proof of (((eq b) (x2 x3)) (Xg x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x2 x3))) as proof of (((eq b) (x2 x3)) (Xg x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x2 x3))) as proof of (forall (x:iS), (((eq b) (x2 x)) (Xg x)))
% Found ((functional_extensionality_dep00000 (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P Xg))
% Found ((functional_extensionality_dep00000 (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P Xg))
% Found (((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (Xg x))))=> ((functional_extensionality_dep0000 x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P Xg))
% Found (((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (Xg x))))=> (((functional_extensionality_dep000 Xg) x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P Xg))
% Found (((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (Xg x))))=> ((((functional_extensionality_dep00 x2) Xg) x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P Xg))
% Found (((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (Xg x))))=> (((((functional_extensionality_dep0 (fun (x5:iS)=> b)) x2) Xg) x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P Xg))
% Found (((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (Xg x))))=> ((((((functional_extensionality_dep iS) (fun (x5:iS)=> b)) x2) Xg) x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> (((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (Xg x))))=> ((((((functional_extensionality_dep iS) (fun (x5:iS)=> b)) x2) Xg) x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) (((eq_ref (iS->b)) x2) P))) as proof of ((P x2)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> (((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (Xg x))))=> ((((((functional_extensionality_dep iS) (fun (x5:iS)=> b)) x2) Xg) x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) (((eq_ref (iS->b)) x2) P))) as proof of (((eq (iS->b)) x2) Xg)
% Found eq_ref00:=(eq_ref0 (x2 x3)):(((eq b) (x2 x3)) (x2 x3))
% Found (eq_ref0 (x2 x3)) as proof of (((eq b) (x2 x3)) (Xg x3))
% Found ((eq_ref b) (x2 x3)) as proof of (((eq b) (x2 x3)) (Xg x3))
% Found ((eq_ref b) (x2 x3)) as proof of (((eq b) (x2 x3)) (Xg x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x2 x3))) as proof of (((eq b) (x2 x3)) (Xg x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x2 x3))) as proof of (forall (x:iS), (((eq b) (x2 x)) (Xg x)))
% Found (functional_extensionality_dep00000 (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of ((P x2)->(P Xg))
% Found (functional_extensionality_dep00000 (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of ((P x2)->(P Xg))
% Found ((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (Xg x))))=> ((functional_extensionality_dep0000 x3) P)) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of ((P x2)->(P Xg))
% Found ((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (Xg x))))=> (((functional_extensionality_dep000 Xg) x3) P)) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of ((P x2)->(P Xg))
% Found ((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (Xg x))))=> ((((functional_extensionality_dep00 x2) Xg) x3) P)) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of ((P x2)->(P Xg))
% Found ((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (Xg x))))=> (((((functional_extensionality_dep0 (fun (x5:iS)=> b)) x2) Xg) x3) P)) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of ((P x2)->(P Xg))
% Found ((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (Xg x))))=> ((((((functional_extensionality_dep iS) (fun (x5:iS)=> b)) x2) Xg) x3) P)) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of ((P x2)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> ((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (Xg x))))=> ((((((functional_extensionality_dep iS) (fun (x5:iS)=> b)) x2) Xg) x3) P)) (fun (x3:iS)=> ((eq_ref b) (x2 x3))))) as proof of ((P x2)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> ((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (Xg x))))=> ((((((functional_extensionality_dep iS) (fun (x5:iS)=> b)) x2) Xg) x3) P)) (fun (x3:iS)=> ((eq_ref b) (x2 x3))))) as proof of (((eq (iS->b)) x2) Xg)
% Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% Found (eq_ref00 P) as proof of (P0 x2)
% Found ((eq_ref0 x2) P) as proof of (P0 x2)
% Found (((eq_ref (iS->b)) x2) P) as proof of (P0 x2)
% Found (((eq_ref (iS->b)) x2) P) as proof of (P0 x2)
% Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% Found (eq_ref00 P) as proof of (P0 x2)
% Found ((eq_ref0 x2) P) as proof of (P0 x2)
% Found (((eq_ref (iS->b)) x2) P) as proof of (P0 x2)
% Found (((eq_ref (iS->b)) x2) P) as proof of (P0 x2)
% Found eq_ref00:=(eq_ref0 x0):(((eq (iS->b)) x0) x0)
% Found (eq_ref0 x0) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P Xg)))
% Found ((eq_ref (iS->b)) x0) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P Xg)))
% Found ((eq_ref (iS->b)) x0) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P Xg)))
% Found ((eq_ref (iS->b)) x0) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P Xg)))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((eq_ref (iS->b)) x0)) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref000:=(eq_ref00 P):((P (x0 ((cP Xx) Xy)))->(P (x0 ((cP Xx) Xy))))
% Found (eq_ref00 P) as proof of (P0 (x0 ((cP Xx) Xy)))
% Found ((eq_ref0 (x0 ((cP Xx) Xy))) P) as proof of (P0 (x0 ((cP Xx) Xy)))
% Found (((eq_ref b) (x0 ((cP Xx) Xy))) P) as proof of (P0 (x0 ((cP Xx) Xy)))
% Found (((eq_ref b) (x0 ((cP Xx) Xy))) P) as proof of (P0 (x0 ((cP Xx) Xy)))
% Found eq_ref00:=(eq_ref0 x2):(((eq (iS->b)) x2) x2)
% Found (eq_ref0 x2) as proof of (forall (P:((iS->b)->Prop)), ((P x2)->(P Xg)))
% Found ((eq_ref (iS->b)) x2) as proof of (forall (P:((iS->b)->Prop)), ((P x2)->(P Xg)))
% Found ((eq_ref (iS->b)) x2) as proof of (forall (P:((iS->b)->Prop)), ((P x2)->(P Xg)))
% Found ((eq_ref (iS->b)) x2) as proof of (forall (P:((iS->b)->Prop)), ((P x2)->(P Xg)))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> ((eq_ref (iS->b)) x2)) as proof of (((eq (iS->b)) x2) Xg)
% Found x5:(P x4)
% Instantiate: x4:=Xg:(iS->b)
% Found (fun (x5:(P x4))=> x5) as proof of (P Xg)
% Found (fun (P:((iS->b)->Prop)) (x5:(P x4))=> x5) as proof of ((P x4)->(P Xg))
% Found (fun (P:((iS->b)->Prop)) (x5:(P x4))=> x5) as proof of (forall (P:((iS->b)->Prop)), ((P x4)->(P Xg)))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop)) (x5:(P x4))=> x5) as proof of (((eq (iS->b)) x4) 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 (iS->b)) x4) P) as proof of ((P x4)->(P Xg))
% Found (((eq_ref (iS->b)) x4) P) as proof of ((P x4)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x4) P)) as proof of ((P x4)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x4) P)) as proof of (forall (P:((iS->b)->Prop)), ((P x4)->(P Xg)))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x4) P)) as proof of (((eq (iS->b)) x4) Xg)
% Found eq_ref00:=(eq_ref0 x4):(((eq (iS->b)) x4) x4)
% Found (eq_ref0 x4) as proof of (forall (P:((iS->b)->Prop)), ((P x4)->(P x')))
% Found ((eq_ref (iS->b)) x4) as proof of (forall (P:((iS->b)->Prop)), ((P x4)->(P x')))
% Found ((eq_ref (iS->b)) x4) as proof of (forall (P:((iS->b)->Prop)), ((P x4)->(P x')))
% Found ((eq_ref (iS->b)) x4) as proof of (forall (P:((iS->b)->Prop)), ((P x4)->(P x')))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((eq_ref (iS->b)) x4)) as proof of (((eq (iS->b)) x4) x')
