%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV204^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n091.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 300.08s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV204^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n091.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 08:28:16 CDT 2014
% % CPUTime  : 300.08 
% 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 0x1186950>, <kernel.Type object at 0x11863b0>) of role type named iS_type
% Using role type
% Declaring iS:Type
% FOF formula (<kernel.Constant object at 0x1368878>, <kernel.Constant object at 0x1186ef0>) of role type named y
% Using role type
% Declaring y:iS
% FOF formula (<kernel.Constant object at 0x11868c0>, <kernel.Constant object at 0x1186ef0>) of role type named x
% Using role type
% Declaring x:iS
% FOF formula (<kernel.Constant object at 0x1186950>, <kernel.DependentProduct object at 0x1186b90>) of role type named cP
% Using role type
% Declaring cP:(iS->(iS->iS))
% FOF formula (<kernel.Constant object at 0x1186a70>, <kernel.Constant object at 0x1186b90>) of role type named c0
% Using role type
% Declaring c0:iS
% FOF formula (((and ((and (forall (Xx0:iS) (Xy0:iS), (not (((eq iS) ((cP Xx0) Xy0)) c0)))) (forall (Xx0:iS) (Xy0:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx0) Xu)) ((cP Xy0) Xv))->((and (((eq iS) Xx0) Xy0)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx0:iS) (Xy0:iS), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(forall (Xx0:iS), (X Xx0)))))->(forall (R:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb) Xc))) ((and (((eq iS) Xb) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2)))))))))))))))->(((R Xa) Xb) Xc))))->(((R ((cP x) c0)) ((cP x) y)) ((cP x) y))))) of role conjecture named cS_INCL_LEM4_pme
% Conjecture to prove = (((and ((and (forall (Xx0:iS) (Xy0:iS), (not (((eq iS) ((cP Xx0) Xy0)) c0)))) (forall (Xx0:iS) (Xy0:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx0) Xu)) ((cP Xy0) Xv))->((and (((eq iS) Xx0) Xy0)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx0:iS) (Xy0:iS), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(forall (Xx0:iS), (X Xx0)))))->(forall (R:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb) Xc))) ((and (((eq iS) Xb) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2)))))))))))))))->(((R Xa) Xb) Xc))))->(((R ((cP x) c0)) ((cP x) y)) ((cP x) y))))):Prop
% We need to prove ['(((and ((and (forall (Xx0:iS) (Xy0:iS), (not (((eq iS) ((cP Xx0) Xy0)) c0)))) (forall (Xx0:iS) (Xy0:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx0) Xu)) ((cP Xy0) Xv))->((and (((eq iS) Xx0) Xy0)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx0:iS) (Xy0:iS), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(forall (Xx0:iS), (X Xx0)))))->(forall (R:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb) Xc))) ((and (((eq iS) Xb) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2)))))))))))))))->(((R Xa) Xb) Xc))))->(((R ((cP x) c0)) ((cP x) y)) ((cP x) y)))))']
% Parameter iS:Type.
% Parameter y:iS.
% Parameter x:iS.
% Parameter cP:(iS->(iS->iS)).
% Parameter c0:iS.
