TSTP Solution File: PUZ106^5 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : PUZ106^5 : TPTP v6.1.0. Bugfixed v5.2.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n116.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 13:28:59 EDT 2014
% Result : Timeout 300.09s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : PUZ106^5 : TPTP v6.1.0. Bugfixed v5.2.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n116.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 08:19:56 CDT 2014
% % CPUTime : 300.09
% 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 0x9fc908>, <kernel.Constant object at 0x9fc2d8>) of role type named c1_type
% Using role type
% Declaring c1:fofType
% FOF formula (<kernel.Constant object at 0x9fd518>, <kernel.DependentProduct object at 0x9fc518>) of role type named s_type
% Using role type
% Declaring s:(fofType->fofType)
% FOF formula (<kernel.Constant object at 0x9fc3b0>, <kernel.DependentProduct object at 0x9fc908>) of role type named cCKB6_BLACK_type
% Using role type
% Declaring cCKB6_BLACK:(fofType->(fofType->Prop))
% FOF formula (<kernel.Constant object at 0x9fc758>, <kernel.DependentProduct object at 0x9fc2d8>) of role type named cCKB_E2_type
% Using role type
% Declaring cCKB_E2:(fofType->(fofType->Prop))
% FOF formula (((eq (fofType->(fofType->Prop))) cCKB6_BLACK) (fun (Xu:fofType) (Xv:fofType)=> (forall (Xw:(fofType->(fofType->Prop))), (((and ((Xw c1) c1)) (forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))->((Xw Xu) Xv))))) of role definition named cCKB6_BLACK_def
% A new definition: (((eq (fofType->(fofType->Prop))) cCKB6_BLACK) (fun (Xu:fofType) (Xv:fofType)=> (forall (Xw:(fofType->(fofType->Prop))), (((and ((Xw c1) c1)) (forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))->((Xw Xu) Xv)))))
% Defined: cCKB6_BLACK:=(fun (Xu:fofType) (Xv:fofType)=> (forall (Xw:(fofType->(fofType->Prop))), (((and ((Xw c1) c1)) (forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))->((Xw Xu) Xv))))
% FOF formula (((eq (fofType->(fofType->Prop))) cCKB_E2) (fun (Xx:fofType) (Xy:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp Xx)) (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu))))))->(Xp Xy))))) of role definition named cCKB_E2_def
% A new definition: (((eq (fofType->(fofType->Prop))) cCKB_E2) (fun (Xx:fofType) (Xy:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp Xx)) (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu))))))->(Xp Xy)))))
% Defined: cCKB_E2:=(fun (Xx:fofType) (Xy:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp Xx)) (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu))))))->(Xp Xy))))
% FOF formula (forall (Xx:fofType) (Xy:fofType), (((cCKB6_BLACK Xx) Xy)->((cCKB_E2 Xx) Xy))) of role conjecture named cCKB_L37000
% Conjecture to prove = (forall (Xx:fofType) (Xy:fofType), (((cCKB6_BLACK Xx) Xy)->((cCKB_E2 Xx) Xy))):Prop
% We need to prove ['(forall (Xx:fofType) (Xy:fofType), (((cCKB6_BLACK Xx) Xy)->((cCKB_E2 Xx) Xy)))']
% Parameter fofType:Type.
% Parameter c1:fofType.
% Parameter s:(fofType->fofType).
% Definition cCKB6_BLACK:=(fun (Xu:fofType) (Xv:fofType)=> (forall (Xw:(fofType->(fofType->Prop))), (((and ((Xw c1) c1)) (forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))->((Xw Xu) Xv)))):(fofType->(fofType->Prop)).
% Definition cCKB_E2:=(fun (Xx:fofType) (Xy:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp Xx)) (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu))))))->(Xp Xy)))):(fofType->(fofType->Prop)).
% Trying to prove (forall (Xx:fofType) (Xy:fofType), (((cCKB6_BLACK Xx) Xy)->((cCKB_E2 Xx) Xy)))
% Found x2:(Xp c1)
% Found (fun (x3:(forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu))))))=> x2) as proof of (Xp c1)
% Found (fun (x2:(Xp c1)) (x3:(forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu))))))=> x2) as proof of ((forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))->(Xp c1))
% Found (fun (x2:(Xp c1)) (x3:(forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu))))))=> x2) as proof of ((Xp c1)->((forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))->(Xp c1)))
% Found (and_rect00 (fun (x2:(Xp c1)) (x3:(forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu))))))=> x2)) as proof of (Xp c1)
% Found ((and_rect0 (Xp c1)) (fun (x2:(Xp c1)) (x3:(forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu))))))=> x2)) as proof of (Xp c1)
% Found (((fun (P:Type) (x2:((Xp c1)->((forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))->P)))=> (((((and_rect (Xp c1)) (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))) P) x2) x1)) (Xp c1)) (fun (x2:(Xp c1)) (x3:(forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu))))))=> x2)) as proof of (Xp c1)
% Found (fun (x1:((and (Xp c1)) (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))))=> (((fun (P:Type) (x2:((Xp c1)->((forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))->P)))=> (((((and_rect (Xp c1)) (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))) P) x2) x1)) (Xp c1)) (fun (x2:(Xp c1)) (x3:(forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu))))))=> x2))) as proof of (Xp c1)
% Found (fun (Xp:(fofType->Prop)) (x1:((and (Xp c1)) (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))))=> (((fun (P:Type) (x2:((Xp c1)->((forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))->P)))=> (((((and_rect (Xp c1)) (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))) P) x2) x1)) (Xp c1)) (fun (x2:(Xp c1)) (x3:(forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu))))))=> x2))) as proof of (((and (Xp c1)) (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu))))))->(Xp c1))
% Found (fun (Xp:(fofType->Prop)) (x1:((and (Xp c1)) (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))))=> (((fun (P:Type) (x2:((Xp c1)->((forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))->P)))=> (((((and_rect (Xp c1)) (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))) P) x2) x1)) (Xp c1)) (fun (x2:(Xp c1)) (x3:(forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu))))))=> x2))) as proof of ((cCKB_E2 c1) c1)
% Found x2:(Xp Xk)
% Found x2 as proof of (Xp Xk)
% Found x3:((forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))->(Xp Xk))
% Found x3 as proof of ((forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))->(Xp Xk))
% Found x4:(Xp Xk)
% Found x4 as proof of (Xp Xk)
% Found x4:(Xp Xk)
% Found x4 as proof of (Xp Xk)
% Found x2:(Xp Xk)
% Found x2 as proof of (Xp Xk)
% Found x4:(forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))
% Found x4 as proof of (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))
% Found x3:(Xp (s Xj))
% Found x3 as proof of (Xp (s Xj))
% Found x40:=(x4 (s Xu)):((Xp (s Xu))->(Xp (s (s (s Xu)))))
% Found (x4 (s Xu)) as proof of ((Xp (s Xu))->(Xp (s (s (s Xu)))))
% Found (fun (Xu:fofType)=> (x4 (s Xu))) as proof of ((Xp (s Xu))->(Xp (s (s (s Xu)))))
% Found (fun (Xu:fofType)=> (x4 (s Xu))) as proof of (forall (Xu:fofType), ((Xp (s Xu))->(Xp (s (s (s Xu))))))
