TSTP Solution File: SEV295^5 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEV295^5 : TPTP v6.2.0. Bugfixed v6.2.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n128.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 2.6.32-504.8.1.el6.x86_64
% CPULimit : 300s
% DateTime : Wed May 6 14:27:24 EDT 2015
% Result : Timeout 296.70s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----No solution output by system
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : SEV295^5 : TPTP v6.2.0. Bugfixed v6.2.0.
% 0.00/0.03 % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.01/1.07 % Computer : n128.star.cs.uiowa.edu
% 0.01/1.07 % Model : x86_64 x86_64
% 0.01/1.07 % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% 0.01/1.07 % Memory : 32286.75MB
% 0.01/1.07 % OS : Linux 2.6.32-504.8.1.el6.x86_64
% 0.01/1.07 % CPULimit : 300
% 0.01/1.07 % DateTime : Thu Apr 16 12:14:57 CDT 2015
% 0.01/1.07 % CPUTime :
% 0.01/1.09 Python 2.7.5
% 0.28/1.41 Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% 0.28/1.41 FOF formula (<kernel.Constant object at 0x2267440>, <kernel.DependentProduct object at 0x2267dd0>) of role type named r_type
% 0.28/1.41 Using role type
% 0.28/1.41 Declaring r:(((fofType->Prop)->Prop)->(((fofType->Prop)->Prop)->Prop))
% 0.28/1.41 FOF formula (<kernel.Constant object at 0x22488c0>, <kernel.Sort object at 0x2174b00>) of role type named cINDUCTION_type
% 0.28/1.41 Using role type
% 0.28/1.41 Declaring cINDUCTION:Prop
% 0.28/1.41 FOF formula (<kernel.Constant object at 0x22671b8>, <kernel.DependentProduct object at 0x2267fc8>) of role type named cNAT_type
% 0.28/1.41 Using role type
% 0.28/1.41 Declaring cNAT:(((fofType->Prop)->Prop)->Prop)
% 0.28/1.41 FOF formula (<kernel.Constant object at 0x22678c0>, <kernel.DependentProduct object at 0x2267560>) of role type named cSUCC_type
% 0.28/1.41 Using role type
% 0.28/1.41 Declaring cSUCC:(((fofType->Prop)->Prop)->((fofType->Prop)->Prop))
% 0.28/1.41 FOF formula (<kernel.Constant object at 0x2267d88>, <kernel.DependentProduct object at 0x2267fc8>) of role type named cZERO_type
% 0.28/1.41 Using role type
% 0.28/1.41 Declaring cZERO:((fofType->Prop)->Prop)
% 0.28/1.41 FOF formula (((eq ((fofType->Prop)->Prop)) cZERO) (fun (Xp:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp Xx)))->False))) of role definition named cZERO_def
% 0.28/1.41 A new definition: (((eq ((fofType->Prop)->Prop)) cZERO) (fun (Xp:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp Xx)))->False)))
% 0.28/1.41 Defined: cZERO:=(fun (Xp:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp Xx)))->False))
% 0.28/1.41 FOF formula (((eq (((fofType->Prop)->Prop)->((fofType->Prop)->Prop))) cSUCC) (fun (Xn:((fofType->Prop)->Prop)) (Xp:(fofType->Prop))=> ((ex fofType) (fun (Xx:fofType)=> ((and (Xp Xx)) (Xn (fun (Xt:fofType)=> ((and (not (((eq fofType) Xt) Xx))) (Xp Xt))))))))) of role definition named cSUCC_def
% 0.28/1.41 A new definition: (((eq (((fofType->Prop)->Prop)->((fofType->Prop)->Prop))) cSUCC) (fun (Xn:((fofType->Prop)->Prop)) (Xp:(fofType->Prop))=> ((ex fofType) (fun (Xx:fofType)=> ((and (Xp Xx)) (Xn (fun (Xt:fofType)=> ((and (not (((eq fofType) Xt) Xx))) (Xp Xt)))))))))
% 0.28/1.41 Defined: cSUCC:=(fun (Xn:((fofType->Prop)->Prop)) (Xp:(fofType->Prop))=> ((ex fofType) (fun (Xx:fofType)=> ((and (Xp Xx)) (Xn (fun (Xt:fofType)=> ((and (not (((eq fofType) Xt) Xx))) (Xp Xt))))))))
% 0.28/1.41 FOF formula (((eq (((fofType->Prop)->Prop)->Prop)) cNAT) (fun (Xn:((fofType->Prop)->Prop))=> (forall (Xp:(((fofType->Prop)->Prop)->Prop)), (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp Xn))))) of role definition named cNAT_def
% 0.28/1.41 A new definition: (((eq (((fofType->Prop)->Prop)->Prop)) cNAT) (fun (Xn:((fofType->Prop)->Prop))=> (forall (Xp:(((fofType->Prop)->Prop)->Prop)), (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp Xn)))))
% 0.28/1.41 Defined: cNAT:=(fun (Xn:((fofType->Prop)->Prop))=> (forall (Xp:(((fofType->Prop)->Prop)->Prop)), (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp Xn))))
% 0.28/1.41 FOF formula (((eq Prop) cINDUCTION) (forall (P:(((fofType->Prop)->Prop)->Prop)), (((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))->(forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M)))))) of role definition named cINDUCTION_def
% 0.28/1.41 A new definition: (((eq Prop) cINDUCTION) (forall (P:(((fofType->Prop)->Prop)->Prop)), (((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))->(forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M))))))
% 0.28/1.41 Defined: cINDUCTION:=(forall (P:(((fofType->Prop)->Prop)->Prop)), (((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))->(forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M)))))
% 0.28/1.41 FOF formula (((and ((and cINDUCTION) ((r cZERO) cZERO))) (forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy)))))->(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r Xx) Xy)))))) of role conjecture named cTHM130_NAT
% 0.28/1.41 Conjecture to prove = (((and ((and cINDUCTION) ((r cZERO) cZERO))) (forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy)))))->(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r Xx) Xy)))))):Prop
% 8.89/10.02 Parameter fofType_DUMMY:fofType.
% 8.89/10.02 We need to prove ['(((and ((and cINDUCTION) ((r cZERO) cZERO))) (forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy)))))->(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r Xx) Xy))))))']
% 8.89/10.02 Parameter fofType:Type.
% 8.89/10.02 Parameter r:(((fofType->Prop)->Prop)->(((fofType->Prop)->Prop)->Prop)).
% 8.89/10.02 Definition cINDUCTION:=(forall (P:(((fofType->Prop)->Prop)->Prop)), (((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))->(forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M))))):Prop.
% 8.89/10.02 Definition cNAT:=(fun (Xn:((fofType->Prop)->Prop))=> (forall (Xp:(((fofType->Prop)->Prop)->Prop)), (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp Xn)))):(((fofType->Prop)->Prop)->Prop).
% 8.89/10.02 Definition cSUCC:=(fun (Xn:((fofType->Prop)->Prop)) (Xp:(fofType->Prop))=> ((ex fofType) (fun (Xx:fofType)=> ((and (Xp Xx)) (Xn (fun (Xt:fofType)=> ((and (not (((eq fofType) Xt) Xx))) (Xp Xt)))))))):(((fofType->Prop)->Prop)->((fofType->Prop)->Prop)).