% Trying to prove (((and ((and (forall (Xx0:iS) (Xy0:iS), (not (((eq iS) ((cP Xx0) Xy0)) c0)))) (forall (Xx0:iS) (Xy0:iS) (Xu:iS) (Xv:iS), ((((eq iS) ((cP Xx0) Xu)) ((cP Xy0) Xv))->((and (((eq iS) Xx0) Xy0)) (((eq iS) Xu) Xv)))))) (forall (X:(iS->Prop)), (((and (X c0)) (forall (Xx0:iS) (Xy0:iS), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(forall (Xx0:iS), (X Xx0)))))->(forall (R:(iS->(iS->(iS->Prop)))), (((and True) (forall (Xa:iS) (Xb:iS) (Xc:iS), (((or ((or ((and (((eq iS) Xa) c0)) (((eq iS) Xb) Xc))) ((and (((eq iS) Xb) c0)) (((eq iS) Xa) Xc)))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) Xa) ((cP Xx1) Xx2))) (((eq iS) Xb) ((cP Xy1) Xy2)))) (((eq iS) Xc) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2)))))))))))))))->(((R Xa) Xb) Xc))))->(((R ((cP x) c0)) ((cP x) y)) ((cP x) y)))))
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 a):(((eq iS) a) a)
% Found (eq_ref0 a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found eq_ref00:=(eq_ref0 a):(((eq iS) a) a)
% Found (eq_ref0 a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found eq_ref00:=(eq_ref0 a):(((eq iS) a) a)
% Found (eq_ref0 a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found eq_ref00:=(eq_ref0 a):(((eq iS) a) a)
% Found (eq_ref0 a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))):(((eq Prop) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2)))))))))))))))
% Found (eq_ref0 ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) as proof of (((eq Prop) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) b)
% Found ((eq_ref Prop) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) as proof of (((eq Prop) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) b)
% Found ((eq_ref Prop) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) as proof of (((eq Prop) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) b)
% Found ((eq_ref Prop) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) as proof of (((eq Prop) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) b)
% Found classic0:=(classic ((or ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y))))):((or ((or ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y))))) (not ((or ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y))))))
% Found (classic ((or ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y))))) as proof of (P b)
% Found (classic ((or ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y))))) as proof of (P b)
% Found (classic ((or ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y))))) as proof of (P b)
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))):(((eq Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0)))))
% Found (eq_ref0 (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) as proof of (((eq Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) b)
% Found ((eq_ref Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) as proof of (((eq Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) b)
% Found ((eq_ref Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) as proof of (((eq Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) b)
% Found ((eq_ref Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) as proof of (((eq Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) b)
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 a):(((eq iS) a) a)
% Found (eq_ref0 a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found eq_ref00:=(eq_ref0 a):(((eq iS) a) a)
% Found (eq_ref0 a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found eq_ref00:=(eq_ref0 a):(((eq iS) a) a)
% Found (eq_ref0 a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found eq_ref00:=(eq_ref0 a):(((eq iS) a) a)
% Found (eq_ref0 a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found eq_ref00:=(eq_ref0 ((cP x) c0)):(((eq iS) ((cP x) c0)) ((cP x) c0))
% Found (eq_ref0 ((cP x) c0)) as proof of (((eq iS) ((cP x) c0)) b)
% Found ((eq_ref iS) ((cP x) c0)) as proof of (((eq iS) ((cP x) c0)) b)
% Found ((eq_ref iS) ((cP x) c0)) as proof of (((eq iS) ((cP x) c0)) b)
% Found ((eq_ref iS) ((cP x) c0)) as proof of (((eq iS) ((cP x) c0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq iS) b) b)
% Found (eq_ref0 b) as proof of (((eq iS) b) c0)
% Found ((eq_ref iS) b) as proof of (((eq iS) b) c0)
% Found ((eq_ref iS) b) as proof of (((eq iS) b) c0)
% Found ((eq_ref iS) b) as proof of (((eq iS) b) c0)
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 a):(((eq iS) a) a)
% Found (eq_ref0 a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found eq_ref00:=(eq_ref0 a):(((eq iS) a) a)
% Found (eq_ref0 a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found eq_ref00:=(eq_ref0 a):(((eq iS) a) a)
% Found (eq_ref0 a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found eq_ref00:=(eq_ref0 a):(((eq iS) a) a)
% Found (eq_ref0 a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b
% Found eq_ref00:=(eq_ref0 ((or ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y))))):(((eq Prop) ((or ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y))))) ((or ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y)))))
% Found (eq_ref0 ((or ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y))))) as proof of (((eq Prop) ((or ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y))))) b)