% Found ((conj20 x3) (fun (Xu:fofType)=> (x4 (s Xu)))) as proof of ((and (Xp (s Xj))) (forall (Xu:fofType), ((Xp (s Xu))->(Xp (s (s (s Xu)))))))
% Found (((conj2 (forall (Xu:fofType), ((Xp (s Xu))->(Xp (s (s (s Xu))))))) x3) (fun (Xu:fofType)=> (x4 (s Xu)))) as proof of ((and (Xp (s Xj))) (forall (Xu:fofType), ((Xp (s Xu))->(Xp (s (s (s Xu)))))))
% Found ((((conj (Xp (s Xj))) (forall (Xu:fofType), ((Xp (s Xu))->(Xp (s (s (s Xu))))))) x3) (fun (Xu:fofType)=> (x4 (s Xu)))) as proof of ((and (Xp (s Xj))) (forall (Xu:fofType), ((Xp (s Xu))->(Xp (s (s (s Xu)))))))
% Found ((((conj (Xp (s Xj))) (forall (Xu:fofType), ((Xp (s Xu))->(Xp (s (s (s Xu))))))) x3) (fun (Xu:fofType)=> (x4 (s Xu)))) as proof of ((and (Xp (s Xj))) (forall (Xu:fofType), ((Xp (s Xu))->(Xp (s (s (s Xu)))))))
% Found (x10 ((((conj (Xp (s Xj))) (forall (Xu:fofType), ((Xp (s Xu))->(Xp (s (s (s Xu))))))) x3) (fun (Xu:fofType)=> (x4 (s Xu))))) as proof of (Xp (s Xk))
% Found ((x1 (fun (x6:fofType)=> (Xp (s x6)))) ((((conj (Xp (s Xj))) (forall (Xu:fofType), ((Xp (s Xu))->(Xp (s (s (s Xu))))))) x3) (fun (Xu:fofType)=> (x4 (s Xu))))) as proof of (Xp (s Xk))
% Found (fun (x4:(forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu))))))=> ((x1 (fun (x6:fofType)=> (Xp (s x6)))) ((((conj (Xp (s Xj))) (forall (Xu:fofType), ((Xp (s Xu))->(Xp (s (s (s Xu))))))) x3) (fun (Xu:fofType)=> (x4 (s Xu)))))) as proof of (Xp (s Xk))
% Found (fun (x3:(Xp (s Xj))) (x4:(forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu))))))=> ((x1 (fun (x6:fofType)=> (Xp (s x6)))) ((((conj (Xp (s Xj))) (forall (Xu:fofType), ((Xp (s Xu))->(Xp (s (s (s Xu))))))) x3) (fun (Xu:fofType)=> (x4 (s Xu)))))) as proof of ((forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))->(Xp (s Xk)))
% Found (fun (x3:(Xp (s Xj))) (x4:(forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu))))))=> ((x1 (fun (x6:fofType)=> (Xp (s x6)))) ((((conj (Xp (s Xj))) (forall (Xu:fofType), ((Xp (s Xu))->(Xp (s (s (s Xu))))))) x3) (fun (Xu:fofType)=> (x4 (s Xu)))))) as proof of ((Xp (s Xj))->((forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))->(Xp (s Xk))))
% Found (and_rect00 (fun (x3:(Xp (s Xj))) (x4:(forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu))))))=> ((x1 (fun (x6:fofType)=> (Xp (s x6)))) ((((conj (Xp (s Xj))) (forall (Xu:fofType), ((Xp (s Xu))->(Xp (s (s (s Xu))))))) x3) (fun (Xu:fofType)=> (x4 (s Xu))))))) as proof of (Xp (s Xk))
% Found ((and_rect0 (Xp (s Xk))) (fun (x3:(Xp (s Xj))) (x4:(forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu))))))=> ((x1 (fun (x6:fofType)=> (Xp (s x6)))) ((((conj (Xp (s Xj))) (forall (Xu:fofType), ((Xp (s Xu))->(Xp (s (s (s Xu))))))) x3) (fun (Xu:fofType)=> (x4 (s Xu))))))) as proof of (Xp (s Xk))
% Found (((fun (P:Type) (x3:((Xp (s Xj))->((forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))->P)))=> (((((and_rect (Xp (s Xj))) (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))) P) x3) x2)) (Xp (s Xk))) (fun (x3:(Xp (s Xj))) (x4:(forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu))))))=> ((x1 (fun (x6:fofType)=> (Xp (s x6)))) ((((conj (Xp (s Xj))) (forall (Xu:fofType), ((Xp (s Xu))->(Xp (s (s (s Xu))))))) x3) (fun (Xu:fofType)=> (x4 (s Xu))))))) as proof of (Xp (s Xk))
% Found (fun (x2:((and (Xp (s Xj))) (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))))=> (((fun (P:Type) (x3:((Xp (s Xj))->((forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))->P)))=> (((((and_rect (Xp (s Xj))) (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))) P) x3) x2)) (Xp (s Xk))) (fun (x3:(Xp (s Xj))) (x4:(forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu))))))=> ((x1 (fun (x6:fofType)=> (Xp (s x6)))) ((((conj (Xp (s Xj))) (forall (Xu:fofType), ((Xp (s Xu))->(Xp (s (s (s Xu))))))) x3) (fun (Xu:fofType)=> (x4 (s Xu)))))))) as proof of (Xp (s Xk))
% Found (fun (Xp:(fofType->Prop)) (x2:((and (Xp (s Xj))) (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))))=> (((fun (P:Type) (x3:((Xp (s Xj))->((forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))->P)))=> (((((and_rect (Xp (s Xj))) (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))) P) x3) x2)) (Xp (s Xk))) (fun (x3:(Xp (s Xj))) (x4:(forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu))))))=> ((x1 (fun (x6:fofType)=> (Xp (s x6)))) ((((conj (Xp (s Xj))) (forall (Xu:fofType), ((Xp (s Xu))->(Xp (s (s (s Xu))))))) x3) (fun (Xu:fofType)=> (x4 (s Xu)))))))) as proof of (((and (Xp (s Xj))) (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu))))))->(Xp (s Xk)))
% Found (fun (Xp:(fofType->Prop)) (x2:((and (Xp (s Xj))) (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))))=> (((fun (P:Type) (x3:((Xp (s Xj))->((forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))->P)))=> (((((and_rect (Xp (s Xj))) (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))) P) x3) x2)) (Xp (s Xk))) (fun (x3:(Xp (s Xj))) (x4:(forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu))))))=> ((x1 (fun (x6:fofType)=> (Xp (s x6)))) ((((conj (Xp (s Xj))) (forall (Xu:fofType), ((Xp (s Xu))->(Xp (s (s (s Xu))))))) x3) (fun (Xu:fofType)=> (x4 (s Xu)))))))) as proof of ((cCKB_E2 (s Xj)) (s Xk))
% Found x4:(forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))
% Found x4 as proof of (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))
% Found x41:=(x4 x40):(Xp Xu)
% Found (x4 x40) as proof of (Xp Xu)
% Found (x4 x40) as proof of (Xp Xu)
% Found (x400 (x4 x40)) as proof of (Xp (s (s Xu)))
% Found ((x40 Xu) (x4 x40)) as proof of (Xp (s (s Xu)))
% Found (fun (x40:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x40 Xu) (x4 x40))) as proof of (Xp (s (s Xu)))
% Found (fun (x4:((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp Xu))) (x40:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x40 Xu) (x4 x40))) as proof of ((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp (s (s Xu))))
% Found (fun (Xu:fofType) (x4:((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp Xu))) (x40:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x40 Xu) (x4 x40))) as proof of (((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp Xu))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp (s (s Xu)))))
% Found (fun (Xu:fofType) (x4:((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp Xu))) (x40:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x40 Xu) (x4 x40))) as proof of (forall (Xu:fofType), (((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp Xu))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp (s (s Xu))))))
% Found x500:=(x50 x3):(Xp (s (s Xu)))
% Found (x50 x3) as proof of (Xp (s (s Xu)))
% Found ((x5 Xu) x3) as proof of (Xp (s (s Xu)))
% Found (fun (x5:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x5 Xu) x3)) as proof of (Xp (s (s Xu)))
% Found (fun (x4:(Xp (s (s Xj)))) (x5:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x5 Xu) x3)) as proof of ((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp (s (s Xu))))