% Found eq_ref00:=(eq_ref0 (x0 x9)):(((eq b) (x0 x9)) (x0 x9))
% Found (eq_ref0 (x0 x9)) as proof of (((eq b) (x0 x9)) (Xg x9))
% Found ((eq_ref b) (x0 x9)) as proof of (((eq b) (x0 x9)) (Xg x9))
% Found ((eq_ref b) (x0 x9)) as proof of (((eq b) (x0 x9)) (Xg x9))
% Found (fun (x9:iS)=> ((eq_ref b) (x0 x9))) as proof of (((eq b) (x0 x9)) (Xg x9))
% Found (fun (x9:iS)=> ((eq_ref b) (x0 x9))) as proof of (forall (x:iS), (((eq b) (x0 x)) (Xg x)))
% Found (functional_extensionality0000 (fun (x9:iS)=> ((eq_ref b) (x0 x9)))) as proof of (((eq (iS->b)) x0) Xg)
% Found ((functional_extensionality000 Xg) (fun (x9:iS)=> ((eq_ref b) (x0 x9)))) as proof of (((eq (iS->b)) x0) Xg)
% Found (((functional_extensionality00 x0) Xg) (fun (x9:iS)=> ((eq_ref b) (x0 x9)))) as proof of (((eq (iS->b)) x0) Xg)
% Found ((((functional_extensionality0 b) x0) Xg) (fun (x9:iS)=> ((eq_ref b) (x0 x9)))) as proof of (((eq (iS->b)) x0) Xg)
% Found (((((functional_extensionality iS) b) x0) Xg) (fun (x9:iS)=> ((eq_ref b) (x0 x9)))) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((((functional_extensionality iS) b) x0) Xg) (fun (x9:iS)=> ((eq_ref b) (x0 x9))))) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 (x0 x9)):(((eq b) (x0 x9)) (x0 x9))
% Found (eq_ref0 (x0 x9)) as proof of (((eq b) (x0 x9)) (Xg x9))
% Found ((eq_ref b) (x0 x9)) as proof of (((eq b) (x0 x9)) (Xg x9))
% Found ((eq_ref b) (x0 x9)) as proof of (((eq b) (x0 x9)) (Xg x9))
% Found (fun (x9:iS)=> ((eq_ref b) (x0 x9))) as proof of (((eq b) (x0 x9)) (Xg x9))
% Found (fun (x9:iS)=> ((eq_ref b) (x0 x9))) as proof of (forall (x:iS), (((eq b) (x0 x)) (Xg x)))
% Found (functional_extensionality_dep0000 (fun (x9:iS)=> ((eq_ref b) (x0 x9)))) as proof of (((eq (iS->b)) x0) Xg)
% Found ((functional_extensionality_dep000 Xg) (fun (x9:iS)=> ((eq_ref b) (x0 x9)))) as proof of (((eq (iS->b)) x0) Xg)
% Found (((functional_extensionality_dep00 x0) Xg) (fun (x9:iS)=> ((eq_ref b) (x0 x9)))) as proof of (((eq (iS->b)) x0) Xg)
% Found ((((functional_extensionality_dep0 (fun (x11:iS)=> b)) x0) Xg) (fun (x9:iS)=> ((eq_ref b) (x0 x9)))) as proof of (((eq (iS->b)) x0) Xg)
% Found (((((functional_extensionality_dep iS) (fun (x11:iS)=> b)) x0) Xg) (fun (x9:iS)=> ((eq_ref b) (x0 x9)))) as proof of (((eq (iS->b)) x0) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((((functional_extensionality_dep iS) (fun (x11:iS)=> b)) x0) Xg) (fun (x9:iS)=> ((eq_ref b) (x0 x9))))) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 (x2 x9)):(((eq b) (x2 x9)) (x2 x9))
% Found (eq_ref0 (x2 x9)) as proof of (((eq b) (x2 x9)) (Xg x9))
% Found ((eq_ref b) (x2 x9)) as proof of (((eq b) (x2 x9)) (Xg x9))
% Found ((eq_ref b) (x2 x9)) as proof of (((eq b) (x2 x9)) (Xg x9))
% Found (fun (x9:iS)=> ((eq_ref b) (x2 x9))) as proof of (((eq b) (x2 x9)) (Xg x9))
% Found (fun (x9:iS)=> ((eq_ref b) (x2 x9))) as proof of (forall (x:iS), (((eq b) (x2 x)) (Xg x)))
% Found (functional_extensionality0000 (fun (x9:iS)=> ((eq_ref b) (x2 x9)))) as proof of (((eq (iS->b)) x2) Xg)
% Found ((functional_extensionality000 Xg) (fun (x9:iS)=> ((eq_ref b) (x2 x9)))) as proof of (((eq (iS->b)) x2) Xg)
% Found (((functional_extensionality00 x2) Xg) (fun (x9:iS)=> ((eq_ref b) (x2 x9)))) as proof of (((eq (iS->b)) x2) Xg)
% Found ((((functional_extensionality0 b) x2) Xg) (fun (x9:iS)=> ((eq_ref b) (x2 x9)))) as proof of (((eq (iS->b)) x2) Xg)
% Found (((((functional_extensionality iS) b) x2) Xg) (fun (x9:iS)=> ((eq_ref b) (x2 x9)))) as proof of (((eq (iS->b)) x2) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((((functional_extensionality iS) b) x2) Xg) (fun (x9:iS)=> ((eq_ref b) (x2 x9))))) as proof of (((eq (iS->b)) x2) Xg)
% Found eq_ref00:=(eq_ref0 (x2 x9)):(((eq b) (x2 x9)) (x2 x9))
% Found (eq_ref0 (x2 x9)) as proof of (((eq b) (x2 x9)) (Xg x9))
% Found ((eq_ref b) (x2 x9)) as proof of (((eq b) (x2 x9)) (Xg x9))
% Found ((eq_ref b) (x2 x9)) as proof of (((eq b) (x2 x9)) (Xg x9))
% Found (fun (x9:iS)=> ((eq_ref b) (x2 x9))) as proof of (((eq b) (x2 x9)) (Xg x9))
% Found (fun (x9:iS)=> ((eq_ref b) (x2 x9))) as proof of (forall (x:iS), (((eq b) (x2 x)) (Xg x)))
% Found (functional_extensionality_dep0000 (fun (x9:iS)=> ((eq_ref b) (x2 x9)))) as proof of (((eq (iS->b)) x2) Xg)
% Found ((functional_extensionality_dep000 Xg) (fun (x9:iS)=> ((eq_ref b) (x2 x9)))) as proof of (((eq (iS->b)) x2) Xg)
% Found (((functional_extensionality_dep00 x2) Xg) (fun (x9:iS)=> ((eq_ref b) (x2 x9)))) as proof of (((eq (iS->b)) x2) Xg)
% Found ((((functional_extensionality_dep0 (fun (x11:iS)=> b)) x2) Xg) (fun (x9:iS)=> ((eq_ref b) (x2 x9)))) as proof of (((eq (iS->b)) x2) Xg)
% Found (((((functional_extensionality_dep iS) (fun (x11:iS)=> b)) x2) Xg) (fun (x9:iS)=> ((eq_ref b) (x2 x9)))) as proof of (((eq (iS->b)) x2) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((((functional_extensionality_dep iS) (fun (x11:iS)=> b)) x2) Xg) (fun (x9:iS)=> ((eq_ref b) (x2 x9))))) as proof of (((eq (iS->b)) x2) Xg)
% Found eq_ref00:=(eq_ref0 b0):(((eq (iS->b)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (iS->b)) b0) x')
% Found ((eq_ref (iS->b)) b0) as proof of (((eq (iS->b)) b0) x')
% Found ((eq_ref (iS->b)) b0) as proof of (((eq (iS->b)) b0) x')
% Found ((eq_ref (iS->b)) b0) as proof of (((eq (iS->b)) b0) x')
% Found eta_expansion000:=(eta_expansion00 x0):(((eq (iS->b)) x0) (fun (x:iS)=> (x0 x)))
% Found (eta_expansion00 x0) as proof of (((eq (iS->b)) x0) b0)
% Found ((eta_expansion0 b) x0) as proof of (((eq (iS->b)) x0) b0)
% Found (((eta_expansion iS) b) x0) as proof of (((eq (iS->b)) x0) b0)
% Found (((eta_expansion iS) b) x0) as proof of (((eq (iS->b)) x0) b0)
% Found (((eta_expansion iS) b) x0) as proof of (((eq (iS->b)) x0) b0)
% Found eq_ref00:=(eq_ref0 x0):(((eq (iS->b)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (iS->b)) x0) b0)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) b0)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) b0)
% Found ((eq_ref (iS->b)) x0) as proof of (((eq (iS->b)) x0) b0)
% Found ((eq_trans0000 ((eq_ref (iS->b)) x0)) ((eq_ref (iS->b)) b0)) as proof of (((eq (iS->b)) x0) x')
% Found (((eq_trans000 x') ((eq_ref (iS->b)) x0)) ((eq_ref (iS->b)) x')) as proof of (((eq (iS->b)) x0) x')
% Found ((((fun (b0:(iS->b))=> ((eq_trans00 b0) x')) x') ((eq_ref (iS->b)) x0)) ((eq_ref (iS->b)) x')) as proof of (((eq (iS->b)) x0) x')
% Found ((((fun (b0:(iS->b))=> (((eq_trans0 x0) b0) x')) x') ((eq_ref (iS->b)) x0)) ((eq_ref (iS->b)) x')) as proof of (((eq (iS->b)) x0) x')
% Found ((((fun (b0:(iS->b))=> ((((eq_trans (iS->b)) x0) b0) x')) x') ((eq_ref (iS->b)) x0)) ((eq_ref (iS->b)) x')) as proof of (((eq (iS->b)) x0) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((((fun (b0:(iS->b))=> ((((eq_trans (iS->b)) x0) b0) x')) x') ((eq_ref (iS->b)) x0)) ((eq_ref (iS->b)) x'))) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 (x4 x9)):(((eq b) (x4 x9)) (x4 x9))
% Found (eq_ref0 (x4 x9)) as proof of (((eq b) (x4 x9)) (Xg x9))
% Found ((eq_ref b) (x4 x9)) as proof of (((eq b) (x4 x9)) (Xg x9))
% Found ((eq_ref b) (x4 x9)) as proof of (((eq b) (x4 x9)) (Xg x9))
% Found (fun (x9:iS)=> ((eq_ref b) (x4 x9))) as proof of (((eq b) (x4 x9)) (Xg x9))
% Found (fun (x9:iS)=> ((eq_ref b) (x4 x9))) as proof of (forall (x:iS), (((eq b) (x4 x)) (Xg x)))
% Found (functional_extensionality0000 (fun (x9:iS)=> ((eq_ref b) (x4 x9)))) as proof of (((eq (iS->b)) x4) Xg)