% 8.89/10.02 Definition cZERO:=(fun (Xp:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp Xx)))->False)):((fofType->Prop)->Prop).
% 8.89/10.02 Trying to prove (((and ((and cINDUCTION) ((r cZERO) cZERO))) (forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy)))))->(forall (Xx:((fofType->Prop)->Prop)), ((cNAT Xx)->((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r Xx) Xy))))))
% 8.89/10.02 Found x4:((r cZERO) cZERO)
% 8.89/10.02 Instantiate: x5:=cZERO:((fofType->Prop)->Prop)
% 8.89/10.02 Found x4 as proof of ((r cZERO) x5)
% 8.89/10.02 Found x5:((r cZERO) cZERO)
% 8.89/10.02 Instantiate: x1:=cZERO:((fofType->Prop)->Prop)
% 8.89/10.02 Found x5 as proof of ((r cZERO) x1)
% 8.89/10.02 Found x5:((r cZERO) cZERO)
% 8.89/10.02 Instantiate: x3:=cZERO:((fofType->Prop)->Prop)
% 8.89/10.02 Found x5 as proof of ((r cZERO) x3)
% 8.89/10.02 Found x5:((r cZERO) cZERO)
% 8.89/10.02 Instantiate: x1:=cZERO:((fofType->Prop)->Prop)
% 8.89/10.02 Found x5 as proof of ((r cZERO) x1)
% 8.89/10.02 Found x5:((r cZERO) cZERO)
% 8.89/10.02 Instantiate: x1:=cZERO:((fofType->Prop)->Prop)
% 8.89/10.02 Found x5 as proof of ((r cZERO) x1)
% 8.89/10.02 Found x5:((r cZERO) cZERO)
% 8.89/10.02 Instantiate: x3:=cZERO:((fofType->Prop)->Prop)
% 8.89/10.02 Found x5 as proof of ((r cZERO) x3)
% 8.89/10.02 Found x50:((r cZERO) cZERO)
% 8.89/10.02 Instantiate: x1:=cZERO:((fofType->Prop)->Prop)
% 8.89/10.02 Found (fun (x50:((r cZERO) cZERO))=> x50) as proof of ((r cZERO) x1)
% 8.89/10.02 Found (fun (x50:((r cZERO) cZERO))=> x50) as proof of (((r cZERO) cZERO)->((r cZERO) x1))
% 8.89/10.02 Found x50:((r cZERO) cZERO)
% 8.89/10.02 Instantiate: x3:=cZERO:((fofType->Prop)->Prop)
% 8.89/10.02 Found (fun (x50:((r cZERO) cZERO))=> x50) as proof of ((r cZERO) x3)
% 8.89/10.02 Found (fun (x50:((r cZERO) cZERO))=> x50) as proof of (((r cZERO) cZERO)->((r cZERO) x3))
% 8.89/10.02 Found x4:((r cZERO) cZERO)
% 8.89/10.02 Instantiate: x1:=cZERO:((fofType->Prop)->Prop)
% 8.89/10.02 Found (fun (x30:(forall (Xx0:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx0) Xy)->((r (cSUCC Xx0)) (cSUCC Xy)))))=> x4) as proof of ((r cZERO) x1)
% 8.89/10.02 Found (fun (x30:(forall (Xx0:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx0) Xy)->((r (cSUCC Xx0)) (cSUCC Xy)))))=> x4) as proof of ((forall (Xx0:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx0) Xy)->((r (cSUCC Xx0)) (cSUCC Xy))))->((r cZERO) x1))
% 8.89/10.02 Found x50:((r cZERO) cZERO)
% 8.89/10.02 Instantiate: x1:=cZERO:((fofType->Prop)->Prop)
% 8.89/10.02 Found (fun (x50:((r cZERO) cZERO))=> x50) as proof of ((r cZERO) x1)
% 8.89/10.02 Found (fun (x40:cINDUCTION) (x50:((r cZERO) cZERO))=> x50) as proof of (((r cZERO) cZERO)->((r cZERO) x1))
% 8.89/10.02 Found (fun (x40:cINDUCTION) (x50:((r cZERO) cZERO))=> x50) as proof of (cINDUCTION->(((r cZERO) cZERO)->((r cZERO) x1)))
% 8.89/10.02 Found x50:((r cZERO) cZERO)
% 8.89/10.02 Instantiate: x3:=cZERO:((fofType->Prop)->Prop)
% 8.89/10.02 Found (fun (x50:((r cZERO) cZERO))=> x50) as proof of ((r cZERO) x3)
% 8.89/10.02 Found (fun (x40:cINDUCTION) (x50:((r cZERO) cZERO))=> x50) as proof of (((r cZERO) cZERO)->((r cZERO) x3))
% 8.89/10.02 Found (fun (x40:cINDUCTION) (x50:((r cZERO) cZERO))=> x50) as proof of (cINDUCTION->(((r cZERO) cZERO)->((r cZERO) x3)))
% 10.78/11.95 Found ex_intro0000:=(ex_intro000 x4):((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 10.78/11.95 Found (ex_intro000 x4) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 10.78/11.95 Found ((ex_intro00 cZERO) x4) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 10.78/11.95 Found (((ex_intro0 (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) x4) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 10.78/11.95 Found ((((ex_intro ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) x4) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 10.78/11.95 Found ((((ex_intro ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) x4) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 10.78/11.95 Found ex_intro0000:=(ex_intro000 x4):((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 10.78/11.95 Found (ex_intro000 x4) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 10.78/11.95 Found ((ex_intro00 cZERO) x4) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 10.78/11.95 Found (((ex_intro0 (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) x4) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 10.78/11.95 Found ((((ex_intro ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) x4) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 10.78/11.95 Found ((((ex_intro ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) x4) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 10.78/11.95 Found ex_intro0000:=(ex_intro000 x4):((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 10.78/11.95 Found (ex_intro000 x4) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 10.78/11.95 Found ((ex_intro00 cZERO) x4) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 10.78/11.95 Found (((ex_intro0 (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) x4) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 10.78/11.95 Found ((((ex_intro ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) x4) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 10.78/11.95 Found ((((ex_intro ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) x4) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 10.78/11.95 Found ex_intro0000:=(ex_intro000 x40):((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 10.78/11.95 Found (ex_intro000 x40) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 10.78/11.95 Found ((ex_intro00 cZERO) x40) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 10.78/11.95 Found (((ex_intro0 (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) x40) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 10.78/11.95 Found ((((ex_intro ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) x40) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 10.78/11.95 Found (fun (x40:((r cZERO) cZERO))=> ((((ex_intro ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) x40)) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 10.78/11.95 Found (fun (x40:((r cZERO) cZERO))=> ((((ex_intro ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) x40)) as proof of (((r cZERO) cZERO)->((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))))
% 12.59/13.70 Found ex_intro0000:=(ex_intro000 x3):((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 12.59/13.70 Found (ex_intro000 x3) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 12.59/13.70 Found ((ex_intro00 cZERO) x3) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 12.59/13.70 Found (((ex_intro0 (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) x3) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 12.59/13.70 Found ((((ex_intro ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) x3) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 12.59/13.70 Found (fun (x20:(forall (Xx0:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx0) Xy)->((r (cSUCC Xx0)) (cSUCC Xy)))))=> ((((ex_intro ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) x3)) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 12.59/13.70 Found (fun (x20:(forall (Xx0:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx0) Xy)->((r (cSUCC Xx0)) (cSUCC Xy)))))=> ((((ex_intro ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) x3)) as proof of ((forall (Xx0:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx0) Xy)->((r (cSUCC Xx0)) (cSUCC Xy))))->((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))))