% Found (fun (x4:(Xp (s (s Xj)))) (x5:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x5 Xu) x3)) as proof of ((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp (s (s Xu)))))
% Found (and_rect00 (fun (x4:(Xp (s (s Xj)))) (x5:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x5 Xu) x3))) as proof of (Xp (s (s Xu)))
% Found ((and_rect0 (Xp (s (s Xu)))) (fun (x4:(Xp (s (s Xj)))) (x5:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x5 Xu) x3))) as proof of (Xp (s (s Xu)))
% Found (((fun (P:Type) (x4:((Xp (s (s Xj)))->((forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))) P) x4) x2)) (Xp (s (s Xu)))) (fun (x4:(Xp (s (s Xj)))) (x5:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x5 Xu) x3))) as proof of (Xp (s (s Xu)))
% Found (fun (x3:(Xp Xu))=> (((fun (P:Type) (x4:((Xp (s (s Xj)))->((forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))) P) x4) x2)) (Xp (s (s Xu)))) (fun (x4:(Xp (s (s Xj)))) (x5:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x5 Xu) x3)))) as proof of (Xp (s (s Xu)))
% Found (fun (Xu:fofType) (x3:(Xp Xu))=> (((fun (P:Type) (x4:((Xp (s (s Xj)))->((forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))) P) x4) x2)) (Xp (s (s Xu)))) (fun (x4:(Xp (s (s Xj)))) (x5:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x5 Xu) x3)))) as proof of ((Xp Xu)->(Xp (s (s Xu))))
% Found (fun (Xu:fofType) (x3:(Xp Xu))=> (((fun (P:Type) (x4:((Xp (s (s Xj)))->((forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))) P) x4) x2)) (Xp (s (s Xu)))) (fun (x4:(Xp (s (s Xj)))) (x5:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x5 Xu) x3)))) as proof of (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))
% Found x310:=(x31 x40):(Xp Xu)
% Found (x31 x40) as proof of (Xp Xu)
% Found ((x3 x30) x40) as proof of (Xp Xu)
% Found ((x3 x30) x40) as proof of (Xp Xu)
% Found (x400 ((x3 x30) x40)) as proof of (Xp (s (s Xu)))
% Found ((x40 Xu) ((x3 x30) x40)) as proof of (Xp (s (s Xu)))
% Found (fun (x40:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x40 Xu) ((x3 x30) x40))) as proof of (Xp (s (s Xu)))
% Found (fun (x30:(Xp (s (s Xj)))) (x40:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x40 Xu) ((x3 x30) x40))) as proof of ((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp (s (s Xu))))
% Found (fun (x3:((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp Xu)))) (x30:(Xp (s (s Xj)))) (x40:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x40 Xu) ((x3 x30) x40))) as proof of ((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp (s (s Xu)))))
% Found (fun (Xu:fofType) (x3:((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp Xu)))) (x30:(Xp (s (s Xj)))) (x40:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x40 Xu) ((x3 x30) x40))) as proof of (((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp Xu)))->((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp (s (s Xu))))))
% Found (fun (Xu:fofType) (x3:((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp Xu)))) (x30:(Xp (s (s Xj)))) (x40:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x40 Xu) ((x3 x30) x40))) as proof of (forall (Xu:fofType), (((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp Xu)))->((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp (s (s Xu)))))))
% Found x21:=(x2 x20):(Xp Xu)
% Found (x2 x20) as proof of (Xp Xu)
% Found (x2 x20) as proof of (Xp Xu)
% Found (x40 (x2 x20)) as proof of (Xp (s (s Xu)))
% Found ((x4 Xu) (x2 x20)) as proof of (Xp (s (s Xu)))
% Found (fun (x4:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x4 Xu) (x2 x20))) as proof of (Xp (s (s Xu)))
% Found (fun (x3:(Xp (s (s Xj)))) (x4:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x4 Xu) (x2 x20))) as proof of ((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp (s (s Xu))))
% Found (fun (x3:(Xp (s (s Xj)))) (x4:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x4 Xu) (x2 x20))) as proof of ((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp (s (s Xu)))))
% Found (and_rect00 (fun (x3:(Xp (s (s Xj)))) (x4:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x4 Xu) (x2 x20)))) as proof of (Xp (s (s Xu)))
% Found ((and_rect0 (Xp (s (s Xu)))) (fun (x3:(Xp (s (s Xj)))) (x4:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x4 Xu) (x2 x20)))) as proof of (Xp (s (s Xu)))
% Found (((fun (P:Type) (x3:((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x3) x20)) (Xp (s (s Xu)))) (fun (x3:(Xp (s (s Xj)))) (x4:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x4 Xu) (x2 x20)))) as proof of (Xp (s (s Xu)))
% Found (fun (x20:((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x3:((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x3) x20)) (Xp (s (s Xu)))) (fun (x3:(Xp (s (s Xj)))) (x4:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x4 Xu) (x2 x20))))) as proof of (Xp (s (s Xu)))
% Found (fun (x2:(((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))->(Xp Xu))) (x20:((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x3:((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x3) x20)) (Xp (s (s Xu)))) (fun (x3:(Xp (s (s Xj)))) (x4:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x4 Xu) (x2 x20))))) as proof of (((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))->(Xp (s (s Xu))))
% Found (fun (Xu:fofType) (x2:(((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))->(Xp Xu))) (x20:((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x3:((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x3) x20)) (Xp (s (s Xu)))) (fun (x3:(Xp (s (s Xj)))) (x4:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x4 Xu) (x2 x20))))) as proof of ((((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))->(Xp Xu))->(((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))->(Xp (s (s Xu)))))
% Found (fun (Xu:fofType) (x2:(((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))->(Xp Xu))) (x20:((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x3:((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x3) x20)) (Xp (s (s Xu)))) (fun (x3:(Xp (s (s Xj)))) (x4:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x4 Xu) (x2 x20))))) as proof of (forall (Xu:fofType), ((((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))->(Xp Xu))->(((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))->(Xp (s (s Xu))))))
% Found x3:(Xp (s Xj))
% Found (fun (x4:(forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu))))))=> x3) as proof of (Xp (s Xj))
% Found (fun (x3:(Xp (s Xj))) (x4:(forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu))))))=> x3) as proof of ((forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))->(Xp (s Xj)))
% Found (fun (x3:(Xp (s Xj))) (x4:(forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu))))))=> x3) as proof of ((Xp (s Xj))->((forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))->(Xp (s Xj))))