% Found ((functional_extensionality000 Xg) (fun (x9:iS)=> ((eq_ref b) (x4 x9)))) as proof of (((eq (iS->b)) x4) Xg)
% Found (((functional_extensionality00 x4) Xg) (fun (x9:iS)=> ((eq_ref b) (x4 x9)))) as proof of (((eq (iS->b)) x4) Xg)
% Found ((((functional_extensionality0 b) x4) Xg) (fun (x9:iS)=> ((eq_ref b) (x4 x9)))) as proof of (((eq (iS->b)) x4) Xg)
% Found (((((functional_extensionality iS) b) x4) Xg) (fun (x9:iS)=> ((eq_ref b) (x4 x9)))) as proof of (((eq (iS->b)) x4) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((((functional_extensionality iS) b) x4) Xg) (fun (x9:iS)=> ((eq_ref b) (x4 x9))))) as proof of (((eq (iS->b)) x4) Xg)
% Found eq_ref00:=(eq_ref0 (x4 x9)):(((eq b) (x4 x9)) (x4 x9))
% Found (eq_ref0 (x4 x9)) as proof of (((eq b) (x4 x9)) (Xg x9))
% Found ((eq_ref b) (x4 x9)) as proof of (((eq b) (x4 x9)) (Xg x9))
% Found ((eq_ref b) (x4 x9)) as proof of (((eq b) (x4 x9)) (Xg x9))
% Found (fun (x9:iS)=> ((eq_ref b) (x4 x9))) as proof of (((eq b) (x4 x9)) (Xg x9))
% Found (fun (x9:iS)=> ((eq_ref b) (x4 x9))) as proof of (forall (x:iS), (((eq b) (x4 x)) (Xg x)))
% Found (functional_extensionality_dep0000 (fun (x9:iS)=> ((eq_ref b) (x4 x9)))) as proof of (((eq (iS->b)) x4) Xg)
% Found ((functional_extensionality_dep000 Xg) (fun (x9:iS)=> ((eq_ref b) (x4 x9)))) as proof of (((eq (iS->b)) x4) Xg)
% Found (((functional_extensionality_dep00 x4) Xg) (fun (x9:iS)=> ((eq_ref b) (x4 x9)))) as proof of (((eq (iS->b)) x4) Xg)
% Found ((((functional_extensionality_dep0 (fun (x11:iS)=> b)) x4) Xg) (fun (x9:iS)=> ((eq_ref b) (x4 x9)))) as proof of (((eq (iS->b)) x4) Xg)
% Found (((((functional_extensionality_dep iS) (fun (x11:iS)=> b)) x4) Xg) (fun (x9:iS)=> ((eq_ref b) (x4 x9)))) as proof of (((eq (iS->b)) x4) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((((functional_extensionality_dep iS) (fun (x11:iS)=> b)) x4) Xg) (fun (x9:iS)=> ((eq_ref b) (x4 x9))))) as proof of (((eq (iS->b)) x4) Xg)
% Found eq_ref00:=(eq_ref0 (x6 x9)):(((eq b) (x6 x9)) (x6 x9))
% Found (eq_ref0 (x6 x9)) as proof of (((eq b) (x6 x9)) (Xg x9))
% Found ((eq_ref b) (x6 x9)) as proof of (((eq b) (x6 x9)) (Xg x9))
% Found ((eq_ref b) (x6 x9)) as proof of (((eq b) (x6 x9)) (Xg x9))
% Found (fun (x9:iS)=> ((eq_ref b) (x6 x9))) as proof of (((eq b) (x6 x9)) (Xg x9))
% Found (fun (x9:iS)=> ((eq_ref b) (x6 x9))) as proof of (forall (x:iS), (((eq b) (x6 x)) (Xg x)))
% Found (functional_extensionality0000 (fun (x9:iS)=> ((eq_ref b) (x6 x9)))) as proof of (((eq (iS->b)) x6) Xg)
% Found ((functional_extensionality000 Xg) (fun (x9:iS)=> ((eq_ref b) (x6 x9)))) as proof of (((eq (iS->b)) x6) Xg)
% Found (((functional_extensionality00 x6) Xg) (fun (x9:iS)=> ((eq_ref b) (x6 x9)))) as proof of (((eq (iS->b)) x6) Xg)
% Found ((((functional_extensionality0 b) x6) Xg) (fun (x9:iS)=> ((eq_ref b) (x6 x9)))) as proof of (((eq (iS->b)) x6) Xg)
% Found (((((functional_extensionality iS) b) x6) Xg) (fun (x9:iS)=> ((eq_ref b) (x6 x9)))) as proof of (((eq (iS->b)) x6) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((((functional_extensionality iS) b) x6) Xg) (fun (x9:iS)=> ((eq_ref b) (x6 x9))))) as proof of (((eq (iS->b)) x6) Xg)
% Found eq_ref00:=(eq_ref0 (x6 x9)):(((eq b) (x6 x9)) (x6 x9))
% Found (eq_ref0 (x6 x9)) as proof of (((eq b) (x6 x9)) (Xg x9))
% Found ((eq_ref b) (x6 x9)) as proof of (((eq b) (x6 x9)) (Xg x9))
% Found ((eq_ref b) (x6 x9)) as proof of (((eq b) (x6 x9)) (Xg x9))
% Found (fun (x9:iS)=> ((eq_ref b) (x6 x9))) as proof of (((eq b) (x6 x9)) (Xg x9))
% Found (fun (x9:iS)=> ((eq_ref b) (x6 x9))) as proof of (forall (x:iS), (((eq b) (x6 x)) (Xg x)))
% Found (functional_extensionality_dep0000 (fun (x9:iS)=> ((eq_ref b) (x6 x9)))) as proof of (((eq (iS->b)) x6) Xg)
% Found ((functional_extensionality_dep000 Xg) (fun (x9:iS)=> ((eq_ref b) (x6 x9)))) as proof of (((eq (iS->b)) x6) Xg)
% Found (((functional_extensionality_dep00 x6) Xg) (fun (x9:iS)=> ((eq_ref b) (x6 x9)))) as proof of (((eq (iS->b)) x6) Xg)
% Found ((((functional_extensionality_dep0 (fun (x11:iS)=> b)) x6) Xg) (fun (x9:iS)=> ((eq_ref b) (x6 x9)))) as proof of (((eq (iS->b)) x6) Xg)
% Found (((((functional_extensionality_dep iS) (fun (x11:iS)=> b)) x6) Xg) (fun (x9:iS)=> ((eq_ref b) (x6 x9)))) as proof of (((eq (iS->b)) x6) Xg)
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((((functional_extensionality_dep iS) (fun (x11:iS)=> b)) x6) Xg) (fun (x9:iS)=> ((eq_ref b) (x6 x9))))) as proof of (((eq (iS->b)) x6) Xg)
% Found eq_ref00:=(eq_ref0 (x0 x3)):(((eq b) (x0 x3)) (x0 x3))
% Found (eq_ref0 (x0 x3)) as proof of (((eq b) (x0 x3)) (Xg x3))
% Found ((eq_ref b) (x0 x3)) as proof of (((eq b) (x0 x3)) (Xg x3))
% Found ((eq_ref b) (x0 x3)) as proof of (((eq b) (x0 x3)) (Xg x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x0 x3))) as proof of (((eq b) (x0 x3)) (Xg x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x0 x3))) as proof of (forall (x:iS), (((eq b) (x0 x)) (Xg x)))
% Found ((functional_extensionality00000 (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found ((functional_extensionality00000 (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((functional_extensionality0000 x3) (fun (x5:(iS->b))=> ((P x0)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> (((functional_extensionality000 Xg) x3) (fun (x5:(iS->b))=> ((P x0)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((((functional_extensionality00 x0) Xg) x3) (fun (x5:(iS->b))=> ((P x0)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> (((((functional_extensionality0 b) x0) Xg) x3) (fun (x5:(iS->b))=> ((P x0)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((((((functional_extensionality iS) b) x0) Xg) x3) (fun (x5:(iS->b))=> ((P x0)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> (((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((((((functional_extensionality iS) b) x0) Xg) x3) (fun (x5:(iS->b))=> ((P x0)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) (((eq_ref (iS->b)) x0) P))) as proof of ((P x0)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> (((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((((((functional_extensionality iS) b) x0) Xg) x3) (fun (x5:(iS->b))=> ((P x0)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) (((eq_ref (iS->b)) x0) P))) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 (x0 x3)):(((eq b) (x0 x3)) (x0 x3))
% Found (eq_ref0 (x0 x3)) as proof of (((eq b) (x0 x3)) (Xg x3))
% Found ((eq_ref b) (x0 x3)) as proof of (((eq b) (x0 x3)) (Xg x3))
% Found ((eq_ref b) (x0 x3)) as proof of (((eq b) (x0 x3)) (Xg x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x0 x3))) as proof of (((eq b) (x0 x3)) (Xg x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x0 x3))) as proof of (forall (x:iS), (((eq b) (x0 x)) (Xg x)))
% Found (functional_extensionality00000 (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of ((P x0)->(P Xg))
% Found (functional_extensionality00000 (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of ((P x0)->(P Xg))
% Found ((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((functional_extensionality0000 x3) P)) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of ((P x0)->(P Xg))
% Found ((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> (((functional_extensionality000 Xg) x3) P)) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of ((P x0)->(P Xg))
% Found ((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((((functional_extensionality00 x0) Xg) x3) P)) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of ((P x0)->(P Xg))