% 12.59/13.70 Found ex_intro0000:=(ex_intro000 x40):((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 12.59/13.70 Found (ex_intro000 x40) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 12.59/13.70 Found ((ex_intro00 cZERO) x40) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 12.59/13.70 Found (((ex_intro0 (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) x40) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 12.59/13.70 Found ((((ex_intro ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) x40) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 12.59/13.70 Found (fun (x40:((r cZERO) cZERO))=> ((((ex_intro ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) x40)) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 12.59/13.70 Found (fun (x30:cINDUCTION) (x40:((r cZERO) cZERO))=> ((((ex_intro ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) x40)) as proof of (((r cZERO) cZERO)->((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))))
% 12.59/13.70 Found (fun (x30:cINDUCTION) (x40:((r cZERO) cZERO))=> ((((ex_intro ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) x40)) as proof of (cINDUCTION->(((r cZERO) cZERO)->((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))))
% 12.59/13.70 Found x200:=(x20 x7):(((r Xx0) x7)->((r (cSUCC Xx0)) (cSUCC x7)))
% 12.59/13.70 Found (x20 x7) as proof of (((r Xx0) x7)->((r (cSUCC Xx0)) x6))
% 12.59/13.70 Found ((x2 Xx0) x7) as proof of (((r Xx0) x7)->((r (cSUCC Xx0)) x6))
% 12.59/13.70 Found ((x2 Xx0) x7) as proof of (((r Xx0) x7)->((r (cSUCC Xx0)) x6))
% 12.59/13.70 Found x3:((r cZERO) cZERO)
% 12.59/13.70 Instantiate: x1:=cZERO:((fofType->Prop)->Prop)
% 12.59/13.70 Found (fun (x3:((r cZERO) cZERO))=> x3) as proof of ((r cZERO) x1)
% 12.59/13.70 Found (fun (x2:cINDUCTION) (x3:((r cZERO) cZERO))=> x3) as proof of (((r cZERO) cZERO)->((r cZERO) x1))
% 12.59/13.70 Found (fun (x2:cINDUCTION) (x3:((r cZERO) cZERO))=> x3) as proof of (cINDUCTION->(((r cZERO) cZERO)->((r cZERO) x1)))
% 12.59/13.70 Found (and_rect10 (fun (x2:cINDUCTION) (x3:((r cZERO) cZERO))=> x3)) as proof of ((r cZERO) x1)
% 12.59/13.70 Found ((and_rect1 ((r cZERO) x1)) (fun (x2:cINDUCTION) (x3:((r cZERO) cZERO))=> x3)) as proof of ((r cZERO) x1)
% 12.59/13.70 Found (((fun (P:Type) (x2:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x2) x20)) ((r cZERO) x1)) (fun (x2:cINDUCTION) (x3:((r cZERO) cZERO))=> x3)) as proof of ((r cZERO) x1)
% 13.79/14.92 Found (fun (x30:(forall (Xx0:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx0) Xy)->((r (cSUCC Xx0)) (cSUCC Xy)))))=> (((fun (P:Type) (x2:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x2) x20)) ((r cZERO) x1)) (fun (x2:cINDUCTION) (x3:((r cZERO) cZERO))=> x3))) as proof of ((r cZERO) x1)
% 13.79/14.92 Found (fun (x20:((and cINDUCTION) ((r cZERO) cZERO))) (x30:(forall (Xx0:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx0) Xy)->((r (cSUCC Xx0)) (cSUCC Xy)))))=> (((fun (P:Type) (x2:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x2) x20)) ((r cZERO) x1)) (fun (x2:cINDUCTION) (x3:((r cZERO) cZERO))=> x3))) as proof of ((forall (Xx0:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx0) Xy)->((r (cSUCC Xx0)) (cSUCC Xy))))->((r cZERO) x1))
% 13.79/14.92 Found (fun (x20:((and cINDUCTION) ((r cZERO) cZERO))) (x30:(forall (Xx0:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx0) Xy)->((r (cSUCC Xx0)) (cSUCC Xy)))))=> (((fun (P:Type) (x2:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x2) x20)) ((r cZERO) x1)) (fun (x2:cINDUCTION) (x3:((r cZERO) cZERO))=> x3))) as proof of (((and cINDUCTION) ((r cZERO) cZERO))->((forall (Xx0:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx0) Xy)->((r (cSUCC Xx0)) (cSUCC Xy))))->((r cZERO) x1)))
% 13.79/14.92 Found x5:((r cZERO) cZERO)
% 13.79/14.92 Instantiate: x1:=cZERO:((fofType->Prop)->Prop)
% 13.79/14.92 Found (fun (x5:((r cZERO) cZERO))=> x5) as proof of ((r cZERO) x1)
% 13.79/14.92 Found (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5) as proof of (((r cZERO) cZERO)->((r cZERO) x1))
% 13.79/14.92 Found (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5) as proof of (cINDUCTION->(((r cZERO) cZERO)->((r cZERO) x1)))
% 13.79/14.92 Found (and_rect10 (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5)) as proof of ((r cZERO) x1)
% 13.79/14.92 Found ((and_rect1 ((r cZERO) x1)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5)) as proof of ((r cZERO) x1)
% 13.79/14.92 Found (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x2)) ((r cZERO) x1)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5)) as proof of ((r cZERO) x1)
% 13.79/14.92 Found (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x2)) ((r cZERO) x1)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5)) as proof of ((r cZERO) x1)
% 13.79/14.92 Found x5:((r cZERO) cZERO)
% 13.79/14.92 Instantiate: x1:=cZERO:((fofType->Prop)->Prop)
% 13.79/14.92 Found (fun (x5:((r cZERO) cZERO))=> x5) as proof of ((r cZERO) x1)
% 13.79/14.92 Found (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5) as proof of (((r cZERO) cZERO)->((r cZERO) x1))
% 13.79/14.92 Found (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5) as proof of (cINDUCTION->(((r cZERO) cZERO)->((r cZERO) x1)))
% 13.79/14.92 Found (and_rect10 (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5)) as proof of ((r cZERO) x1)
% 13.79/14.92 Found ((and_rect1 ((r cZERO) x1)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5)) as proof of ((r cZERO) x1)
% 13.79/14.92 Found (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x2)) ((r cZERO) x1)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5)) as proof of ((r cZERO) x1)
% 13.79/14.92 Found (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x2)) ((r cZERO) x1)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5)) as proof of ((r cZERO) x1)
% 13.79/14.92 Found x5:((r cZERO) cZERO)
% 13.79/14.92 Instantiate: x3:=cZERO:((fofType->Prop)->Prop)
% 13.79/14.92 Found (fun (x5:((r cZERO) cZERO))=> x5) as proof of ((r cZERO) x3)
% 13.79/14.92 Found (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5) as proof of (((r cZERO) cZERO)->((r cZERO) x3))
% 13.79/14.92 Found (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5) as proof of (cINDUCTION->(((r cZERO) cZERO)->((r cZERO) x3)))
% 13.79/14.92 Found (and_rect10 (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5)) as proof of ((r cZERO) x3)
% 13.79/14.92 Found ((and_rect1 ((r cZERO) x3)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5)) as proof of ((r cZERO) x3)
% 13.79/14.92 Found (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x1)) ((r cZERO) x3)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5)) as proof of ((r cZERO) x3)