% Found ((eq_ref Prop) ((or ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y))))) as proof of (((eq Prop) ((or ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y))))) b)
% Found ((eq_ref Prop) ((or ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y))))) as proof of (((eq Prop) ((or ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y))))) b)
% Found ((eq_ref Prop) ((or ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y))))) as proof of (((eq Prop) ((or ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y))))) b)
% Found classic0:=(classic ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))):((or ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) (not ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))))
% Found (classic ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) as proof of (P b)
% Found (classic ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) as proof of (P b)
% Found (classic ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) as proof of (P b)
% Found eq_ref00:=(eq_ref0 ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y)))):(((eq Prop) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y)))) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y))))
% Found (eq_ref0 ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y)))) as proof of (((eq Prop) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y)))) b)
% Found ((eq_ref Prop) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y)))) as proof of (((eq Prop) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y)))) b)
% Found ((eq_ref Prop) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y)))) as proof of (((eq Prop) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y)))) b)
% Found ((eq_ref Prop) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y)))) as proof of (((eq Prop) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y)))) b)
% Found classic0:=(classic ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))):((or ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) (not ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))))
% Found (classic ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) as proof of (P b)
% Found (classic ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) as proof of (P b)
% Found (classic ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2)))))))))))))):(((eq (iS->Prop)) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2)))))))))))))) (fun (x0:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP x0) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R x0) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))
% Found (eta_expansion_dep00 (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2)))))))))))))) as proof of (((eq (iS->Prop)) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2)))))))))))))) b)
% Found ((eta_expansion_dep0 (fun (x2:iS)=> Prop)) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2)))))))))))))) as proof of (((eq (iS->Prop)) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2)))))))))))))) b)
% Found (((eta_expansion_dep iS) (fun (x2:iS)=> Prop)) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2)))))))))))))) as proof of (((eq (iS->Prop)) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2)))))))))))))) b)
% Found (((eta_expansion_dep iS) (fun (x2:iS)=> Prop)) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2)))))))))))))) as proof of (((eq (iS->Prop)) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2)))))))))))))) b)
% Found (((eta_expansion_dep iS) (fun (x2:iS)=> Prop)) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2)))))))))))))) as proof of (((eq (iS->Prop)) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2)))))))))))))) b)
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))):(((eq Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0)))))
% Found (eq_ref0 (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) as proof of (((eq Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) b)
% Found ((eq_ref Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) as proof of (((eq Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) b)
% Found ((eq_ref Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) as proof of (((eq Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) b)
% Found ((eq_ref Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) as proof of (((eq Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) b)
% Found eq_ref00:=(eq_ref0 ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))):(((eq Prop) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2)))))))))))))))
% Found (eq_ref0 ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) as proof of (((eq Prop) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) b)
% Found ((eq_ref Prop) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) as proof of (((eq Prop) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) b)
% Found ((eq_ref Prop) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) as proof of (((eq Prop) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) b)
% Found ((eq_ref Prop) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) as proof of (((eq Prop) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) b)