% Found ((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> (((((functional_extensionality0 b) x0) Xg) x3) P)) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of ((P x0)->(P Xg))
% Found ((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((((((functional_extensionality iS) b) x0) Xg) x3) P)) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of ((P x0)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> ((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((((((functional_extensionality iS) b) x0) Xg) x3) P)) (fun (x3:iS)=> ((eq_ref b) (x0 x3))))) as proof of ((P x0)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> ((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((((((functional_extensionality iS) b) x0) Xg) x3) P)) (fun (x3:iS)=> ((eq_ref b) (x0 x3))))) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 (x0 x3)):(((eq b) (x0 x3)) (x0 x3))
% Found (eq_ref0 (x0 x3)) as proof of (((eq b) (x0 x3)) (Xg x3))
% Found ((eq_ref b) (x0 x3)) as proof of (((eq b) (x0 x3)) (Xg x3))
% Found ((eq_ref b) (x0 x3)) as proof of (((eq b) (x0 x3)) (Xg x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x0 x3))) as proof of (((eq b) (x0 x3)) (Xg x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x0 x3))) as proof of (forall (x:iS), (((eq b) (x0 x)) (Xg x)))
% Found (functional_extensionality_dep00000 (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of ((P x0)->(P Xg))
% Found (functional_extensionality_dep00000 (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of ((P x0)->(P Xg))
% Found ((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((functional_extensionality_dep0000 x3) P)) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of ((P x0)->(P Xg))
% Found ((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> (((functional_extensionality_dep000 Xg) x3) P)) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of ((P x0)->(P Xg))
% Found ((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((((functional_extensionality_dep00 x0) Xg) x3) P)) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of ((P x0)->(P Xg))
% Found ((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> (((((functional_extensionality_dep0 (fun (x5:iS)=> b)) x0) Xg) x3) P)) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of ((P x0)->(P Xg))
% Found ((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((((((functional_extensionality_dep iS) (fun (x5:iS)=> b)) x0) Xg) x3) P)) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of ((P x0)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> ((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((((((functional_extensionality_dep iS) (fun (x5:iS)=> b)) x0) Xg) x3) P)) (fun (x3:iS)=> ((eq_ref b) (x0 x3))))) as proof of ((P x0)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> ((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((((((functional_extensionality_dep iS) (fun (x5:iS)=> b)) x0) Xg) x3) P)) (fun (x3:iS)=> ((eq_ref b) (x0 x3))))) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 (x0 x3)):(((eq b) (x0 x3)) (x0 x3))
% Found (eq_ref0 (x0 x3)) as proof of (((eq b) (x0 x3)) (Xg x3))
% Found ((eq_ref b) (x0 x3)) as proof of (((eq b) (x0 x3)) (Xg x3))
% Found ((eq_ref b) (x0 x3)) as proof of (((eq b) (x0 x3)) (Xg x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x0 x3))) as proof of (((eq b) (x0 x3)) (Xg x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x0 x3))) as proof of (forall (x:iS), (((eq b) (x0 x)) (Xg x)))
% Found ((functional_extensionality_dep00000 (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found ((functional_extensionality_dep00000 (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((functional_extensionality_dep0000 x3) (fun (x5:(iS->b))=> ((P x0)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> (((functional_extensionality_dep000 Xg) x3) (fun (x5:(iS->b))=> ((P x0)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((((functional_extensionality_dep00 x0) Xg) x3) (fun (x5:(iS->b))=> ((P x0)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> (((((functional_extensionality_dep0 (fun (x5:iS)=> b)) x0) Xg) x3) (fun (x5:(iS->b))=> ((P x0)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((((((functional_extensionality_dep iS) (fun (x5:iS)=> b)) x0) Xg) x3) (fun (x5:(iS->b))=> ((P x0)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> (((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((((((functional_extensionality_dep iS) (fun (x5:iS)=> b)) x0) Xg) x3) (fun (x5:(iS->b))=> ((P x0)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) (((eq_ref (iS->b)) x0) P))) as proof of ((P x0)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> (((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (Xg x))))=> ((((((functional_extensionality_dep iS) (fun (x5:iS)=> b)) x0) Xg) x3) (fun (x5:(iS->b))=> ((P x0)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) (((eq_ref (iS->b)) x0) P))) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) ((cP2 (x0 Xx)) (x0 Xy)))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) ((cP2 (x0 Xx)) (x0 Xy)))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) ((cP2 (x0 Xx)) (x0 Xy)))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) ((cP2 (x0 Xx)) (x0 Xy)))
% Found eq_ref00:=(eq_ref0 (x0 ((cP Xx) Xy))):(((eq b) (x0 ((cP Xx) Xy))) (x0 ((cP Xx) Xy)))
% Found (eq_ref0 (x0 ((cP Xx) Xy))) as proof of (((eq b) (x0 ((cP Xx) Xy))) b0)
% Found ((eq_ref b) (x0 ((cP Xx) Xy))) as proof of (((eq b) (x0 ((cP Xx) Xy))) b0)
% Found ((eq_ref b) (x0 ((cP Xx) Xy))) as proof of (((eq b) (x0 ((cP Xx) Xy))) b0)
% Found ((eq_ref b) (x0 ((cP Xx) Xy))) as proof of (((eq b) (x0 ((cP Xx) Xy))) b0)
% Found eq_ref00:=(eq_ref0 (x0 c0)):(((eq b) (x0 c0)) (x0 c0))
% Found (eq_ref0 (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found eq_ref00:=(eq_ref0 (x2 c0)):(((eq b) (x2 c0)) (x2 c0))
% Found (eq_ref0 (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found ((eq_ref b) (x2 c0)) as proof of (((eq b) (x2 c0)) c02)
% Found eq_ref00:=(eq_ref0 x'):(((eq (iS->b)) x') x')
% Found (eq_ref0 x') as proof of (((eq (iS->b)) x') x0)
% Found ((eq_ref (iS->b)) x') as proof of (((eq (iS->b)) x') x0)
% Found ((eq_ref (iS->b)) x') as proof of (((eq (iS->b)) x') x0)
% Found ((eq_ref (iS->b)) x') as proof of (((eq (iS->b)) x') x0)
% Found (eq_sym000 ((eq_ref (iS->b)) x')) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P x')))
% Found ((eq_sym00 x0) ((eq_ref (iS->b)) x')) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P x')))
% Found (((eq_sym0 x') x0) ((eq_ref (iS->b)) x')) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P x')))
% Found ((((eq_sym (iS->b)) x') x0) ((eq_ref (iS->b)) x')) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P x')))
% Found ((((eq_sym (iS->b)) x') x0) ((eq_ref (iS->b)) x')) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P x')))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((((eq_sym (iS->b)) x') x0) ((eq_ref (iS->b)) x'))) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 (x4 c0)):(((eq b) (x4 c0)) (x4 c0))
% Found (eq_ref0 (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found ((eq_ref b) (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found ((eq_ref b) (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found ((eq_ref b) (x4 c0)) as proof of (((eq b) (x4 c0)) c02)
% Found eq_ref00:=(eq_ref0 (x6 c0)):(((eq b) (x6 c0)) (x6 c0))
% Found (eq_ref0 (x6 c0)) as proof of (((eq b) (x6 c0)) c02)