% 15.99/17.13 Found (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x1)) ((r cZERO) x3)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5)) as proof of ((r cZERO) x3)
% 15.99/17.13 Found x200:=(x20 x7):(((r Xx0) x7)->((r (cSUCC Xx0)) (cSUCC x7)))
% 15.99/17.13 Found (x20 x7) as proof of (((r Xx0) x7)->((r (cSUCC Xx0)) x6))
% 15.99/17.13 Found ((x2 Xx0) x7) as proof of (((r Xx0) x7)->((r (cSUCC Xx0)) x6))
% 15.99/17.13 Found ((x2 Xx0) x7) as proof of (((r Xx0) x7)->((r (cSUCC Xx0)) x6))
% 15.99/17.13 Found x200:=(x20 x7):(((r Xx0) x7)->((r (cSUCC Xx0)) (cSUCC x7)))
% 15.99/17.13 Found (x20 x7) as proof of (((r Xx0) x7)->((r (cSUCC Xx0)) x6))
% 15.99/17.13 Found ((x2 Xx0) x7) as proof of (((r Xx0) x7)->((r (cSUCC Xx0)) x6))
% 15.99/17.13 Found ((x2 Xx0) x7) as proof of (((r Xx0) x7)->((r (cSUCC Xx0)) x6))
% 15.99/17.13 Found x4:((r cZERO) cZERO)
% 15.99/17.13 Instantiate: x1:=cZERO:((fofType->Prop)->Prop)
% 15.99/17.15 Found (fun (x4:((r cZERO) cZERO))=> x4) as proof of ((r cZERO) x1)
% 15.99/17.15 Found (fun (x3:cINDUCTION) (x4:((r cZERO) cZERO))=> x4) as proof of (((r cZERO) cZERO)->((r cZERO) x1))
% 15.99/17.15 Found (fun (x3:cINDUCTION) (x4:((r cZERO) cZERO))=> x4) as proof of (cINDUCTION->(((r cZERO) cZERO)->((r cZERO) x1)))
% 15.99/17.15 Found (and_rect10 (fun (x3:cINDUCTION) (x4:((r cZERO) cZERO))=> x4)) as proof of ((r cZERO) x1)
% 15.99/17.15 Found ((and_rect1 ((r cZERO) x1)) (fun (x3:cINDUCTION) (x4:((r cZERO) cZERO))=> x4)) as proof of ((r cZERO) x1)
% 15.99/17.15 Found (((fun (P:Type) (x3:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x3) x2)) ((r cZERO) x1)) (fun (x3:cINDUCTION) (x4:((r cZERO) cZERO))=> x4)) as proof of ((r cZERO) x1)
% 15.99/17.15 Found (fun (x30:(forall (Xx0:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx0) Xy)->((r (cSUCC Xx0)) (cSUCC Xy)))))=> (((fun (P:Type) (x3:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x3) x2)) ((r cZERO) x1)) (fun (x3:cINDUCTION) (x4:((r cZERO) cZERO))=> x4))) as proof of ((r cZERO) x1)
% 15.99/17.15 Found (fun (x30:(forall (Xx0:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx0) Xy)->((r (cSUCC Xx0)) (cSUCC Xy)))))=> (((fun (P:Type) (x3:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x3) x2)) ((r cZERO) x1)) (fun (x3:cINDUCTION) (x4:((r cZERO) cZERO))=> x4))) as proof of ((forall (Xx0:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx0) Xy)->((r (cSUCC Xx0)) (cSUCC Xy))))->((r cZERO) x1))
% 15.99/17.15 Found x3:((r cZERO) cZERO)
% 15.99/17.15 Instantiate: x1:=cZERO:((fofType->Prop)->Prop)
% 15.99/17.15 Found (fun (x3:((r cZERO) cZERO))=> x3) as proof of ((r cZERO) x1)
% 15.99/17.15 Found (fun (x2:cINDUCTION) (x3:((r cZERO) cZERO))=> x3) as proof of (((r cZERO) cZERO)->((r cZERO) x1))
% 15.99/17.15 Found (fun (x2:cINDUCTION) (x3:((r cZERO) cZERO))=> x3) as proof of (cINDUCTION->(((r cZERO) cZERO)->((r cZERO) x1)))
% 15.99/17.15 Found (and_rect10 (fun (x2:cINDUCTION) (x3:((r cZERO) cZERO))=> x3)) as proof of ((r cZERO) x1)
% 15.99/17.15 Found ((and_rect1 ((r cZERO) x1)) (fun (x2:cINDUCTION) (x3:((r cZERO) cZERO))=> x3)) as proof of ((r cZERO) x1)
% 15.99/17.15 Found (((fun (P:Type) (x2:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x2) x10)) ((r cZERO) x1)) (fun (x2:cINDUCTION) (x3:((r cZERO) cZERO))=> x3)) as proof of ((r cZERO) x1)
% 15.99/17.15 Found (((fun (P:Type) (x2:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x2) x10)) ((r cZERO) x1)) (fun (x2:cINDUCTION) (x3:((r cZERO) cZERO))=> x3)) as proof of ((r cZERO) x1)
% 15.99/17.15 Found (ex_intro000 (((fun (P:Type) (x2:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x2) x10)) ((r cZERO) x1)) (fun (x2:cINDUCTION) (x3:((r cZERO) cZERO))=> x3))) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 15.99/17.15 Found ((ex_intro00 cZERO) (((fun (P:Type) (x2:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x2) x10)) ((r cZERO) cZERO)) (fun (x2:cINDUCTION) (x3:((r cZERO) cZERO))=> x3))) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 18.29/19.43 Found (((ex_intro0 (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) (((fun (P:Type) (x2:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x2) x10)) ((r cZERO) cZERO)) (fun (x2:cINDUCTION) (x3:((r cZERO) cZERO))=> x3))) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 18.29/19.43 Found ((((ex_intro ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) (((fun (P:Type) (x2:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x2) x10)) ((r cZERO) cZERO)) (fun (x2:cINDUCTION) (x3:((r cZERO) cZERO))=> x3))) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 18.29/19.43 Found (fun (x20:(forall (Xx0:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx0) Xy)->((r (cSUCC Xx0)) (cSUCC Xy)))))=> ((((ex_intro ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) (((fun (P:Type) (x2:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x2) x10)) ((r cZERO) cZERO)) (fun (x2:cINDUCTION) (x3:((r cZERO) cZERO))=> x3)))) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 18.29/19.43 Found (fun (x10:((and cINDUCTION) ((r cZERO) cZERO))) (x20:(forall (Xx0:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx0) Xy)->((r (cSUCC Xx0)) (cSUCC Xy)))))=> ((((ex_intro ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) (((fun (P:Type) (x2:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x2) x10)) ((r cZERO) cZERO)) (fun (x2:cINDUCTION) (x3:((r cZERO) cZERO))=> x3)))) as proof of ((forall (Xx0:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx0) Xy)->((r (cSUCC Xx0)) (cSUCC Xy))))->((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))))
% 18.29/19.43 Found (fun (x10:((and cINDUCTION) ((r cZERO) cZERO))) (x20:(forall (Xx0:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx0) Xy)->((r (cSUCC Xx0)) (cSUCC Xy)))))=> ((((ex_intro ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) (((fun (P:Type) (x2:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x2) x10)) ((r cZERO) cZERO)) (fun (x2:cINDUCTION) (x3:((r cZERO) cZERO))=> x3)))) as proof of (((and cINDUCTION) ((r cZERO) cZERO))->((forall (Xx0:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx0) Xy)->((r (cSUCC Xx0)) (cSUCC Xy))))->((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))))
% 18.29/19.43 Found x2000:=(x200 x3):(((r Xx0) x3)->((r (cSUCC Xx0)) (cSUCC x3)))
% 18.29/19.43 Found (x200 x3) as proof of (((r Xx0) x3)->((r (cSUCC Xx0)) x2))
% 18.29/19.43 Found ((x20 Xx0) x3) as proof of (((r Xx0) x3)->((r (cSUCC Xx0)) x2))
% 18.29/19.43 Found ((x20 Xx0) x3) as proof of (((r Xx0) x3)->((r (cSUCC Xx0)) x2))
% 18.29/19.43 Found x5:((r cZERO) cZERO)
% 18.29/19.43 Instantiate: x3:=cZERO:((fofType->Prop)->Prop)
% 18.29/19.43 Found (fun (x5:((r cZERO) cZERO))=> x5) as proof of ((r cZERO) x3)
% 18.29/19.43 Found (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5) as proof of (((r cZERO) cZERO)->((r cZERO) x3))
% 18.29/19.43 Found (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5) as proof of (cINDUCTION->(((r cZERO) cZERO)->((r cZERO) x3)))
% 18.29/19.43 Found (and_rect10 (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5)) as proof of ((r cZERO) x3)
% 18.29/19.43 Found ((and_rect1 ((r cZERO) x3)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5)) as proof of ((r cZERO) x3)