% Found classic0:=(classic ((or ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y))))):((or ((or ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y))))) (not ((or ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y))))))
% Found (classic ((or ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y))))) as proof of (P b)
% Found (classic ((or ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y))))) as proof of (P b)
% Found (classic ((or ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y))))) as proof of (P b)
% Found eq_ref00:=(eq_ref0 ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))):(((eq Prop) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2)))))))))))))))
% Found (eq_ref0 ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) as proof of (((eq Prop) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) b)
% Found ((eq_ref Prop) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) as proof of (((eq Prop) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) b)
% Found ((eq_ref Prop) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) as proof of (((eq Prop) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) b)
% Found ((eq_ref Prop) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) as proof of (((eq Prop) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) b)
% Found classic0:=(classic ((or ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y))))):((or ((or ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y))))) (not ((or ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y))))))
% Found (classic ((or ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y))))) as proof of (P b)
% Found (classic ((or ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y))))) as proof of (P b)
% Found (classic ((or ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y))))) as proof of (P b)
% Found eq_ref00:=(eq_ref0 a):(((eq iS) a) a)
% Found (eq_ref0 a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (iS->Prop)) a) (fun (x:iS)=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq (iS->Prop)) a) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))
% Found ((eta_expansion_dep0 (fun (x2:iS)=> Prop)) a) as proof of (((eq (iS->Prop)) a) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))
% Found (((eta_expansion_dep iS) (fun (x2:iS)=> Prop)) a) as proof of (((eq (iS->Prop)) a) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))
% Found (((eta_expansion_dep iS) (fun (x2:iS)=> Prop)) a) as proof of (((eq (iS->Prop)) a) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))
% Found (((eta_expansion_dep iS) (fun (x2:iS)=> Prop)) a) as proof of (((eq (iS->Prop)) a) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))
% Found eta_expansion000:=(eta_expansion00 a):(((eq (iS->Prop)) a) (fun (x:iS)=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq (iS->Prop)) a) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))
% Found ((eta_expansion0 Prop) a) as proof of (((eq (iS->Prop)) a) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))
% Found (((eta_expansion iS) Prop) a) as proof of (((eq (iS->Prop)) a) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))
% Found (((eta_expansion iS) Prop) a) as proof of (((eq (iS->Prop)) a) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))
% Found (((eta_expansion iS) Prop) a) as proof of (((eq (iS->Prop)) a) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))
% Found eq_ref00:=(eq_ref0 a):(((eq iS) a) a)
% Found (eq_ref0 a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found eq_ref00:=(eq_ref0 a):(((eq iS) a) a)
% Found (eq_ref0 a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found eq_ref00:=(eq_ref0 a):(((eq iS) a) a)
% Found (eq_ref0 a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 (f x03)):(((eq Prop) (f x03)) (f x03))
% Found (eq_ref0 (f x03)) as proof of (((eq Prop) (f x03)) ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP x03) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R x03) Xy1) Xz1))) (((R Xx2) Xy2) Xz2)))))))))))))
% Found ((eq_ref Prop) (f x03)) as proof of (((eq Prop) (f x03)) ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP x03) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R x03) Xy1) Xz1))) (((R Xx2) Xy2) Xz2)))))))))))))
% Found ((eq_ref Prop) (f x03)) as proof of (((eq Prop) (f x03)) ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP x03) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R x03) Xy1) Xz1))) (((R Xx2) Xy2) Xz2)))))))))))))
% Found (fun (x03:iS)=> ((eq_ref Prop) (f x03))) as proof of (((eq Prop) (f x03)) ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP x03) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R x03) Xy1) Xz1))) (((R Xx2) Xy2) Xz2)))))))))))))
% Found (fun (x03:iS)=> ((eq_ref Prop) (f x03))) as proof of (forall (x0:iS), (((eq Prop) (f x0)) ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP x0) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R x0) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))
% Found eq_ref00:=(eq_ref0 (f x03)):(((eq Prop) (f x03)) (f x03))
% Found (eq_ref0 (f x03)) as proof of (((eq Prop) (f x03)) ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP x03) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R x03) Xy1) Xz1))) (((R Xx2) Xy2) Xz2)))))))))))))
% Found ((eq_ref Prop) (f x03)) as proof of (((eq Prop) (f x03)) ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP x03) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R x03) Xy1) Xz1))) (((R Xx2) Xy2) Xz2)))))))))))))