% Found ((eq_ref b) (x6 c0)) as proof of (((eq b) (x6 c0)) c02)
% Found ((eq_ref b) (x6 c0)) as proof of (((eq b) (x6 c0)) c02)
% Found ((eq_ref b) (x6 c0)) as proof of (((eq b) (x6 c0)) c02)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (iS->b)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (iS->b)) x0) P) as proof of (P0 x0)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (iS->b)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (iS->b)) x0) P) as proof of (P0 x0)
% Found eq_ref00:=(eq_ref0 (x0 c0)):(((eq b) (x0 c0)) (x0 c0))
% Found (eq_ref0 (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found ((eq_ref b) (x0 c0)) as proof of (((eq b) (x0 c0)) c02)
% Found eq_ref000:=(eq_ref00 P):((P (x0 ((cP Xx) Xy)))->(P (x0 ((cP Xx) Xy))))
% Found (eq_ref00 P) as proof of (P0 (x0 ((cP Xx) Xy)))
% Found ((eq_ref0 (x0 ((cP Xx) Xy))) P) as proof of (P0 (x0 ((cP Xx) Xy)))
% Found (((eq_ref b) (x0 ((cP Xx) Xy))) P) as proof of (P0 (x0 ((cP Xx) Xy)))
% Found (((eq_ref b) (x0 ((cP Xx) Xy))) P) as proof of (P0 (x0 ((cP Xx) Xy)))
% Found eq_ref00:=(eq_ref0 (x0 ((cP Xx) Xy))):(((eq b) (x0 ((cP Xx) Xy))) (x0 ((cP Xx) Xy)))
% Found (eq_ref0 (x0 ((cP Xx) Xy))) as proof of (((eq b) (x0 ((cP Xx) Xy))) b0)
% Found ((eq_ref b) (x0 ((cP Xx) Xy))) as proof of (((eq b) (x0 ((cP Xx) Xy))) b0)
% Found ((eq_ref b) (x0 ((cP Xx) Xy))) as proof of (((eq b) (x0 ((cP Xx) Xy))) b0)
% Found ((eq_ref b) (x0 ((cP Xx) Xy))) as proof of (((eq b) (x0 ((cP Xx) Xy))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) ((cP2 (x0 Xx)) (x0 Xy)))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) ((cP2 (x0 Xx)) (x0 Xy)))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) ((cP2 (x0 Xx)) (x0 Xy)))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) ((cP2 (x0 Xx)) (x0 Xy)))
% Found eq_ref00:=(eq_ref0 (x2 x3)):(((eq b) (x2 x3)) (x2 x3))
% Found (eq_ref0 (x2 x3)) as proof of (((eq b) (x2 x3)) (x' x3))
% Found ((eq_ref b) (x2 x3)) as proof of (((eq b) (x2 x3)) (x' x3))
% Found ((eq_ref b) (x2 x3)) as proof of (((eq b) (x2 x3)) (x' x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x2 x3))) as proof of (((eq b) (x2 x3)) (x' x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x2 x3))) as proof of (forall (x:iS), (((eq b) (x2 x)) (x' x)))
% Found (functional_extensionality00000 (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of ((P x2)->(P x'))
% Found (functional_extensionality00000 (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of ((P x2)->(P x'))
% Found ((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (x' x))))=> ((functional_extensionality0000 x3) P)) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of ((P x2)->(P x'))
% Found ((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (x' x))))=> (((functional_extensionality000 x') x3) P)) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of ((P x2)->(P x'))
% Found ((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (x' x))))=> ((((functional_extensionality00 x2) x') x3) P)) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of ((P x2)->(P x'))
% Found ((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (x' x))))=> (((((functional_extensionality0 b) x2) x') x3) P)) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of ((P x2)->(P x'))
% Found ((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (x' x))))=> ((((((functional_extensionality iS) b) x2) x') x3) P)) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of ((P x2)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> ((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (x' x))))=> ((((((functional_extensionality iS) b) x2) x') x3) P)) (fun (x3:iS)=> ((eq_ref b) (x2 x3))))) as proof of ((P x2)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop))=> ((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (x' x))))=> ((((((functional_extensionality iS) b) x2) x') x3) P)) (fun (x3:iS)=> ((eq_ref b) (x2 x3))))) as proof of (((eq (iS->b)) x2) x')
% Found eq_ref00:=(eq_ref0 (x2 x3)):(((eq b) (x2 x3)) (x2 x3))
% Found (eq_ref0 (x2 x3)) as proof of (((eq b) (x2 x3)) (x' x3))
% Found ((eq_ref b) (x2 x3)) as proof of (((eq b) (x2 x3)) (x' x3))
% Found ((eq_ref b) (x2 x3)) as proof of (((eq b) (x2 x3)) (x' x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x2 x3))) as proof of (((eq b) (x2 x3)) (x' x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x2 x3))) as proof of (forall (x:iS), (((eq b) (x2 x)) (x' x)))
% Found ((functional_extensionality00000 (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P x'))
% Found ((functional_extensionality00000 (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P x'))
% Found (((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (x' x))))=> ((functional_extensionality0000 x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P x'))
% Found (((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (x' x))))=> (((functional_extensionality000 x') x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P x'))
% Found (((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (x' x))))=> ((((functional_extensionality00 x2) x') x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P x'))
% Found (((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (x' x))))=> (((((functional_extensionality0 b) x2) x') x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P x'))
% Found (((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (x' x))))=> ((((((functional_extensionality iS) b) x2) x') x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> (((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (x' x))))=> ((((((functional_extensionality iS) b) x2) x') x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) (((eq_ref (iS->b)) x2) P))) as proof of ((P x2)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop))=> (((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (x' x))))=> ((((((functional_extensionality iS) b) x2) x') x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) (((eq_ref (iS->b)) x2) P))) as proof of (((eq (iS->b)) x2) x')
% Found eq_ref00:=(eq_ref0 (x2 x3)):(((eq b) (x2 x3)) (x2 x3))
% Found (eq_ref0 (x2 x3)) as proof of (((eq b) (x2 x3)) (x' x3))
% Found ((eq_ref b) (x2 x3)) as proof of (((eq b) (x2 x3)) (x' x3))
% Found ((eq_ref b) (x2 x3)) as proof of (((eq b) (x2 x3)) (x' x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x2 x3))) as proof of (((eq b) (x2 x3)) (x' x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x2 x3))) as proof of (forall (x:iS), (((eq b) (x2 x)) (x' x)))
% Found ((functional_extensionality_dep00000 (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P x'))
% Found ((functional_extensionality_dep00000 (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P x'))
% Found (((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (x' x))))=> ((functional_extensionality_dep0000 x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P x'))
% Found (((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (x' x))))=> (((functional_extensionality_dep000 x') x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P x'))
% Found (((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (x' x))))=> ((((functional_extensionality_dep00 x2) x') x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P x'))
% Found (((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (x' x))))=> (((((functional_extensionality_dep0 (fun (x5:iS)=> b)) x2) x') x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P x'))