% 18.29/19.43 Found (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x1)) ((r cZERO) x3)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5)) as proof of ((r cZERO) x3)
% 18.29/19.43 Found (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x1)) ((r cZERO) x3)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5)) as proof of ((r cZERO) x3)
% 18.29/19.43 Found (ex_intro000 (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x1)) ((r cZERO) x3)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5))) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 19.09/20.26 Found ((ex_intro00 cZERO) (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x1)) ((r cZERO) cZERO)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5))) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 19.09/20.26 Found (((ex_intro0 (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x1)) ((r cZERO) cZERO)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5))) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 19.09/20.26 Found ((((ex_intro ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x1)) ((r cZERO) cZERO)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5))) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 19.09/20.26 Found ((((ex_intro ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x1)) ((r cZERO) cZERO)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5))) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 19.09/20.26 Found x200:=(x20 x5):(((r Xx0) x5)->((r (cSUCC Xx0)) (cSUCC x5)))
% 19.09/20.26 Found (x20 x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 19.09/20.26 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 19.09/20.26 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 19.09/20.26 Found x200:=(x20 x7):(((r Xx0) x7)->((r (cSUCC Xx0)) (cSUCC x7)))
% 19.09/20.26 Found (x20 x7) as proof of (((r Xx0) x7)->((r (cSUCC Xx0)) x6))
% 19.09/20.26 Found ((x2 Xx0) x7) as proof of (((r Xx0) x7)->((r (cSUCC Xx0)) x6))
% 19.09/20.26 Found ((x2 Xx0) x7) as proof of (((r Xx0) x7)->((r (cSUCC Xx0)) x6))
% 19.09/20.26 Found x5:((r cZERO) cZERO)
% 19.09/20.26 Instantiate: x3:=cZERO:((fofType->Prop)->Prop)
% 19.09/20.26 Found (fun (x5:((r cZERO) cZERO))=> x5) as proof of ((r cZERO) x3)
% 19.09/20.26 Found (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5) as proof of (((r cZERO) cZERO)->((r cZERO) x3))
% 19.09/20.26 Found (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5) as proof of (cINDUCTION->(((r cZERO) cZERO)->((r cZERO) x3)))
% 19.09/20.26 Found (and_rect10 (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5)) as proof of ((r cZERO) x3)
% 19.09/20.26 Found ((and_rect1 ((r cZERO) x3)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5)) as proof of ((r cZERO) x3)
% 19.09/20.26 Found (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x1)) ((r cZERO) x3)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5)) as proof of ((r cZERO) x3)
% 19.09/20.26 Found (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x1)) ((r cZERO) x3)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5)) as proof of ((r cZERO) x3)
% 19.09/20.26 Found (ex_intro000 (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x1)) ((r cZERO) x3)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5))) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 19.09/20.26 Found ((ex_intro00 cZERO) (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x1)) ((r cZERO) cZERO)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5))) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 19.09/20.26 Found (((ex_intro0 (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x1)) ((r cZERO) cZERO)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5))) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 20.19/21.31 Found ((((ex_intro ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x1)) ((r cZERO) cZERO)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5))) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 20.19/21.31 Found ((((ex_intro ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x1)) ((r cZERO) cZERO)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5))) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 20.19/21.31 Found x200:=(x20 x5):(((r Xx0) x5)->((r (cSUCC Xx0)) (cSUCC x5)))
% 20.19/21.31 Found (x20 x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 20.19/21.31 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 20.19/21.31 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 20.19/21.31 Found x200:=(x20 x7):(((r Xx0) x7)->((r (cSUCC Xx0)) (cSUCC x7)))
% 20.19/21.31 Found (x20 x7) as proof of (((r Xx0) x7)->((r (cSUCC Xx0)) x6))
% 20.19/21.31 Found ((x2 Xx0) x7) as proof of (((r Xx0) x7)->((r (cSUCC Xx0)) x6))
% 20.19/21.31 Found ((x2 Xx0) x7) as proof of (((r Xx0) x7)->((r (cSUCC Xx0)) x6))
% 20.19/21.31 Found x200:=(x20 x7):(((r Xx0) x7)->((r (cSUCC Xx0)) (cSUCC x7)))
% 20.19/21.31 Found (x20 x7) as proof of (((r Xx0) x7)->((r (cSUCC Xx0)) x4))
% 20.19/21.31 Found ((x2 Xx0) x7) as proof of (((r Xx0) x7)->((r (cSUCC Xx0)) x4))
% 20.19/21.31 Found ((x2 Xx0) x7) as proof of (((r Xx0) x7)->((r (cSUCC Xx0)) x4))
% 20.19/21.31 Found x5:((r cZERO) cZERO)
% 20.19/21.31 Instantiate: x1:=cZERO:((fofType->Prop)->Prop)
% 20.19/21.31 Found (fun (x5:((r cZERO) cZERO))=> x5) as proof of ((r cZERO) x1)
% 20.19/21.31 Found (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5) as proof of (((r cZERO) cZERO)->((r cZERO) x1))
% 20.19/21.31 Found (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5) as proof of (cINDUCTION->(((r cZERO) cZERO)->((r cZERO) x1)))
% 20.19/21.31 Found (and_rect10 (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5)) as proof of ((r cZERO) x1)
% 20.19/21.31 Found ((and_rect1 ((r cZERO) x1)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5)) as proof of ((r cZERO) x1)
% 20.19/21.31 Found (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x2)) ((r cZERO) x1)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5)) as proof of ((r cZERO) x1)
% 20.19/21.31 Found (fun (x3:(forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy)))))=> (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x2)) ((r cZERO) x1)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5))) as proof of ((r cZERO) x1)
% 20.19/21.31 Found (fun (x2:((and cINDUCTION) ((r cZERO) cZERO))) (x3:(forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy)))))=> (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x2)) ((r cZERO) x1)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5))) as proof of ((forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy))))->((r cZERO) x1))
% 20.19/21.31 Found (fun (x2:((and cINDUCTION) ((r cZERO) cZERO))) (x3:(forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy)))))=> (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x2)) ((r cZERO) x1)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5))) as proof of (((and cINDUCTION) ((r cZERO) cZERO))->((forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy))))->((r cZERO) x1)))
% 20.19/21.31 Found (and_rect00 (fun (x2:((and cINDUCTION) ((r cZERO) cZERO))) (x3:(forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy)))))=> (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x2)) ((r cZERO) x1)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5)))) as proof of ((r cZERO) x1)