% Found (and_rect00 (fun (x3:(Xp (s Xj))) (x4:(forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu))))))=> x3)) as proof of (Xp (s Xj))
% Found ((and_rect0 (Xp (s Xj))) (fun (x3:(Xp (s Xj))) (x4:(forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu))))))=> x3)) as proof of (Xp (s Xj))
% Found (((fun (P:Type) (x3:((Xp (s Xj))->((forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))->P)))=> (((((and_rect (Xp (s Xj))) (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))) P) x3) x2)) (Xp (s Xj))) (fun (x3:(Xp (s Xj))) (x4:(forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu))))))=> x3)) as proof of (Xp (s Xj))
% Found (fun (x2:((and (Xp (s Xj))) (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))))=> (((fun (P:Type) (x3:((Xp (s Xj))->((forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))->P)))=> (((((and_rect (Xp (s Xj))) (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))) P) x3) x2)) (Xp (s Xj))) (fun (x3:(Xp (s Xj))) (x4:(forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu))))))=> x3))) as proof of (Xp (s Xj))
% Found (fun (Xp:(fofType->Prop)) (x2:((and (Xp (s Xj))) (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))))=> (((fun (P:Type) (x3:((Xp (s Xj))->((forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))->P)))=> (((((and_rect (Xp (s Xj))) (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))) P) x3) x2)) (Xp (s Xj))) (fun (x3:(Xp (s Xj))) (x4:(forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu))))))=> x3))) as proof of (((and (Xp (s Xj))) (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu))))))->(Xp (s Xj)))
% Found (fun (Xp:(fofType->Prop)) (x2:((and (Xp (s Xj))) (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))))=> (((fun (P:Type) (x3:((Xp (s Xj))->((forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))->P)))=> (((((and_rect (Xp (s Xj))) (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))) P) x3) x2)) (Xp (s Xj))) (fun (x3:(Xp (s Xj))) (x4:(forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu))))))=> x3))) as proof of ((cCKB_E2 (s Xj)) (s Xj))
% Found x200:=(x20 x3):(Xp Xu)
% Found (x20 x3) as proof of (Xp Xu)
% Found ((x2 Xp) x3) as proof of (Xp Xu)
% Found ((x2 Xp) x3) as proof of (Xp Xu)
% Found (x50 ((x2 Xp) x3)) as proof of (Xp (s (s Xu)))
% Found ((x5 Xu) ((x2 Xp) x3)) as proof of (Xp (s (s Xu)))
% Found (fun (x5:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x5 Xu) ((x2 Xp) x3))) as proof of (Xp (s (s Xu)))
% Found (fun (x4:(Xp (s (s Xj)))) (x5:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x5 Xu) ((x2 Xp) x3))) as proof of ((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp (s (s Xu))))
% Found (fun (x4:(Xp (s (s Xj)))) (x5:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x5 Xu) ((x2 Xp) x3))) as proof of ((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp (s (s Xu)))))
% Found (and_rect00 (fun (x4:(Xp (s (s Xj)))) (x5:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x5 Xu) ((x2 Xp) x3)))) as proof of (Xp (s (s Xu)))
% Found ((and_rect0 (Xp (s (s Xu)))) (fun (x4:(Xp (s (s Xj)))) (x5:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x5 Xu) ((x2 Xp) x3)))) as proof of (Xp (s (s Xu)))
% Found (((fun (P:Type) (x4:((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x4) x3)) (Xp (s (s Xu)))) (fun (x4:(Xp (s (s Xj)))) (x5:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x5 Xu) ((x2 Xp) x3)))) as proof of (Xp (s (s Xu)))
% Found (fun (x3:((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x4:((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x4) x3)) (Xp (s (s Xu)))) (fun (x4:(Xp (s (s Xj)))) (x5:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x5 Xu) ((x2 Xp) x3))))) as proof of (Xp (s (s Xu)))
% Found (fun (Xp:(fofType->Prop)) (x3:((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x4:((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x4) x3)) (Xp (s (s Xu)))) (fun (x4:(Xp (s (s Xj)))) (x5:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x5 Xu) ((x2 Xp) x3))))) as proof of (((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))->(Xp (s (s Xu))))
% Found (fun (x2:((cCKB_E2 (s (s Xj))) Xu)) (Xp:(fofType->Prop)) (x3:((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x4:((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x4) x3)) (Xp (s (s Xu)))) (fun (x4:(Xp (s (s Xj)))) (x5:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x5 Xu) ((x2 Xp) x3))))) as proof of ((cCKB_E2 (s (s Xj))) (s (s Xu)))
% Found (fun (Xu:fofType) (x2:((cCKB_E2 (s (s Xj))) Xu)) (Xp:(fofType->Prop)) (x3:((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x4:((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x4) x3)) (Xp (s (s Xu)))) (fun (x4:(Xp (s (s Xj)))) (x5:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x5 Xu) ((x2 Xp) x3))))) as proof of (((cCKB_E2 (s (s Xj))) Xu)->((cCKB_E2 (s (s Xj))) (s (s Xu))))
% Found (fun (Xu:fofType) (x2:((cCKB_E2 (s (s Xj))) Xu)) (Xp:(fofType->Prop)) (x3:((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x4:((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x4) x3)) (Xp (s (s Xu)))) (fun (x4:(Xp (s (s Xj)))) (x5:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x5 Xu) ((x2 Xp) x3))))) as proof of (forall (Xu:fofType), (((cCKB_E2 (s (s Xj))) Xu)->((cCKB_E2 (s (s Xj))) (s (s Xu)))))
% Found x2:(Xp Xk)
% Found x2 as proof of (Xp Xk)
% Found x3:((forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))->(Xp Xk))
% Found x3 as proof of ((forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))->(Xp Xk))
% Found x4:(Xp Xk)
% Found x4 as proof of (Xp Xk)
% Found x4:(Xp Xk)
% Found x4 as proof of (Xp Xk)
% Found x4:(forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))
% Found x4 as proof of (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))
% Found x4:(forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))
% Found x4 as proof of (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))
% Found x2:(Xp Xk)
% Found x2 as proof of (Xp Xk)
% Found x41:=(x4 x40):(Xp Xu)
% Found (x4 x40) as proof of (Xp Xu)
% Found (x4 x40) as proof of (Xp Xu)
% Found (x400 (x4 x40)) as proof of (Xp (s (s Xu)))
% Found ((x40 Xu) (x4 x40)) as proof of (Xp (s (s Xu)))
% Found (fun (x40:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x40 Xu) (x4 x40))) as proof of (Xp (s (s Xu)))
% Found (fun (x4:((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp Xu))) (x40:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x40 Xu) (x4 x40))) as proof of ((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp (s (s Xu))))
% Found (fun (Xu:fofType) (x4:((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp Xu))) (x40:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x40 Xu) (x4 x40))) as proof of (((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp Xu))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp (s (s Xu)))))