% Found ((eq_ref Prop) (f x03)) as proof of (((eq Prop) (f x03)) ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP x03) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R x03) Xy1) Xz1))) (((R Xx2) Xy2) Xz2)))))))))))))
% Found (fun (x03:iS)=> ((eq_ref Prop) (f x03))) as proof of (((eq Prop) (f x03)) ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP x03) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R x03) Xy1) Xz1))) (((R Xx2) Xy2) Xz2)))))))))))))
% Found (fun (x03:iS)=> ((eq_ref Prop) (f x03))) as proof of (forall (x0:iS), (((eq Prop) (f x0)) ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP x0) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R x0) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))):(((eq Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0)))))
% Found (eq_ref0 (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) as proof of (((eq Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) b)
% Found ((eq_ref Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) as proof of (((eq Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) b)
% Found ((eq_ref Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) as proof of (((eq Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) b)
% Found ((eq_ref Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) as proof of (((eq Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))):(((eq Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0)))))
% Found (eq_ref0 (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) as proof of (((eq Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) b)
% Found ((eq_ref Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) as proof of (((eq Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) b)
% Found ((eq_ref Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) as proof of (((eq Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) b)
% Found ((eq_ref Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) as proof of (((eq Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))):(((eq Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0)))))
% Found (eq_ref0 (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) as proof of (((eq Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) b)
% Found ((eq_ref Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) as proof of (((eq Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) b)
% Found ((eq_ref Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) as proof of (((eq Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) b)
% Found ((eq_ref Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) as proof of (((eq Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) b)
% Found eq_ref000:=(eq_ref00 P):((P ((cP x) y))->(P ((cP x) y)))
% Found (eq_ref00 P) as proof of (P0 ((cP x) y))
% Found ((eq_ref0 ((cP x) y)) P) as proof of (P0 ((cP x) y))
% Found (((eq_ref iS) ((cP x) y)) P) as proof of (P0 ((cP x) y))
% Found (((eq_ref iS) ((cP x) y)) P) as proof of (P0 ((cP x) y))
% Found eq_ref000:=(eq_ref00 P):((P ((cP x) c0))->(P ((cP x) c0)))
% Found (eq_ref00 P) as proof of (P0 ((cP x) c0))
% Found ((eq_ref0 ((cP x) c0)) P) as proof of (P0 ((cP x) c0))
% Found (((eq_ref iS) ((cP x) c0)) P) as proof of (P0 ((cP x) c0))
% Found (((eq_ref iS) ((cP x) c0)) P) as proof of (P0 ((cP x) c0))
% Found eq_ref00:=(eq_ref0 ((cP x) c0)):(((eq iS) ((cP x) c0)) ((cP x) c0))
% Found (eq_ref0 ((cP x) c0)) as proof of (((eq iS) ((cP x) c0)) b)
% Found ((eq_ref iS) ((cP x) c0)) as proof of (((eq iS) ((cP x) c0)) b)
% Found ((eq_ref iS) ((cP x) c0)) as proof of (((eq iS) ((cP x) c0)) b)
% Found ((eq_ref iS) ((cP x) c0)) as proof of (((eq iS) ((cP x) c0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq iS) b) b)
% Found (eq_ref0 b) as proof of (((eq iS) b) c0)
% Found ((eq_ref iS) b) as proof of (((eq iS) b) c0)
% Found ((eq_ref iS) b) as proof of (((eq iS) b) c0)
% Found ((eq_ref iS) b) as proof of (((eq iS) b) c0)
% Found eq_ref00:=(eq_ref0 ((cP x) c0)):(((eq iS) ((cP x) c0)) ((cP x) c0))
% Found (eq_ref0 ((cP x) c0)) as proof of (((eq iS) ((cP x) c0)) b)
% Found ((eq_ref iS) ((cP x) c0)) as proof of (((eq iS) ((cP x) c0)) b)
% Found ((eq_ref iS) ((cP x) c0)) as proof of (((eq iS) ((cP x) c0)) b)
% Found ((eq_ref iS) ((cP x) c0)) as proof of (((eq iS) ((cP x) c0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq iS) b) b)
% Found (eq_ref0 b) as proof of (((eq iS) b) c0)
% Found ((eq_ref iS) b) as proof of (((eq iS) b) c0)
% Found ((eq_ref iS) b) as proof of (((eq iS) b) c0)
% Found ((eq_ref iS) b) as proof of (((eq iS) b) c0)
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))):(((eq Prop) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2)))))))))))))))
% Found (eq_ref0 ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) as proof of (((eq Prop) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) b)
% Found ((eq_ref Prop) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) as proof of (((eq Prop) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) b)
% Found ((eq_ref Prop) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) as proof of (((eq Prop) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) b)