% 20.50/21.67 Found ((and_rect0 ((r cZERO) x1)) (fun (x2:((and cINDUCTION) ((r cZERO) cZERO))) (x3:(forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy)))))=> (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x2)) ((r cZERO) x1)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5)))) as proof of ((r cZERO) x1)
% 20.50/21.67 Found (((fun (P:Type) (x2:(((and cINDUCTION) ((r cZERO) cZERO))->((forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy))))->P)))=> (((((and_rect ((and cINDUCTION) ((r cZERO) cZERO))) (forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy))))) P) x2) x)) ((r cZERO) x1)) (fun (x2:((and cINDUCTION) ((r cZERO) cZERO))) (x3:(forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy)))))=> (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x2)) ((r cZERO) x1)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5)))) as proof of ((r cZERO) x1)
% 20.50/21.67 Found (((fun (P:Type) (x2:(((and cINDUCTION) ((r cZERO) cZERO))->((forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy))))->P)))=> (((((and_rect ((and cINDUCTION) ((r cZERO) cZERO))) (forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy))))) P) x2) x)) ((r cZERO) x1)) (fun (x2:((and cINDUCTION) ((r cZERO) cZERO))) (x3:(forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy)))))=> (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x2)) ((r cZERO) x1)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5)))) as proof of ((r cZERO) x1)
% 20.50/21.67 Found x200:=(x20 x7):(((r Xx0) x7)->((r (cSUCC Xx0)) (cSUCC x7)))
% 20.50/21.67 Found (x20 x7) as proof of (((r Xx0) x7)->((r (cSUCC Xx0)) x4))
% 20.50/21.67 Found ((x2 Xx0) x7) as proof of (((r Xx0) x7)->((r (cSUCC Xx0)) x4))
% 20.50/21.67 Found ((x2 Xx0) x7) as proof of (((r Xx0) x7)->((r (cSUCC Xx0)) x4))
% 20.50/21.67 Found x4:((r cZERO) cZERO)
% 20.50/21.67 Instantiate: x2:=cZERO:((fofType->Prop)->Prop)
% 20.50/21.67 Found (fun (x4:((r cZERO) cZERO))=> x4) as proof of ((r cZERO) x2)
% 20.50/21.67 Found (fun (x3:cINDUCTION) (x4:((r cZERO) cZERO))=> x4) as proof of (((r cZERO) cZERO)->((r cZERO) x2))
% 20.50/21.67 Found (fun (x3:cINDUCTION) (x4:((r cZERO) cZERO))=> x4) as proof of (cINDUCTION->(((r cZERO) cZERO)->((r cZERO) x2)))
% 20.50/21.67 Found (and_rect10 (fun (x3:cINDUCTION) (x4:((r cZERO) cZERO))=> x4)) as proof of ((r cZERO) x2)
% 20.50/21.67 Found ((and_rect1 ((r cZERO) x2)) (fun (x3:cINDUCTION) (x4:((r cZERO) cZERO))=> x4)) as proof of ((r cZERO) x2)
% 20.50/21.67 Found (((fun (P:Type) (x3:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x3) x1)) ((r cZERO) x2)) (fun (x3:cINDUCTION) (x4:((r cZERO) cZERO))=> x4)) as proof of ((r cZERO) x2)
% 20.50/21.67 Found (((fun (P:Type) (x3:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x3) x1)) ((r cZERO) x2)) (fun (x3:cINDUCTION) (x4:((r cZERO) cZERO))=> x4)) as proof of ((r cZERO) x2)
% 20.50/21.67 Found (ex_intro000 (((fun (P:Type) (x3:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x3) x1)) ((r cZERO) x2)) (fun (x3:cINDUCTION) (x4:((r cZERO) cZERO))=> x4))) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 20.50/21.67 Found ((ex_intro00 cZERO) (((fun (P:Type) (x3:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x3) x1)) ((r cZERO) cZERO)) (fun (x3:cINDUCTION) (x4:((r cZERO) cZERO))=> x4))) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 20.50/21.67 Found (((ex_intro0 (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) (((fun (P:Type) (x3:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x3) x1)) ((r cZERO) cZERO)) (fun (x3:cINDUCTION) (x4:((r cZERO) cZERO))=> x4))) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 24.70/25.82 Found ((((ex_intro ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) (((fun (P:Type) (x3:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x3) x1)) ((r cZERO) cZERO)) (fun (x3:cINDUCTION) (x4:((r cZERO) cZERO))=> x4))) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 24.70/25.82 Found (fun (x20:(forall (Xx0:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx0) Xy)->((r (cSUCC Xx0)) (cSUCC Xy)))))=> ((((ex_intro ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) (((fun (P:Type) (x3:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x3) x1)) ((r cZERO) cZERO)) (fun (x3:cINDUCTION) (x4:((r cZERO) cZERO))=> x4)))) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 24.70/25.82 Found (fun (x20:(forall (Xx0:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx0) Xy)->((r (cSUCC Xx0)) (cSUCC Xy)))))=> ((((ex_intro ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) (((fun (P:Type) (x3:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x3) x1)) ((r cZERO) cZERO)) (fun (x3:cINDUCTION) (x4:((r cZERO) cZERO))=> x4)))) as proof of ((forall (Xx0:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx0) Xy)->((r (cSUCC Xx0)) (cSUCC Xy))))->((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))))
% 24.70/25.82 Found x2000:=(x200 x4):(((r Xx0) x4)->((r (cSUCC Xx0)) (cSUCC x4)))
% 24.70/25.82 Found (x200 x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 24.70/25.82 Found ((x20 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 24.70/25.82 Found ((x20 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 24.70/25.82 Found x500:=(x50 x4):(P M)
% 24.70/25.82 Found (x50 x4) as proof of (P M)
% 24.70/25.82 Found ((x5 P) x4) as proof of (P M)
% 24.70/25.82 Found (fun (x5:(cNAT M))=> ((x5 P) x4)) as proof of (P M)
% 24.70/25.82 Found (fun (M:((fofType->Prop)->Prop)) (x5:(cNAT M))=> ((x5 P) x4)) as proof of ((cNAT M)->(P M))
% 24.70/25.82 Found (fun (x4:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x5:(cNAT M))=> ((x5 P) x4)) as proof of (forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M)))
% 24.70/25.82 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x4:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x5:(cNAT M))=> ((x5 P) x4)) as proof of (((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))->(forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M))))
% 24.70/25.82 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x4:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x5:(cNAT M))=> ((x5 P) x4)) as proof of cINDUCTION
% 24.70/25.82 Found x200:=(x20 x6):(((r Xx0) x6)->((r (cSUCC Xx0)) (cSUCC x6)))
% 24.70/25.82 Found (x20 x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 24.70/25.82 Found ((x2 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 24.70/25.82 Found ((x2 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 24.70/25.82 Found x00:=(x0 Xp):(((and (Xp cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp Xx0)->(Xp (cSUCC Xx0)))))->(Xp Xx))
% 24.70/25.82 Found (x0 Xp) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 24.70/25.82 Found (x0 Xp) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 24.70/25.82 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop))=> (x0 Xp)) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 24.70/25.82 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop))=> (x0 Xp)) as proof of (cNAT M)
% 24.70/25.82 Found x2000:=(x200 x6):(((r Xx0) x6)->((r (cSUCC Xx0)) (cSUCC x6)))