% Found (fun (Xu:fofType) (x4:((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp Xu))) (x40:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x40 Xu) (x4 x40))) as proof of (forall (Xu:fofType), (((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp Xu))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp (s (s Xu))))))
% Found x300:=(x30 x5):(Xp Xu)
% Found (x30 x5) as proof of (Xp Xu)
% Found ((x3 Xp) x5) as proof of (Xp Xu)
% Found ((x3 Xp) x5) as proof of (Xp Xu)
% Found (x70 ((x3 Xp) x5)) as proof of (Xp (s (s Xu)))
% Found ((x7 Xu) ((x3 Xp) x5)) as proof of (Xp (s (s Xu)))
% Found (fun (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 Xu) ((x3 Xp) x5))) as proof of (Xp (s (s Xu)))
% Found (fun (x6:(Xp (s (s Xj)))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 Xu) ((x3 Xp) x5))) as proof of ((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp (s (s Xu))))
% Found (fun (x6:(Xp (s (s Xj)))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 Xu) ((x3 Xp) x5))) as proof of ((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp (s (s Xu)))))
% Found (and_rect10 (fun (x6:(Xp (s (s Xj)))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 Xu) ((x3 Xp) x5)))) as proof of (Xp (s (s Xu)))
% Found ((and_rect1 (Xp (s (s Xu)))) (fun (x6:(Xp (s (s Xj)))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 Xu) ((x3 Xp) x5)))) as proof of (Xp (s (s Xu)))
% Found (((fun (P:Type) (x6:((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x6) x5)) (Xp (s (s Xu)))) (fun (x6:(Xp (s (s Xj)))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 Xu) ((x3 Xp) x5)))) as proof of (Xp (s (s Xu)))
% Found (fun (x5:((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x6:((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x6) x5)) (Xp (s (s Xu)))) (fun (x6:(Xp (s (s Xj)))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 Xu) ((x3 Xp) x5))))) as proof of (Xp (s (s Xu)))
% Found (fun (Xp:(fofType->Prop)) (x5:((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x6:((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x6) x5)) (Xp (s (s Xu)))) (fun (x6:(Xp (s (s Xj)))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 Xu) ((x3 Xp) x5))))) as proof of (((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))->(Xp (s (s Xu))))
% Found (fun (Xp:(fofType->Prop)) (x5:((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x6:((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x6) x5)) (Xp (s (s Xu)))) (fun (x6:(Xp (s (s Xj)))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 Xu) ((x3 Xp) x5))))) as proof of ((cCKB_E2 (s (s Xj))) (s (s Xu)))
% Found x400:=(x40 x5):(Xp (s Xu))
% Found (x40 x5) as proof of (Xp (s Xu))
% Found ((x4 Xp) x5) as proof of (Xp (s Xu))
% Found ((x4 Xp) x5) as proof of (Xp (s Xu))
% Found (x70 ((x4 Xp) x5)) as proof of (Xp (s (s (s Xu))))
% Found ((x7 (s Xu)) ((x4 Xp) x5)) as proof of (Xp (s (s (s Xu))))
% Found (fun (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 (s Xu)) ((x4 Xp) x5))) as proof of (Xp (s (s (s Xu))))
% Found (fun (x6:(Xp (s Xj))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 (s Xu)) ((x4 Xp) x5))) as proof of ((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp (s (s (s Xu)))))
% Found (fun (x6:(Xp (s Xj))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 (s Xu)) ((x4 Xp) x5))) as proof of ((Xp (s Xj))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp (s (s (s Xu))))))
% Found (and_rect10 (fun (x6:(Xp (s Xj))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 (s Xu)) ((x4 Xp) x5)))) as proof of (Xp (s (s (s Xu))))
% Found ((and_rect1 (Xp (s (s (s Xu))))) (fun (x6:(Xp (s Xj))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 (s Xu)) ((x4 Xp) x5)))) as proof of (Xp (s (s (s Xu))))
% Found (((fun (P:Type) (x6:((Xp (s Xj))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s Xj))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x6) x5)) (Xp (s (s (s Xu))))) (fun (x6:(Xp (s Xj))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 (s Xu)) ((x4 Xp) x5)))) as proof of (Xp (s (s (s Xu))))
% Found (fun (x5:((and (Xp (s Xj))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x6:((Xp (s Xj))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s Xj))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x6) x5)) (Xp (s (s (s Xu))))) (fun (x6:(Xp (s Xj))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 (s Xu)) ((x4 Xp) x5))))) as proof of (Xp (s (s (s Xu))))
% Found (fun (Xp:(fofType->Prop)) (x5:((and (Xp (s Xj))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x6:((Xp (s Xj))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s Xj))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x6) x5)) (Xp (s (s (s Xu))))) (fun (x6:(Xp (s Xj))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 (s Xu)) ((x4 Xp) x5))))) as proof of (((and (Xp (s Xj))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))->(Xp (s (s (s Xu)))))
% Found (fun (Xp:(fofType->Prop)) (x5:((and (Xp (s Xj))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x6:((Xp (s Xj))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s Xj))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x6) x5)) (Xp (s (s (s Xu))))) (fun (x6:(Xp (s Xj))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 (s Xu)) ((x4 Xp) x5))))) as proof of ((cCKB_E2 (s Xj)) (s (s (s Xu))))
% Found ((conj20 (fun (Xp:(fofType->Prop)) (x5:((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x6:((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x6) x5)) (Xp (s (s Xu)))) (fun (x6:(Xp (s (s Xj)))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 Xu) ((x3 Xp) x5)))))) (fun (Xp:(fofType->Prop)) (x5:((and (Xp (s Xj))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x6:((Xp (s Xj))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s Xj))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x6) x5)) (Xp (s (s (s Xu))))) (fun (x6:(Xp (s Xj))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 (s Xu)) ((x4 Xp) x5)))))) as proof of ((and ((cCKB_E2 (s (s Xj))) (s (s Xu)))) ((cCKB_E2 (s Xj)) (s (s (s Xu)))))
% Found (((conj2 ((cCKB_E2 (s Xj)) (s (s (s Xu))))) (fun (Xp:(fofType->Prop)) (x5:((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x6:((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x6) x5)) (Xp (s (s Xu)))) (fun (x6:(Xp (s (s Xj)))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 Xu) ((x3 Xp) x5)))))) (fun (Xp:(fofType->Prop)) (x5:((and (Xp (s Xj))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x6:((Xp (s Xj))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s Xj))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x6) x5)) (Xp (s (s (s Xu))))) (fun (x6:(Xp (s Xj))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 (s Xu)) ((x4 Xp) x5)))))) as proof of ((and ((cCKB_E2 (s (s Xj))) (s (s Xu)))) ((cCKB_E2 (s Xj)) (s (s (s Xu)))))