% Found (((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (x' x))))=> ((((((functional_extensionality_dep iS) (fun (x5:iS)=> b)) x2) x') x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> (((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (x' x))))=> ((((((functional_extensionality_dep iS) (fun (x5:iS)=> b)) x2) x') x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) (((eq_ref (iS->b)) x2) P))) as proof of ((P x2)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop))=> (((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (x' x))))=> ((((((functional_extensionality_dep iS) (fun (x5:iS)=> b)) x2) x') x3) (fun (x5:(iS->b))=> ((P x2)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) (((eq_ref (iS->b)) x2) P))) as proof of (((eq (iS->b)) x2) x')
% Found eq_ref00:=(eq_ref0 (x2 x3)):(((eq b) (x2 x3)) (x2 x3))
% Found (eq_ref0 (x2 x3)) as proof of (((eq b) (x2 x3)) (x' x3))
% Found ((eq_ref b) (x2 x3)) as proof of (((eq b) (x2 x3)) (x' x3))
% Found ((eq_ref b) (x2 x3)) as proof of (((eq b) (x2 x3)) (x' x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x2 x3))) as proof of (((eq b) (x2 x3)) (x' x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x2 x3))) as proof of (forall (x:iS), (((eq b) (x2 x)) (x' x)))
% Found (functional_extensionality_dep00000 (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of ((P x2)->(P x'))
% Found (functional_extensionality_dep00000 (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of ((P x2)->(P x'))
% Found ((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (x' x))))=> ((functional_extensionality_dep0000 x3) P)) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of ((P x2)->(P x'))
% Found ((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (x' x))))=> (((functional_extensionality_dep000 x') x3) P)) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of ((P x2)->(P x'))
% Found ((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (x' x))))=> ((((functional_extensionality_dep00 x2) x') x3) P)) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of ((P x2)->(P x'))
% Found ((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (x' x))))=> (((((functional_extensionality_dep0 (fun (x5:iS)=> b)) x2) x') x3) P)) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of ((P x2)->(P x'))
% Found ((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (x' x))))=> ((((((functional_extensionality_dep iS) (fun (x5:iS)=> b)) x2) x') x3) P)) (fun (x3:iS)=> ((eq_ref b) (x2 x3)))) as proof of ((P x2)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> ((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (x' x))))=> ((((((functional_extensionality_dep iS) (fun (x5:iS)=> b)) x2) x') x3) P)) (fun (x3:iS)=> ((eq_ref b) (x2 x3))))) as proof of ((P x2)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop))=> ((fun (x3:(forall (x:iS), (((eq b) (x2 x)) (x' x))))=> ((((((functional_extensionality_dep iS) (fun (x5:iS)=> b)) x2) x') x3) P)) (fun (x3:iS)=> ((eq_ref b) (x2 x3))))) as proof of (((eq (iS->b)) x2) x')
% Found x5:(P x0)
% Instantiate: x0:=Xg:(iS->b)
% Found (fun (x5:(P x0))=> x5) as proof of (P Xg)
% Found (fun (P:((iS->b)->Prop)) (x5:(P x0))=> x5) as proof of ((P x0)->(P Xg))
% Found (fun (P:((iS->b)->Prop)) (x5:(P x0))=> x5) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P Xg)))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop)) (x5:(P x0))=> x5) as proof of (((eq (iS->b)) 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 (iS->b)) x0) P) as proof of ((P x0)->(P Xg))
% Found (((eq_ref (iS->b)) x0) P) as proof of ((P x0)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x0) P)) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P Xg)))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x0) P)) as proof of (((eq (iS->b)) x0) Xg)
% Found x5:(P x2)
% Instantiate: x2:=Xg:(iS->b)
% Found (fun (x5:(P x2))=> x5) as proof of (P Xg)
% Found (fun (P:((iS->b)->Prop)) (x5:(P x2))=> x5) as proof of ((P x2)->(P Xg))
% Found (fun (P:((iS->b)->Prop)) (x5:(P x2))=> x5) as proof of (forall (P:((iS->b)->Prop)), ((P x2)->(P Xg)))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop)) (x5:(P x2))=> x5) as proof of (((eq (iS->b)) 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 (iS->b)) x2) P) as proof of ((P x2)->(P Xg))
% Found (((eq_ref (iS->b)) x2) P) as proof of ((P x2)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x2) P)) as proof of ((P x2)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x2) P)) as proof of (forall (P:((iS->b)->Prop)), ((P x2)->(P Xg)))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x2) P)) as proof of (((eq (iS->b)) x2) Xg)
% Found eq_ref000:=(eq_ref00 P):((P (x0 ((cP Xx) Xy)))->(P (x0 ((cP Xx) Xy))))
% Found (eq_ref00 P) as proof of (P0 (x0 ((cP Xx) Xy)))
% Found ((eq_ref0 (x0 ((cP Xx) Xy))) P) as proof of (P0 (x0 ((cP Xx) Xy)))
% Found (((eq_ref b) (x0 ((cP Xx) Xy))) P) as proof of (P0 (x0 ((cP Xx) Xy)))
% Found (((eq_ref b) (x0 ((cP Xx) Xy))) P) as proof of (P0 (x0 ((cP Xx) Xy)))
% Found eq_ref00:=(eq_ref0 x0):(((eq (iS->b)) x0) x0)
% Found (eq_ref0 x0) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P x')))
% Found ((eq_ref (iS->b)) x0) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P x')))
% Found ((eq_ref (iS->b)) x0) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P x')))
% Found ((eq_ref (iS->b)) x0) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P x')))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((eq_ref (iS->b)) x0)) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 x2):(((eq (iS->b)) x2) x2)
% Found (eq_ref0 x2) as proof of (forall (P:((iS->b)->Prop)), ((P x2)->(P x')))
% Found ((eq_ref (iS->b)) x2) as proof of (forall (P:((iS->b)->Prop)), ((P x2)->(P x')))
% Found ((eq_ref (iS->b)) x2) as proof of (forall (P:((iS->b)->Prop)), ((P x2)->(P x')))
% Found ((eq_ref (iS->b)) x2) as proof of (forall (P:((iS->b)->Prop)), ((P x2)->(P x')))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> ((eq_ref (iS->b)) x2)) as proof of (((eq (iS->b)) x2) x')
% Found eq_ref00:=(eq_ref0 (x8 x9)):(((eq b) (x8 x9)) (x8 x9))
% Found (eq_ref0 (x8 x9)) as proof of (((eq b) (x8 x9)) (x' x9))
% Found ((eq_ref b) (x8 x9)) as proof of (((eq b) (x8 x9)) (x' x9))
% Found ((eq_ref b) (x8 x9)) as proof of (((eq b) (x8 x9)) (x' x9))
% Found (fun (x9:iS)=> ((eq_ref b) (x8 x9))) as proof of (((eq b) (x8 x9)) (x' x9))
% Found (fun (x9:iS)=> ((eq_ref b) (x8 x9))) as proof of (forall (x:iS), (((eq b) (x8 x)) (x' x)))
% Found (functional_extensionality0000 (fun (x9:iS)=> ((eq_ref b) (x8 x9)))) as proof of (((eq (iS->b)) x8) x')
% Found ((functional_extensionality000 x') (fun (x9:iS)=> ((eq_ref b) (x8 x9)))) as proof of (((eq (iS->b)) x8) x')
% Found (((functional_extensionality00 x8) x') (fun (x9:iS)=> ((eq_ref b) (x8 x9)))) as proof of (((eq (iS->b)) x8) x')
% Found ((((functional_extensionality0 b) x8) x') (fun (x9:iS)=> ((eq_ref b) (x8 x9)))) as proof of (((eq (iS->b)) x8) x')
% Found (((((functional_extensionality iS) b) x8) x') (fun (x9:iS)=> ((eq_ref b) (x8 x9)))) as proof of (((eq (iS->b)) x8) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((((functional_extensionality iS) b) x8) x') (fun (x9:iS)=> ((eq_ref b) (x8 x9))))) as proof of (((eq (iS->b)) x8) x')
% Found eq_ref00:=(eq_ref0 (x8 x9)):(((eq b) (x8 x9)) (x8 x9))
% Found (eq_ref0 (x8 x9)) as proof of (((eq b) (x8 x9)) (x' x9))
% Found ((eq_ref b) (x8 x9)) as proof of (((eq b) (x8 x9)) (x' x9))