% 24.70/25.82 Found (x200 x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 24.70/25.82 Found ((x20 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 24.70/25.82 Found ((x20 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 25.99/27.19 Found x5:((r cZERO) cZERO)
% 25.99/27.19 Instantiate: x1:=cZERO:((fofType->Prop)->Prop)
% 25.99/27.19 Found (fun (x5:((r cZERO) cZERO))=> x5) as proof of ((r cZERO) x1)
% 25.99/27.19 Found (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5) as proof of (((r cZERO) cZERO)->((r cZERO) x1))
% 25.99/27.19 Found (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5) as proof of (cINDUCTION->(((r cZERO) cZERO)->((r cZERO) x1)))
% 25.99/27.19 Found (and_rect10 (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5)) as proof of ((r cZERO) x1)
% 25.99/27.19 Found ((and_rect1 ((r cZERO) x1)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5)) as proof of ((r cZERO) x1)
% 25.99/27.19 Found (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x2)) ((r cZERO) x1)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5)) as proof of ((r cZERO) x1)
% 25.99/27.19 Found (fun (x3:(forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy)))))=> (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x2)) ((r cZERO) x1)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5))) as proof of ((r cZERO) x1)
% 25.99/27.19 Found (fun (x2:((and cINDUCTION) ((r cZERO) cZERO))) (x3:(forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy)))))=> (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x2)) ((r cZERO) x1)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5))) as proof of ((forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy))))->((r cZERO) x1))
% 25.99/27.19 Found (fun (x2:((and cINDUCTION) ((r cZERO) cZERO))) (x3:(forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy)))))=> (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x2)) ((r cZERO) x1)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5))) as proof of (((and cINDUCTION) ((r cZERO) cZERO))->((forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy))))->((r cZERO) x1)))
% 25.99/27.19 Found (and_rect00 (fun (x2:((and cINDUCTION) ((r cZERO) cZERO))) (x3:(forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy)))))=> (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x2)) ((r cZERO) x1)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5)))) as proof of ((r cZERO) x1)
% 25.99/27.19 Found ((and_rect0 ((r cZERO) x1)) (fun (x2:((and cINDUCTION) ((r cZERO) cZERO))) (x3:(forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy)))))=> (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x2)) ((r cZERO) x1)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5)))) as proof of ((r cZERO) x1)
% 25.99/27.19 Found (((fun (P:Type) (x2:(((and cINDUCTION) ((r cZERO) cZERO))->((forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy))))->P)))=> (((((and_rect ((and cINDUCTION) ((r cZERO) cZERO))) (forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy))))) P) x2) x)) ((r cZERO) x1)) (fun (x2:((and cINDUCTION) ((r cZERO) cZERO))) (x3:(forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy)))))=> (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x2)) ((r cZERO) x1)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5)))) as proof of ((r cZERO) x1)
% 25.99/27.19 Found (((fun (P:Type) (x2:(((and cINDUCTION) ((r cZERO) cZERO))->((forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy))))->P)))=> (((((and_rect ((and cINDUCTION) ((r cZERO) cZERO))) (forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy))))) P) x2) x)) ((r cZERO) x1)) (fun (x2:((and cINDUCTION) ((r cZERO) cZERO))) (x3:(forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy)))))=> (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x2)) ((r cZERO) x1)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5)))) as proof of ((r cZERO) x1)
% 26.09/27.21 Found (ex_intro000 (((fun (P:Type) (x2:(((and cINDUCTION) ((r cZERO) cZERO))->((forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy))))->P)))=> (((((and_rect ((and cINDUCTION) ((r cZERO) cZERO))) (forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy))))) P) x2) x)) ((r cZERO) x1)) (fun (x2:((and cINDUCTION) ((r cZERO) cZERO))) (x3:(forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy)))))=> (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x2)) ((r cZERO) x1)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5))))) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 26.09/27.21 Found ((ex_intro00 cZERO) (((fun (P:Type) (x2:(((and cINDUCTION) ((r cZERO) cZERO))->((forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy))))->P)))=> (((((and_rect ((and cINDUCTION) ((r cZERO) cZERO))) (forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy))))) P) x2) x)) ((r cZERO) cZERO)) (fun (x2:((and cINDUCTION) ((r cZERO) cZERO))) (x3:(forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy)))))=> (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x2)) ((r cZERO) cZERO)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5))))) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 26.09/27.21 Found (((ex_intro0 (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) (((fun (P:Type) (x2:(((and cINDUCTION) ((r cZERO) cZERO))->((forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy))))->P)))=> (((((and_rect ((and cINDUCTION) ((r cZERO) cZERO))) (forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy))))) P) x2) x)) ((r cZERO) cZERO)) (fun (x2:((and cINDUCTION) ((r cZERO) cZERO))) (x3:(forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy)))))=> (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x2)) ((r cZERO) cZERO)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5))))) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 26.09/27.21 Found ((((ex_intro ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) (((fun (P:Type) (x2:(((and cINDUCTION) ((r cZERO) cZERO))->((forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy))))->P)))=> (((((and_rect ((and cINDUCTION) ((r cZERO) cZERO))) (forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy))))) P) x2) x)) ((r cZERO) cZERO)) (fun (x2:((and cINDUCTION) ((r cZERO) cZERO))) (x3:(forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy)))))=> (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x2)) ((r cZERO) cZERO)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5))))) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 26.09/27.21 Found ((((ex_intro ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy))) cZERO) (((fun (P:Type) (x2:(((and cINDUCTION) ((r cZERO) cZERO))->((forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy))))->P)))=> (((((and_rect ((and cINDUCTION) ((r cZERO) cZERO))) (forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy))))) P) x2) x)) ((r cZERO) cZERO)) (fun (x2:((and cINDUCTION) ((r cZERO) cZERO))) (x3:(forall (Xx:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx) Xy)->((r (cSUCC Xx)) (cSUCC Xy)))))=> (((fun (P:Type) (x4:(cINDUCTION->(((r cZERO) cZERO)->P)))=> (((((and_rect cINDUCTION) ((r cZERO) cZERO)) P) x4) x2)) ((r cZERO) cZERO)) (fun (x4:cINDUCTION) (x5:((r cZERO) cZERO))=> x5))))) as proof of ((ex ((fofType->Prop)->Prop)) (fun (Xy:((fofType->Prop)->Prop))=> ((r cZERO) Xy)))