% Found ((((conj ((cCKB_E2 (s (s Xj))) (s (s Xu)))) ((cCKB_E2 (s Xj)) (s (s (s Xu))))) (fun (Xp:(fofType->Prop)) (x5:((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x6:((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x6) x5)) (Xp (s (s Xu)))) (fun (x6:(Xp (s (s Xj)))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 Xu) ((x3 Xp) x5)))))) (fun (Xp:(fofType->Prop)) (x5:((and (Xp (s Xj))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x6:((Xp (s Xj))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s Xj))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x6) x5)) (Xp (s (s (s Xu))))) (fun (x6:(Xp (s Xj))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 (s Xu)) ((x4 Xp) x5)))))) as proof of ((and ((cCKB_E2 (s (s Xj))) (s (s Xu)))) ((cCKB_E2 (s Xj)) (s (s (s Xu)))))
% Found (fun (x4:((cCKB_E2 (s Xj)) (s Xu)))=> ((((conj ((cCKB_E2 (s (s Xj))) (s (s Xu)))) ((cCKB_E2 (s Xj)) (s (s (s Xu))))) (fun (Xp:(fofType->Prop)) (x5:((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x6:((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x6) x5)) (Xp (s (s Xu)))) (fun (x6:(Xp (s (s Xj)))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 Xu) ((x3 Xp) x5)))))) (fun (Xp:(fofType->Prop)) (x5:((and (Xp (s Xj))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x6:((Xp (s Xj))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s Xj))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x6) x5)) (Xp (s (s (s Xu))))) (fun (x6:(Xp (s Xj))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 (s Xu)) ((x4 Xp) x5))))))) as proof of ((and ((cCKB_E2 (s (s Xj))) (s (s Xu)))) ((cCKB_E2 (s Xj)) (s (s (s Xu)))))
% Found (fun (x3:((cCKB_E2 (s (s Xj))) Xu)) (x4:((cCKB_E2 (s Xj)) (s Xu)))=> ((((conj ((cCKB_E2 (s (s Xj))) (s (s Xu)))) ((cCKB_E2 (s Xj)) (s (s (s Xu))))) (fun (Xp:(fofType->Prop)) (x5:((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x6:((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x6) x5)) (Xp (s (s Xu)))) (fun (x6:(Xp (s (s Xj)))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 Xu) ((x3 Xp) x5)))))) (fun (Xp:(fofType->Prop)) (x5:((and (Xp (s Xj))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x6:((Xp (s Xj))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s Xj))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x6) x5)) (Xp (s (s (s Xu))))) (fun (x6:(Xp (s Xj))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 (s Xu)) ((x4 Xp) x5))))))) as proof of (((cCKB_E2 (s Xj)) (s Xu))->((and ((cCKB_E2 (s (s Xj))) (s (s Xu)))) ((cCKB_E2 (s Xj)) (s (s (s Xu))))))
% Found (fun (x3:((cCKB_E2 (s (s Xj))) Xu)) (x4:((cCKB_E2 (s Xj)) (s Xu)))=> ((((conj ((cCKB_E2 (s (s Xj))) (s (s Xu)))) ((cCKB_E2 (s Xj)) (s (s (s Xu))))) (fun (Xp:(fofType->Prop)) (x5:((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x6:((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x6) x5)) (Xp (s (s Xu)))) (fun (x6:(Xp (s (s Xj)))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 Xu) ((x3 Xp) x5)))))) (fun (Xp:(fofType->Prop)) (x5:((and (Xp (s Xj))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x6:((Xp (s Xj))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s Xj))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x6) x5)) (Xp (s (s (s Xu))))) (fun (x6:(Xp (s Xj))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 (s Xu)) ((x4 Xp) x5))))))) as proof of (((cCKB_E2 (s (s Xj))) Xu)->(((cCKB_E2 (s Xj)) (s Xu))->((and ((cCKB_E2 (s (s Xj))) (s (s Xu)))) ((cCKB_E2 (s Xj)) (s (s (s Xu)))))))
% Found (and_rect00 (fun (x3:((cCKB_E2 (s (s Xj))) Xu)) (x4:((cCKB_E2 (s Xj)) (s Xu)))=> ((((conj ((cCKB_E2 (s (s Xj))) (s (s Xu)))) ((cCKB_E2 (s Xj)) (s (s (s Xu))))) (fun (Xp:(fofType->Prop)) (x5:((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x6:((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x6) x5)) (Xp (s (s Xu)))) (fun (x6:(Xp (s (s Xj)))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 Xu) ((x3 Xp) x5)))))) (fun (Xp:(fofType->Prop)) (x5:((and (Xp (s Xj))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x6:((Xp (s Xj))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s Xj))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x6) x5)) (Xp (s (s (s Xu))))) (fun (x6:(Xp (s Xj))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 (s Xu)) ((x4 Xp) x5)))))))) as proof of ((and ((cCKB_E2 (s (s Xj))) (s (s Xu)))) ((cCKB_E2 (s Xj)) (s (s (s Xu)))))
% Found ((and_rect0 ((and ((cCKB_E2 (s (s Xj))) (s (s Xu)))) ((cCKB_E2 (s Xj)) (s (s (s Xu)))))) (fun (x3:((cCKB_E2 (s (s Xj))) Xu)) (x4:((cCKB_E2 (s Xj)) (s Xu)))=> ((((conj ((cCKB_E2 (s (s Xj))) (s (s Xu)))) ((cCKB_E2 (s Xj)) (s (s (s Xu))))) (fun (Xp:(fofType->Prop)) (x5:((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x6:((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x6) x5)) (Xp (s (s Xu)))) (fun (x6:(Xp (s (s Xj)))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 Xu) ((x3 Xp) x5)))))) (fun (Xp:(fofType->Prop)) (x5:((and (Xp (s Xj))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x6:((Xp (s Xj))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s Xj))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x6) x5)) (Xp (s (s (s Xu))))) (fun (x6:(Xp (s Xj))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 (s Xu)) ((x4 Xp) x5)))))))) as proof of ((and ((cCKB_E2 (s (s Xj))) (s (s Xu)))) ((cCKB_E2 (s Xj)) (s (s (s Xu)))))
% Found (((fun (P:Type) (x3:(((cCKB_E2 (s (s Xj))) Xu)->(((cCKB_E2 (s Xj)) (s Xu))->P)))=> (((((and_rect ((cCKB_E2 (s (s Xj))) Xu)) ((cCKB_E2 (s Xj)) (s Xu))) P) x3) x2)) ((and ((cCKB_E2 (s (s Xj))) (s (s Xu)))) ((cCKB_E2 (s Xj)) (s (s (s Xu)))))) (fun (x3:((cCKB_E2 (s (s Xj))) Xu)) (x4:((cCKB_E2 (s Xj)) (s Xu)))=> ((((conj ((cCKB_E2 (s (s Xj))) (s (s Xu)))) ((cCKB_E2 (s Xj)) (s (s (s Xu))))) (fun (Xp:(fofType->Prop)) (x5:((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x6:((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x6) x5)) (Xp (s (s Xu)))) (fun (x6:(Xp (s (s Xj)))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 Xu) ((x3 Xp) x5)))))) (fun (Xp:(fofType->Prop)) (x5:((and (Xp (s Xj))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x6:((Xp (s Xj))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s Xj))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x6) x5)) (Xp (s (s (s Xu))))) (fun (x6:(Xp (s Xj))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 (s Xu)) ((x4 Xp) x5)))))))) as proof of ((and ((cCKB_E2 (s (s Xj))) (s (s Xu)))) ((cCKB_E2 (s Xj)) (s (s (s Xu)))))
% Found (fun (x2:((and ((cCKB_E2 (s (s Xj))) Xu)) ((cCKB_E2 (s Xj)) (s Xu))))=> (((fun (P:Type) (x3:(((cCKB_E2 (s (s Xj))) Xu)->(((cCKB_E2 (s Xj)) (s Xu))->P)))=> (((((and_rect ((cCKB_E2 (s (s Xj))) Xu)) ((cCKB_E2 (s Xj)) (s Xu))) P) x3) x2)) ((and ((cCKB_E2 (s (s Xj))) (s (s Xu)))) ((cCKB_E2 (s Xj)) (s (s (s Xu)))))) (fun (x3:((cCKB_E2 (s (s Xj))) Xu)) (x4:((cCKB_E2 (s Xj)) (s Xu)))=> ((((conj ((cCKB_E2 (s (s Xj))) (s (s Xu)))) ((cCKB_E2 (s Xj)) (s (s (s Xu))))) (fun (Xp:(fofType->Prop)) (x5:((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x6:((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x6) x5)) (Xp (s (s Xu)))) (fun (x6:(Xp (s (s Xj)))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 Xu) ((x3 Xp) x5)))))) (fun (Xp:(fofType->Prop)) (x5:((and (Xp (s Xj))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x6:((Xp (s Xj))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s Xj))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x6) x5)) (Xp (s (s (s Xu))))) (fun (x6:(Xp (s Xj))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 (s Xu)) ((x4 Xp) x5))))))))) as proof of ((and ((cCKB_E2 (s (s Xj))) (s (s Xu)))) ((cCKB_E2 (s Xj)) (s (s (s Xu)))))