% Found ((eq_ref b) (x8 x9)) as proof of (((eq b) (x8 x9)) (x' x9))
% Found (fun (x9:iS)=> ((eq_ref b) (x8 x9))) as proof of (((eq b) (x8 x9)) (x' x9))
% Found (fun (x9:iS)=> ((eq_ref b) (x8 x9))) as proof of (forall (x:iS), (((eq b) (x8 x)) (x' x)))
% Found (functional_extensionality_dep0000 (fun (x9:iS)=> ((eq_ref b) (x8 x9)))) as proof of (((eq (iS->b)) x8) x')
% Found ((functional_extensionality_dep000 x') (fun (x9:iS)=> ((eq_ref b) (x8 x9)))) as proof of (((eq (iS->b)) x8) x')
% Found (((functional_extensionality_dep00 x8) x') (fun (x9:iS)=> ((eq_ref b) (x8 x9)))) as proof of (((eq (iS->b)) x8) x')
% Found ((((functional_extensionality_dep0 (fun (x11:iS)=> b)) x8) x') (fun (x9:iS)=> ((eq_ref b) (x8 x9)))) as proof of (((eq (iS->b)) x8) x')
% Found (((((functional_extensionality_dep iS) (fun (x11:iS)=> b)) x8) x') (fun (x9:iS)=> ((eq_ref b) (x8 x9)))) as proof of (((eq (iS->b)) x8) x')
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy))))))=> (((((functional_extensionality_dep iS) (fun (x11:iS)=> b)) x8) x') (fun (x9:iS)=> ((eq_ref b) (x8 x9))))) as proof of (((eq (iS->b)) x8) x')
% Found eq_ref000:=(eq_ref00 P):((P x4)->(P x4))
% Found (eq_ref00 P) as proof of (P0 x4)
% Found ((eq_ref0 x4) P) as proof of (P0 x4)
% Found (((eq_ref (iS->b)) x4) P) as proof of (P0 x4)
% Found (((eq_ref (iS->b)) x4) P) as proof of (P0 x4)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq b) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq b) (x0 x1)) (Xg x1))
% Found ((eq_ref b) (x0 x1)) as proof of (((eq b) (x0 x1)) (Xg x1))
% Found ((eq_ref b) (x0 x1)) as proof of (((eq b) (x0 x1)) (Xg x1))
% Found (fun (x1:iS)=> ((eq_ref b) (x0 x1))) as proof of (((eq b) (x0 x1)) (Xg x1))
% Found (fun (x1:iS)=> ((eq_ref b) (x0 x1))) as proof of (forall (x:iS), (((eq b) (x0 x)) (Xg x)))
% Found (functional_extensionality0000 (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P Xg)))
% Found ((functional_extensionality000 Xg) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P Xg)))
% Found (((functional_extensionality00 x0) Xg) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P Xg)))
% Found ((((functional_extensionality0 b) x0) Xg) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P Xg)))
% Found (((((functional_extensionality iS) b) x0) Xg) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P Xg)))
% Found (((((functional_extensionality iS) b) x0) Xg) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P Xg)))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((((functional_extensionality iS) b) x0) Xg) (fun (x1:iS)=> ((eq_ref b) (x0 x1))))) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq b) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq b) (x0 x1)) (Xg x1))
% Found ((eq_ref b) (x0 x1)) as proof of (((eq b) (x0 x1)) (Xg x1))
% Found ((eq_ref b) (x0 x1)) as proof of (((eq b) (x0 x1)) (Xg x1))
% Found (fun (x1:iS)=> ((eq_ref b) (x0 x1))) as proof of (((eq b) (x0 x1)) (Xg x1))
% Found (fun (x1:iS)=> ((eq_ref b) (x0 x1))) as proof of (forall (x:iS), (((eq b) (x0 x)) (Xg x)))
% Found (functional_extensionality_dep0000 (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P Xg)))
% Found ((functional_extensionality_dep000 Xg) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P Xg)))
% Found (((functional_extensionality_dep00 x0) Xg) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P Xg)))
% Found ((((functional_extensionality_dep0 (fun (x3:iS)=> b)) x0) Xg) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P Xg)))
% Found (((((functional_extensionality_dep iS) (fun (x3:iS)=> b)) x0) Xg) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P Xg)))
% Found (((((functional_extensionality_dep iS) (fun (x3:iS)=> b)) x0) Xg) (fun (x1:iS)=> ((eq_ref b) (x0 x1)))) as proof of (forall (P:((iS->b)->Prop)), ((P x0)->(P Xg)))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy))))))=> (((((functional_extensionality_dep iS) (fun (x3:iS)=> b)) x0) Xg) (fun (x1:iS)=> ((eq_ref b) (x0 x1))))) as proof of (((eq (iS->b)) x0) Xg)
% Found eq_ref000:=(eq_ref00 P):((P x4)->(P x4))
% Found (eq_ref00 P) as proof of ((P x4)->(P x'))
% Found ((eq_ref0 x4) P) as proof of ((P x4)->(P x'))
% Found (((eq_ref (iS->b)) x4) P) as proof of ((P x4)->(P x'))
% Found (((eq_ref (iS->b)) x4) P) as proof of ((P x4)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x4) P)) as proof of ((P x4)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x4) P)) as proof of (forall (P:((iS->b)->Prop)), ((P x4)->(P x')))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop))=> (((eq_ref (iS->b)) x4) P)) as proof of (((eq (iS->b)) x4) x')
% Found x5:(P x4)
% Instantiate: x4:=x':(iS->b)
% Found (fun (x5:(P x4))=> x5) as proof of (P x')
% Found (fun (P:((iS->b)->Prop)) (x5:(P x4))=> x5) as proof of ((P x4)->(P x'))
% Found (fun (P:((iS->b)->Prop)) (x5:(P x4))=> x5) as proof of (forall (P:((iS->b)->Prop)), ((P x4)->(P x')))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop)) (x5:(P x4))=> x5) as proof of (((eq (iS->b)) x4) x')
% Found eq_ref00:=(eq_ref0 Xg):(((eq (iS->b)) Xg) Xg)
% Found (eq_ref0 Xg) as proof of (((eq (iS->b)) Xg) x4)
% Found ((eq_ref (iS->b)) Xg) as proof of (((eq (iS->b)) Xg) x4)
% Found ((eq_ref (iS->b)) Xg) as proof of (((eq (iS->b)) Xg) x4)
% Found ((eq_ref (iS->b)) Xg) as proof of (((eq (iS->b)) Xg) x4)
% Found ((eq_sym0000 ((eq_ref (iS->b)) Xg)) (((eq_ref (iS->b)) x4) P)) as proof of ((P x4)->(P Xg))
% Found ((eq_sym0000 ((eq_ref (iS->b)) Xg)) (((eq_ref (iS->b)) x4) P)) as proof of ((P x4)->(P Xg))
% Found (((fun (x5:(((eq (iS->b)) Xg) x4))=> ((eq_sym000 x5) (fun (x7:(iS->b))=> ((P x4)->(P x7))))) ((eq_ref (iS->b)) Xg)) (((eq_ref (iS->b)) x4) P)) as proof of ((P x4)->(P Xg))
% Found (((fun (x5:(((eq (iS->b)) Xg) x4))=> (((eq_sym00 x4) x5) (fun (x7:(iS->b))=> ((P x4)->(P x7))))) ((eq_ref (iS->b)) Xg)) (((eq_ref (iS->b)) x4) P)) as proof of ((P x4)->(P Xg))
% Found (((fun (x5:(((eq (iS->b)) Xg) x4))=> ((((eq_sym0 Xg) x4) x5) (fun (x7:(iS->b))=> ((P x4)->(P x7))))) ((eq_ref (iS->b)) Xg)) (((eq_ref (iS->b)) x4) P)) as proof of ((P x4)->(P Xg))
% Found (((fun (x5:(((eq (iS->b)) Xg) x4))=> (((((eq_sym (iS->b)) Xg) x4) x5) (fun (x7:(iS->b))=> ((P x4)->(P x7))))) ((eq_ref (iS->b)) Xg)) (((eq_ref (iS->b)) x4) P)) as proof of ((P x4)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> (((fun (x5:(((eq (iS->b)) Xg) x4))=> (((((eq_sym (iS->b)) Xg) x4) x5) (fun (x7:(iS->b))=> ((P x4)->(P x7))))) ((eq_ref (iS->b)) Xg)) (((eq_ref (iS->b)) x4) P))) as proof of ((P x4)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> (((fun (x5:(((eq (iS->b)) Xg) x4))=> (((((eq_sym (iS->b)) Xg) x4) x5) (fun (x7:(iS->b))=> ((P x4)->(P x7))))) ((eq_ref (iS->b)) Xg)) (((eq_ref (iS->b)) x4) P))) as proof of (((eq (iS->b)) x4) Xg)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xg):(((eq (iS->b)) Xg) (fun (x:iS)=> (Xg x)))
% Found (eta_expansion_dep00 Xg) as proof of (((eq (iS->b)) Xg) x4)
% Found ((eta_expansion_dep0 (fun (x6:iS)=> b)) Xg) as proof of (((eq (iS->b)) Xg) x4)
% Found (((eta_expansion_dep iS) (fun (x6:iS)=> b)) Xg) as proof of (((eq (iS->b)) Xg) x4)
% Found (((eta_expansion_dep iS) (fun (x6:iS)=> b)) Xg) as proof of (((eq (iS->b)) Xg) x4)
% Found (((eta_expansion_dep iS) (fun (x6:iS)=> b)) Xg) as proof of (((eq (iS->b)) Xg) x4)
% Found (eq_sym0000 (((eta_expansion_dep iS) (fun (x6:iS)=> b)) Xg)) as proof of ((P x4)->(P Xg))
% Found (eq_sym0000 (((eta_expansion_dep iS) (fun (x6:iS)=> b)) Xg)) as proof of ((P x4)->(P Xg))