% Found ((eq_ref Prop) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) as proof of (((eq Prop) ((ex iS) (fun (Xx1:iS)=> ((ex iS) (fun (Xx2:iS)=> ((ex iS) (fun (Xy1:iS)=> ((ex iS) (fun (Xy2:iS)=> ((ex iS) (fun (Xz1:iS)=> ((ex iS) (fun (Xz2:iS)=> ((and ((and ((and ((and (((eq iS) ((cP x) c0)) ((cP Xx1) Xx2))) (((eq iS) ((cP x) y)) ((cP Xy1) Xy2)))) (((eq iS) ((cP x) y)) ((cP Xz1) Xz2)))) (((R Xx1) Xy1) Xz1))) (((R Xx2) Xy2) Xz2))))))))))))))) b)
% Found classic0:=(classic ((or ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y))))):((or ((or ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y))))) (not ((or ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y))))))
% Found (classic ((or ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y))))) as proof of (P b)
% Found (classic ((or ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y))))) as proof of (P b)
% Found (classic ((or ((and (((eq iS) ((cP x) c0)) c0)) (((eq iS) ((cP x) y)) ((cP x) y)))) ((and (((eq iS) ((cP x) y)) c0)) (((eq iS) ((cP x) c0)) ((cP x) y))))) as proof of (P b)
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 a):(((eq iS) a) a)
% Found (eq_ref0 a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found eq_ref00:=(eq_ref0 a):(((eq iS) a) a)
% Found (eq_ref0 a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found eq_ref00:=(eq_ref0 a):(((eq iS) a) a)
% Found (eq_ref0 a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found eq_ref00:=(eq_ref0 a):(((eq iS) a) a)
% Found (eq_ref0 a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found eq_ref00:=(eq_ref0 a):(((eq iS) a) a)
% Found (eq_ref0 a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found eq_ref00:=(eq_ref0 a):(((eq iS) a) a)
% Found (eq_ref0 a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found eq_ref00:=(eq_ref0 a):(((eq iS) a) a)
% Found (eq_ref0 a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found eq_ref00:=(eq_ref0 a):(((eq iS) a) a)
% Found (eq_ref0 a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found ((eq_ref iS) a) as proof of (((eq iS) a) y)
% Found eq_ref00:=(eq_ref0 (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))):(((eq Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0)))))
% Found (eq_ref0 (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) as proof of (((eq Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) b)
% Found ((eq_ref Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) as proof of (((eq Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) b)
% Found ((eq_ref Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) as proof of (((eq Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) b)
% Found ((eq_ref Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) as proof of (((eq Prop) (forall (Xx0:iS) (Xy0:iS), (((and (((R ((cP x) c0)) ((cP x) y)) Xx0)) (((R ((cP x) c0)) ((cP x) y)) Xy0))->(((R ((cP x) c0)) ((cP x) y)) ((cP Xx0) Xy0))))) b)
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq iS) b) b)
% Found (eq_ref0 b) as proof of (((eq iS) b) c0)
% Found ((eq_ref iS) b) as proof of (((eq iS) b) c0)
% Found ((eq_ref iS) b) as proof of (((eq iS) b) c0)
% Found ((eq_ref iS) b) as proof of (((eq iS) b) c0)
% Found eq_ref00:=(eq_ref0 ((cP x) c0)):(((eq iS) ((cP x) c0)) ((cP x) c0))
% Found (eq_ref0 ((cP x) c0)) as proof of (((eq iS) ((cP x) c0)) b)
% Found ((eq_ref iS) ((cP x) c0)) as proof of (((eq iS) ((cP x) c0)) b)
% Found ((eq_ref iS) ((cP x) c0)) as proof of (((eq iS) ((cP x) c0)) b)
% Found ((eq_ref iS) ((cP x) c0)) as proof of (((eq iS) ((cP x) c0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq iS) b) b)
% Found (eq_ref0 b) as proof of (((eq iS) b) ((cP x) y))
% Found ((eq_ref iS) b) as proof of (((eq iS) b) ((cP x) y))
% Found ((eq_ref iS) b) as proof of (((eq iS) b) ((cP x) y))
% Found ((eq_ref iS) b) as proof of (((eq iS) b) ((cP x) y))
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) b)
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found eq_ref00:=(eq_ref0 c0):(((eq iS) c0) c0)
% Found (eq_ref0 c0) as proof of (((eq iS) c0) b)
% Found ((eq_ref iS) c0) as proof of (((eq iS) c0) b)
% Found ((eq_ref iS) c0) as proof of (((eq iS) c0) b)
% Found ((eq_ref iS) c0) as proof of (((eq iS) c0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq iS) b) b)
% Found (eq_ref0 b) as proof of (((eq iS) b) ((cP x) c0))
% Found ((eq_ref iS) b) as proof of (((eq iS) b) ((cP x) c0))
% Found ((eq_ref iS) b) as proof of (((eq iS) b) ((cP x) c0))
% Found ((eq_ref iS) b) as proof of (((eq iS) b) ((cP x) c0))
% Found eq_ref000:=(eq_ref00 ((R ((cP x) c0)) ((cP x) y))):((((R ((cP x) c0)) ((cP x) y)) Xy0)->(((R ((cP x) c0)) ((cP x) y)) Xy0))
% Found (eq_ref00 ((R ((cP x) c0)) ((cP x) y))) as proof of (P Xy0)
% Found ((eq_ref0 Xy0) ((R ((cP x) c0)) ((cP x) y))) as proof of (P Xy0)
% Found (((eq_ref iS) Xy0) ((R ((cP x) c0)) ((cP x) y))) as proof of (P Xy0)
% Found (((eq_ref iS) Xy0) ((R ((cP x) c0)) ((cP x) y))) as proof of (P Xy0)
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b
% Found eq_ref00:=(eq_ref0 ((cP x) y)):(((eq iS) ((cP x) y)) ((cP x) y))
% Found (eq_ref0 ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Found ((eq_ref iS) ((cP x) y)) as proof of (((eq iS) ((cP x) y)) ((cP x) y))
% Foun
% EOF
%------------------------------------------------------------------------------