% Found (fun (Xu:fofType) (x2:((and ((cCKB_E2 (s (s Xj))) Xu)) ((cCKB_E2 (s Xj)) (s Xu))))=> (((fun (P:Type) (x3:(((cCKB_E2 (s (s Xj))) Xu)->(((cCKB_E2 (s Xj)) (s Xu))->P)))=> (((((and_rect ((cCKB_E2 (s (s Xj))) Xu)) ((cCKB_E2 (s Xj)) (s Xu))) P) x3) x2)) ((and ((cCKB_E2 (s (s Xj))) (s (s Xu)))) ((cCKB_E2 (s Xj)) (s (s (s Xu)))))) (fun (x3:((cCKB_E2 (s (s Xj))) Xu)) (x4:((cCKB_E2 (s Xj)) (s Xu)))=> ((((conj ((cCKB_E2 (s (s Xj))) (s (s Xu)))) ((cCKB_E2 (s Xj)) (s (s (s Xu))))) (fun (Xp:(fofType->Prop)) (x5:((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x6:((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x6) x5)) (Xp (s (s Xu)))) (fun (x6:(Xp (s (s Xj)))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 Xu) ((x3 Xp) x5)))))) (fun (Xp:(fofType->Prop)) (x5:((and (Xp (s Xj))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x6:((Xp (s Xj))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s Xj))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x6) x5)) (Xp (s (s (s Xu))))) (fun (x6:(Xp (s Xj))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 (s Xu)) ((x4 Xp) x5))))))))) as proof of (((and ((cCKB_E2 (s (s Xj))) Xu)) ((cCKB_E2 (s Xj)) (s Xu)))->((and ((cCKB_E2 (s (s Xj))) (s (s Xu)))) ((cCKB_E2 (s Xj)) (s (s (s Xu))))))
% Found (fun (Xu:fofType) (x2:((and ((cCKB_E2 (s (s Xj))) Xu)) ((cCKB_E2 (s Xj)) (s Xu))))=> (((fun (P:Type) (x3:(((cCKB_E2 (s (s Xj))) Xu)->(((cCKB_E2 (s Xj)) (s Xu))->P)))=> (((((and_rect ((cCKB_E2 (s (s Xj))) Xu)) ((cCKB_E2 (s Xj)) (s Xu))) P) x3) x2)) ((and ((cCKB_E2 (s (s Xj))) (s (s Xu)))) ((cCKB_E2 (s Xj)) (s (s (s Xu)))))) (fun (x3:((cCKB_E2 (s (s Xj))) Xu)) (x4:((cCKB_E2 (s Xj)) (s Xu)))=> ((((conj ((cCKB_E2 (s (s Xj))) (s (s Xu)))) ((cCKB_E2 (s Xj)) (s (s (s Xu))))) (fun (Xp:(fofType->Prop)) (x5:((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x6:((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x6) x5)) (Xp (s (s Xu)))) (fun (x6:(Xp (s (s Xj)))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 Xu) ((x3 Xp) x5)))))) (fun (Xp:(fofType->Prop)) (x5:((and (Xp (s Xj))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x6:((Xp (s Xj))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s Xj))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x6) x5)) (Xp (s (s (s Xu))))) (fun (x6:(Xp (s Xj))) (x7:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x7 (s Xu)) ((x4 Xp) x5))))))))) as proof of (forall (Xu:fofType), (((and ((cCKB_E2 (s (s Xj))) Xu)) ((cCKB_E2 (s Xj)) (s Xu)))->((and ((cCKB_E2 (s (s Xj))) (s (s Xu)))) ((cCKB_E2 (s Xj)) (s (s (s Xu)))))))
% Found x500:=(x50 x3):(Xp (s (s Xu)))
% Found (x50 x3) as proof of (Xp (s (s Xu)))
% Found ((x5 Xu) x3) as proof of (Xp (s (s Xu)))
% Found (fun (x5:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x5 Xu) x3)) as proof of (Xp (s (s Xu)))
% Found (fun (x4:(Xp (s (s Xj)))) (x5:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x5 Xu) x3)) as proof of ((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp (s (s Xu))))
% Found (fun (x4:(Xp (s (s Xj)))) (x5:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x5 Xu) x3)) as proof of ((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp (s (s Xu)))))
% Found (and_rect00 (fun (x4:(Xp (s (s Xj)))) (x5:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x5 Xu) x3))) as proof of (Xp (s (s Xu)))
% Found ((and_rect0 (Xp (s (s Xu)))) (fun (x4:(Xp (s (s Xj)))) (x5:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x5 Xu) x3))) as proof of (Xp (s (s Xu)))
% Found (((fun (P:Type) (x4:((Xp (s (s Xj)))->((forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))) P) x4) x2)) (Xp (s (s Xu)))) (fun (x4:(Xp (s (s Xj)))) (x5:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x5 Xu) x3))) as proof of (Xp (s (s Xu)))
% Found (fun (x3:(Xp Xu))=> (((fun (P:Type) (x4:((Xp (s (s Xj)))->((forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))) P) x4) x2)) (Xp (s (s Xu)))) (fun (x4:(Xp (s (s Xj)))) (x5:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x5 Xu) x3)))) as proof of (Xp (s (s Xu)))
% Found (fun (Xu:fofType) (x3:(Xp Xu))=> (((fun (P:Type) (x4:((Xp (s (s Xj)))->((forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))) P) x4) x2)) (Xp (s (s Xu)))) (fun (x4:(Xp (s (s Xj)))) (x5:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x5 Xu) x3)))) as proof of ((Xp Xu)->(Xp (s (s Xu))))
% Found (fun (Xu:fofType) (x3:(Xp Xu))=> (((fun (P:Type) (x4:((Xp (s (s Xj)))->((forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))) P) x4) x2)) (Xp (s (s Xu)))) (fun (x4:(Xp (s (s Xj)))) (x5:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x5 Xu) x3)))) as proof of (forall (Xu:fofType), ((Xp Xu)->(Xp (s (s Xu)))))
% Found x310:=(x31 x40):(Xp Xu)
% Found (x31 x40) as proof of (Xp Xu)
% Found ((x3 x30) x40) as proof of (Xp Xu)
% Found ((x3 x30) x40) as proof of (Xp Xu)
% Found (x400 ((x3 x30) x40)) as proof of (Xp (s (s Xu)))
% Found ((x40 Xu) ((x3 x30) x40)) as proof of (Xp (s (s Xu)))
% Found (fun (x40:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x40 Xu) ((x3 x30) x40))) as proof of (Xp (s (s Xu)))
% Found (fun (x30:(Xp (s (s Xj)))) (x40:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x40 Xu) ((x3 x30) x40))) as proof of ((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp (s (s Xu))))
% Found (fun (x3:((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp Xu)))) (x30:(Xp (s (s Xj)))) (x40:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x40 Xu) ((x3 x30) x40))) as proof of ((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp (s (s Xu)))))
% Found (fun (Xu:fofType) (x3:((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp Xu)))) (x30:(Xp (s (s Xj)))) (x40:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x40 Xu) ((x3 x30) x40))) as proof of (((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp Xu)))->((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp (s (s Xu))))))
% Found (fun (Xu:fofType) (x3:((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp Xu)))) (x30:(Xp (s (s Xj)))) (x40:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x40 Xu) ((x3 x30) x40))) as proof of (forall (Xu:fofType), (((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp Xu)))->((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp (s (s Xu)))))))
% Found x21:=(x2 x20):(Xp Xu)
% Found (x2 x20) as proof of (Xp Xu)
% Found (x2 x20) as proof of (Xp Xu)
% Found (x40 (x2 x20)) as proof of (Xp (s (s Xu)))
% Found ((x4 Xu) (x2 x20)) as proof of (Xp (s (s Xu)))