% Found ((fun (x5:(((eq (iS->b)) Xg) x4))=> ((eq_sym000 x5) P)) (((eta_expansion_dep iS) (fun (x6:iS)=> b)) Xg)) as proof of ((P x4)->(P Xg))
% Found ((fun (x5:(((eq (iS->b)) Xg) x4))=> (((eq_sym00 x4) x5) P)) (((eta_expansion_dep iS) (fun (x6:iS)=> b)) Xg)) as proof of ((P x4)->(P Xg))
% Found ((fun (x5:(((eq (iS->b)) Xg) x4))=> ((((eq_sym0 Xg) x4) x5) P)) (((eta_expansion_dep iS) (fun (x6:iS)=> b)) Xg)) as proof of ((P x4)->(P Xg))
% Found ((fun (x5:(((eq (iS->b)) Xg) x4))=> (((((eq_sym (iS->b)) Xg) x4) x5) P)) (((eta_expansion_dep iS) (fun (x6:iS)=> b)) Xg)) as proof of ((P x4)->(P Xg))
% Found (fun (P:((iS->b)->Prop))=> ((fun (x5:(((eq (iS->b)) Xg) x4))=> (((((eq_sym (iS->b)) Xg) x4) x5) P)) (((eta_expansion_dep iS) (fun (x6:iS)=> b)) Xg))) as proof of ((P x4)->(P Xg))
% Found (fun (x00:((and (((eq b) (Xg c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (Xg ((cP Xx) Xy))) ((cP2 (Xg Xx)) (Xg Xy)))))) (P:((iS->b)->Prop))=> ((fun (x5:(((eq (iS->b)) Xg) x4))=> (((((eq_sym (iS->b)) Xg) x4) x5) P)) (((eta_expansion_dep iS) (fun (x6:iS)=> b)) Xg))) as proof of (((eq (iS->b)) x4) Xg)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) ((cP2 (x0 Xx)) (x0 Xy)))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) ((cP2 (x0 Xx)) (x0 Xy)))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) ((cP2 (x0 Xx)) (x0 Xy)))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) ((cP2 (x0 Xx)) (x0 Xy)))
% Found eq_ref00:=(eq_ref0 (x0 ((cP Xx) Xy))):(((eq b) (x0 ((cP Xx) Xy))) (x0 ((cP Xx) Xy)))
% Found (eq_ref0 (x0 ((cP Xx) Xy))) as proof of (((eq b) (x0 ((cP Xx) Xy))) b0)
% Found ((eq_ref b) (x0 ((cP Xx) Xy))) as proof of (((eq b) (x0 ((cP Xx) Xy))) b0)
% Found ((eq_ref b) (x0 ((cP Xx) Xy))) as proof of (((eq b) (x0 ((cP Xx) Xy))) b0)
% Found ((eq_ref b) (x0 ((cP Xx) Xy))) as proof of (((eq b) (x0 ((cP Xx) Xy))) b0)
% Found eq_ref00:=(eq_ref0 (x0 x3)):(((eq b) (x0 x3)) (x0 x3))
% Found (eq_ref0 (x0 x3)) as proof of (((eq b) (x0 x3)) (x' x3))
% Found ((eq_ref b) (x0 x3)) as proof of (((eq b) (x0 x3)) (x' x3))
% Found ((eq_ref b) (x0 x3)) as proof of (((eq b) (x0 x3)) (x' x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x0 x3))) as proof of (((eq b) (x0 x3)) (x' x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x0 x3))) as proof of (forall (x:iS), (((eq b) (x0 x)) (x' x)))
% Found ((functional_extensionality00000 (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found ((functional_extensionality00000 (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found (((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> ((functional_extensionality0000 x3) (fun (x5:(iS->b))=> ((P x0)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found (((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> (((functional_extensionality000 x') x3) (fun (x5:(iS->b))=> ((P x0)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found (((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> ((((functional_extensionality00 x0) x') x3) (fun (x5:(iS->b))=> ((P x0)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found (((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> (((((functional_extensionality0 b) x0) x') x3) (fun (x5:(iS->b))=> ((P x0)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found (((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> ((((((functional_extensionality iS) b) x0) x') x3) (fun (x5:(iS->b))=> ((P x0)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> (((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> ((((((functional_extensionality iS) b) x0) x') x3) (fun (x5:(iS->b))=> ((P x0)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) (((eq_ref (iS->b)) x0) P))) as proof of ((P x0)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop))=> (((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> ((((((functional_extensionality iS) b) x0) x') x3) (fun (x5:(iS->b))=> ((P x0)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) (((eq_ref (iS->b)) x0) P))) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 (x0 x3)):(((eq b) (x0 x3)) (x0 x3))
% Found (eq_ref0 (x0 x3)) as proof of (((eq b) (x0 x3)) (x' x3))
% Found ((eq_ref b) (x0 x3)) as proof of (((eq b) (x0 x3)) (x' x3))
% Found ((eq_ref b) (x0 x3)) as proof of (((eq b) (x0 x3)) (x' x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x0 x3))) as proof of (((eq b) (x0 x3)) (x' x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x0 x3))) as proof of (forall (x:iS), (((eq b) (x0 x)) (x' x)))
% Found (functional_extensionality00000 (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of ((P x0)->(P x'))
% Found (functional_extensionality00000 (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of ((P x0)->(P x'))
% Found ((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> ((functional_extensionality0000 x3) P)) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of ((P x0)->(P x'))
% Found ((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> (((functional_extensionality000 x') x3) P)) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of ((P x0)->(P x'))
% Found ((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> ((((functional_extensionality00 x0) x') x3) P)) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of ((P x0)->(P x'))
% Found ((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> (((((functional_extensionality0 b) x0) x') x3) P)) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of ((P x0)->(P x'))
% Found ((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> ((((((functional_extensionality iS) b) x0) x') x3) P)) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) as proof of ((P x0)->(P x'))
% Found (fun (P:((iS->b)->Prop))=> ((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> ((((((functional_extensionality iS) b) x0) x') x3) P)) (fun (x3:iS)=> ((eq_ref b) (x0 x3))))) as proof of ((P x0)->(P x'))
% Found (fun (x00:((and (((eq b) (x' c0)) c02)) (forall (Xx:iS) (Xy:iS), (((eq b) (x' ((cP Xx) Xy))) ((cP2 (x' Xx)) (x' Xy)))))) (P:((iS->b)->Prop))=> ((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> ((((((functional_extensionality iS) b) x0) x') x3) P)) (fun (x3:iS)=> ((eq_ref b) (x0 x3))))) as proof of (((eq (iS->b)) x0) x')
% Found eq_ref00:=(eq_ref0 (x0 x3)):(((eq b) (x0 x3)) (x0 x3))
% Found (eq_ref0 (x0 x3)) as proof of (((eq b) (x0 x3)) (x' x3))
% Found ((eq_ref b) (x0 x3)) as proof of (((eq b) (x0 x3)) (x' x3))
% Found ((eq_ref b) (x0 x3)) as proof of (((eq b) (x0 x3)) (x' x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x0 x3))) as proof of (((eq b) (x0 x3)) (x' x3))
% Found (fun (x3:iS)=> ((eq_ref b) (x0 x3))) as proof of (forall (x:iS), (((eq b) (x0 x)) (x' x)))
% Found ((functional_extensionality_dep00000 (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found ((functional_extensionality_dep00000 (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found (((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> ((functional_extensionality_dep0000 x3) (fun (x5:(iS->b))=> ((P x0)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found (((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> (((functional_extensionality_dep000 x') x3) (fun (x5:(iS->b))=> ((P x0)->(P x5))))) (fun (x3:iS)=> ((eq_ref b) (x0 x3)))) (((eq_ref (iS->b)) x0) P)) as proof of ((P x0)->(P x'))
% Found (((fun (x3:(forall (x:iS), (((eq b) (x0 x)) (x' x))))=> ((((fu
% EOF
%------------------------------------------------------------------------------