% Found (fun (x4:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x4 Xu) (x2 x20))) as proof of (Xp (s (s Xu)))
% Found (fun (x3:(Xp (s (s Xj)))) (x4:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x4 Xu) (x2 x20))) as proof of ((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp (s (s Xu))))
% Found (fun (x3:(Xp (s (s Xj)))) (x4:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x4 Xu) (x2 x20))) as proof of ((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->(Xp (s (s Xu)))))
% Found (and_rect00 (fun (x3:(Xp (s (s Xj)))) (x4:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x4 Xu) (x2 x20)))) as proof of (Xp (s (s Xu)))
% Found ((and_rect0 (Xp (s (s Xu)))) (fun (x3:(Xp (s (s Xj)))) (x4:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x4 Xu) (x2 x20)))) as proof of (Xp (s (s Xu)))
% Found (((fun (P:Type) (x3:((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x3) x20)) (Xp (s (s Xu)))) (fun (x3:(Xp (s (s Xj)))) (x4:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x4 Xu) (x2 x20)))) as proof of (Xp (s (s Xu)))
% Found (fun (x20:((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x3:((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x3) x20)) (Xp (s (s Xu)))) (fun (x3:(Xp (s (s Xj)))) (x4:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x4 Xu) (x2 x20))))) as proof of (Xp (s (s Xu)))
% Found (fun (x2:(((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))->(Xp Xu))) (x20:((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x3:((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x3) x20)) (Xp (s (s Xu)))) (fun (x3:(Xp (s (s Xj)))) (x4:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x4 Xu) (x2 x20))))) as proof of (((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))->(Xp (s (s Xu))))
% Found (fun (Xu:fofType) (x2:(((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))->(Xp Xu))) (x20:((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x3:((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x3) x20)) (Xp (s (s Xu)))) (fun (x3:(Xp (s (s Xj)))) (x4:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x4 Xu) (x2 x20))))) as proof of ((((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))->(Xp Xu))->(((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))->(Xp (s (s Xu)))))
% Found (fun (Xu:fofType) (x2:(((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))->(Xp Xu))) (x20:((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))))=> (((fun (P:Type) (x3:((Xp (s (s Xj)))->((forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))->P)))=> (((((and_rect (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0)))))) P) x3) x20)) (Xp (s (s Xu)))) (fun (x3:(Xp (s (s Xj)))) (x4:(forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))=> ((x4 Xu) (x2 x20))))) as proof of (forall (Xu:fofType), ((((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))->(Xp Xu))->(((and (Xp (s (s Xj)))) (forall (Xu0:fofType), ((Xp Xu0)->(Xp (s (s Xu0))))))->(Xp (s (s Xu))))))
% Found x210:=(x21 x20):(Xp0 Xu)
% Found (x21 x20) as proof of (Xp0 Xu)
% Found ((x2 Xp0) x20) as proof of (Xp0 Xu)
% Found ((x2 Xp0) x20) as proof of (Xp0 Xu)
% Found (x40 ((x2 Xp0) x20)) as proof of (Xp0 (s (s Xu)))
% Found ((x4 Xu) ((x2 Xp0) x20)) as proof of (Xp0 (s (s Xu)))
% Found (fun (x4:(forall (Xu0:fofType), ((Xp0 Xu0)->(Xp0 (s (s Xu0))))))=> ((x4 Xu) ((x2 Xp0) x20))) as proof of (Xp0 (s (s Xu)))
% Found (fun (x3:(Xp0 (s (s Xj)))) (x4:(forall (Xu0:fofType), ((Xp0 Xu0)->(Xp0 (s (s Xu0))))))=> ((x4 Xu) ((x2 Xp0) x20))) as proof of ((forall (Xu0:fofType), ((Xp0 Xu0)->(Xp0 (s (s Xu0)))))->(Xp0 (s (s Xu))))
% Found (fun (x3:(Xp0 (s (s Xj)))) (x4:(forall (Xu0:fofType), ((Xp0 Xu0)->(Xp0 (s (s Xu0))))))=> ((x4 Xu) ((x2 Xp0) x20))) as proof of ((Xp0 (s (s Xj)))->((forall (Xu0:fofType), ((Xp0 Xu0)->(Xp0 (s (s Xu0)))))->(Xp0 (s (s Xu)))))
% Found (and_rect00 (fun (x3:(Xp0 (s (s Xj)))) (x4:(forall (Xu0:fofType), ((Xp0 Xu0)->(Xp0 (s (s Xu0))))))=> ((x4 Xu) ((x2 Xp0) x20)))) as proof of (Xp0 (s (s Xu)))
% Found ((and_rect0 (Xp0 (s (s Xu)))) (fun (x3:(Xp0 (s (s Xj)))) (x4:(forall (Xu0:fofType), ((Xp0 Xu0)->(Xp0 (s (s Xu0))))))=> ((x4 Xu) ((x2 Xp0) x20)))) as proof of (Xp0 (s (s Xu)))
% Found (((fun (P:Type) (x3:((Xp0 (s (s Xj)))->((forall (Xu0:fofType), ((Xp0 Xu0)->(Xp0 (s (s Xu0)))))->P)))=> (((((and_rect (Xp0 (s (s Xj)))) (forall (Xu0:fofType), ((Xp0 Xu0)->(Xp0 (s (s Xu0)))))) P) x3) x20)) (Xp0 (s (s Xu)))) (fun (x3:(Xp0 (s (s Xj)))) (x4:(forall (Xu0:fofType), ((Xp0 Xu0)->(Xp0 (s (s Xu0))))))=> ((x4 Xu) ((x2 Xp0) x20)))) as proof of (Xp0 (s (s Xu)))
% Found (fun (x20:((and (Xp0 (s (s Xj)))) (forall (Xu0:fofType), ((Xp0 Xu0)->(Xp0 (s (s Xu0)))))))=> (((fun (P:Type) (x3:((Xp0 (s (s Xj)))->((forall (Xu0:fofType), ((Xp0 Xu0)->(Xp0 (s (s Xu0)))))->P)))=> (((((and_rect (Xp0 (s (s Xj)))) (forall (Xu0:fofType), ((Xp0 Xu0)->(Xp0 (s (s Xu0)))))) P) x3) x20)) (Xp0 (s (s Xu)))) (fun (x3:(Xp0 (s (s Xj)))) (x4:(forall (Xu0:fofType), ((Xp0 Xu0)->(Xp0 (s (s Xu0))))))=> ((x4 Xu) ((x2 Xp0) x20))))) as proof of (Xp0 (s (s Xu)))
% Found (fun (Xp0:(fofType->Prop)) (x20:((and (Xp0 (s (s Xj)))) (forall (Xu0:fofType), ((Xp0 Xu0)->(Xp0 (s (s Xu0)))))))=> (((fun (P:Type) (x3:((Xp0 (s (s Xj)))->((forall (Xu0:fofType), ((Xp0 Xu0)->(Xp0 (s (s Xu0)))))->P)))=> (((((and_rect (Xp0 (s (s Xj)))) (forall (Xu0:fofType), ((Xp0 Xu0)->(Xp0 (s (s Xu0)))))) P) x3) x20)) (Xp0 (s (s Xu)))) (fun (x3:(Xp0 (s (s Xj)))) (x4:(forall (Xu0:fofType), ((Xp0 Xu0)->(Xp0 (s (s Xu0))))))=> ((x4 Xu) ((x2 Xp0) x20))))) as proof of (((and (Xp0 (s (s Xj)))) (forall (Xu0:fofType), ((Xp0 Xu0)->(Xp0 (s (s Xu0))))))->(Xp0 (s (s Xu))))
% Found (fun (x2:(forall (Xp0:(fofType->Prop)), (((and (Xp0 (s (s Xj)))) (forall (Xu0:fofType), ((Xp0 Xu0)->(Xp0 (s (s Xu0))))))->(Xp0 Xu)))) (Xp0:(fofType->Prop)) (x20:((and (Xp0 (s (s Xj)))) (forall (Xu0:fofType), ((Xp0 Xu0)->(Xp0 (s (s Xu0)))))))=> (((fun (P:Type) (x3:((Xp0 (s (s Xj)))->((forall (Xu0:fofType), ((Xp0 Xu0)->(Xp0 (s (s Xu0)))))->P)))=> (((((and_rect (Xp0 (s (s Xj)))) (forall (Xu0:fofType), ((Xp0 Xu0)->(Xp0 (s (s Xu0)))))) P) x3) x20)) (Xp0 (s (s Xu)))) (fun (x3:(Xp0 (s (s Xj)))) (x4:(forall (Xu0:fofType), ((Xp0 Xu0)->(Xp0 (s (s Xu0))))))=> ((x4 Xu) ((x2 Xp0) x20))))) as proof of (forall (Xp0:(fofType->Prop)), (((and (Xp0 (s (s Xj)))) (forall (Xu0:fofType), ((Xp0 Xu0)->(Xp0 (s (s Xu0))))))->(Xp0 (s (s Xu)))))
% Found (fun (Xu:fofType) (x2:(forall (Xp0:(fofType->Prop)), (((and (Xp0 (s (s Xj)))) (forall (Xu0:fofType), ((Xp0 Xu0)->(Xp0 (s (s Xu0))))))->(Xp0 Xu)))) (Xp0:(fofType->Prop)) (x20:((and (Xp0 (s (s Xj)))) (forall (Xu0:fofType), ((Xp0 Xu0)->(Xp0 (s (s Xu0)))))))=> (((fun (P:Type) (x3:((Xp0 (s (s Xj)))->((forall (Xu0:fofType), ((Xp0 Xu0)->(Xp0 (s (s Xu0)))))->P)))=> (((((and_rect (Xp0 (s (s Xj)))) (forall (Xu0:fofType), ((Xp0 Xu0)->(Xp0 (s (s Xu0)))))) P) x3) x20)) (Xp0 (s (s Xu)))) (fun (x3
% EOF
%------------------------------------------------------------------------------