% 29.09/30.29 Found x300:=(x30 x5):(((r Xx0) x5)->((r (cSUCC Xx0)) (cSUCC x5)))
% 29.09/30.29 Found (x30 x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 29.09/30.29 Found ((x3 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 29.09/30.29 Found ((x3 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 29.09/30.29 Found x400:=(x40 x5):(((r Xx0) x5)->((r (cSUCC Xx0)) (cSUCC x5)))
% 29.09/30.29 Found (x40 x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x2))
% 29.09/30.29 Found ((x4 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x2))
% 29.09/30.29 Found ((x4 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x2))
% 29.09/30.29 Found x300:=(x30 x7):(((r Xx0) x7)->((r (cSUCC Xx0)) (cSUCC x7)))
% 29.09/30.29 Found (x30 x7) as proof of (((r Xx0) x7)->((r (cSUCC Xx0)) x6))
% 29.09/30.29 Found ((x3 Xx0) x7) as proof of (((r Xx0) x7)->((r (cSUCC Xx0)) x6))
% 29.09/30.29 Found ((x3 Xx0) x7) as proof of (((r Xx0) x7)->((r (cSUCC Xx0)) x6))
% 29.09/30.29 Found x400:=(x40 x7):(((r Xx0) x7)->((r (cSUCC Xx0)) (cSUCC x7)))
% 29.09/30.29 Found (x40 x7) as proof of (((r Xx0) x7)->((r (cSUCC Xx0)) x2))
% 29.09/30.29 Found ((x4 Xx0) x7) as proof of (((r Xx0) x7)->((r (cSUCC Xx0)) x2))
% 29.09/30.29 Found ((x4 Xx0) x7) as proof of (((r Xx0) x7)->((r (cSUCC Xx0)) x2))
% 29.09/30.29 Found x300:=(x30 x7):(((r Xx0) x7)->((r (cSUCC Xx0)) (cSUCC x7)))
% 29.09/30.29 Found (x30 x7) as proof of (((r Xx0) x7)->((r (cSUCC Xx0)) x4))
% 29.09/30.29 Found ((x3 Xx0) x7) as proof of (((r Xx0) x7)->((r (cSUCC Xx0)) x4))
% 29.09/30.29 Found ((x3 Xx0) x7) as proof of (((r Xx0) x7)->((r (cSUCC Xx0)) x4))
% 29.09/30.29 Found x00:=(x0 Xp):(((and (Xp cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp Xx0)->(Xp (cSUCC Xx0)))))->(Xp Xx))
% 29.09/30.29 Found (x0 Xp) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 29.09/30.29 Found (x0 Xp) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 29.09/30.29 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop))=> (x0 Xp)) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 29.09/30.29 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop))=> (x0 Xp)) as proof of (cNAT M)
% 29.09/30.29 Found x00:=(x0 Xp):(((and (Xp cZERO)) (forall (Xx0:((fofType->Prop)->Prop)), ((Xp Xx0)->(Xp (cSUCC Xx0)))))->(Xp Xx))
% 29.09/30.29 Found (x0 Xp) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 29.09/30.29 Found (x0 Xp) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 29.09/30.29 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop))=> (x0 Xp)) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 29.09/30.29 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop))=> (x0 Xp)) as proof of (cNAT M)
% 29.09/30.29 Found x200:=(x20 x5):(((r Xx0) x5)->((r (cSUCC Xx0)) (cSUCC x5)))
% 29.09/30.29 Found (x20 x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 29.09/30.29 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 29.09/30.29 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 29.09/30.29 Found x2000:=(x200 x5):(((r Xx0) x5)->((r (cSUCC Xx0)) (cSUCC x5)))
% 29.09/30.29 Found (x200 x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 29.09/30.29 Found ((x20 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 29.09/30.29 Found ((x20 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 29.09/30.29 Found x2000:=(x200 x6):(((r Xx0) x6)->((r (cSUCC Xx0)) (cSUCC x6)))
% 29.09/30.29 Found (x200 x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 29.09/30.29 Found ((x20 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 29.09/30.29 Found ((x20 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 39.11/40.26 Found x100:((and cINDUCTION) ((r cZERO) cZERO))
% 39.11/40.26 Found x100 as proof of ((and cINDUCTION) ((r cZERO) cZERO))
% 39.11/40.26 Found x2000:=(x200 x5):(((r Xx0) x5)->((r (cSUCC Xx0)) (cSUCC x5)))
% 39.11/40.26 Found (x200 x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x2))
% 39.11/40.26 Found ((x20 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x2))
% 39.11/40.26 Found ((x20 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x2))
% 39.11/40.26 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 39.11/40.26 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 39.11/40.26 Found x2000:=(x200 x6):(((r Xx0) x6)->((r (cSUCC Xx0)) (cSUCC x6)))
% 39.11/40.26 Found (x200 x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x3))
% 39.11/40.26 Found ((x20 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x3))
% 39.11/40.26 Found ((x20 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x3))
% 39.11/40.26 Found x300:cINDUCTION
% 39.11/40.26 Found x300 as proof of cINDUCTION
% 39.11/40.26 Found x400:((r cZERO) cZERO)
% 39.11/40.26 Found x400 as proof of ((r cZERO) cZERO)
% 39.11/40.26 Found x300:=(x30 x2):(P M)
% 39.11/40.26 Found (x30 x2) as proof of (P M)
% 39.11/40.26 Found ((x3 P) x2) as proof of (P M)
% 39.11/40.26 Found (fun (x3:(cNAT M))=> ((x3 P) x2)) as proof of (P M)
% 39.11/40.26 Found (fun (M:((fofType->Prop)->Prop)) (x3:(cNAT M))=> ((x3 P) x2)) as proof of ((cNAT M)->(P M))
% 39.11/40.26 Found (fun (x2:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x3:(cNAT M))=> ((x3 P) x2)) as proof of (forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M)))
% 39.11/40.26 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x2:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x3:(cNAT M))=> ((x3 P) x2)) as proof of (((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))->(forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M))))
% 39.11/40.26 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x2:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x3:(cNAT M))=> ((x3 P) x2)) as proof of cINDUCTION
% 39.11/40.26 Found x20:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 39.11/40.26 Found x20 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 39.11/40.26 Found x600:=(x60 x5):(P M)
% 39.11/40.26 Found (x60 x5) as proof of (P M)
% 39.11/40.26 Found ((x6 P) x5) as proof of (P M)
% 39.11/40.26 Found (fun (x6:(cNAT M))=> ((x6 P) x5)) as proof of (P M)
% 39.11/40.26 Found (fun (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of ((cNAT M)->(P M))
% 39.11/40.26 Found (fun (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of (forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M)))
% 39.11/40.26 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of (((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))->(forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M))))
% 39.11/40.26 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of cINDUCTION
% 39.11/40.26 Found x600:=(x60 x5):(P M)
% 39.11/40.26 Found (x60 x5) as proof of (P M)
% 39.11/40.26 Found ((x6 P) x5) as proof of (P M)
% 39.11/40.26 Found (fun (x6:(cNAT M))=> ((x6 P) x5)) as proof of (P M)
% 39.11/40.26 Found (fun (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of ((cNAT M)->(P M))
% 39.11/40.26 Found (fun (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of (forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M)))
% 39.11/40.26 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of (((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))->(forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M))))
% 44.61/45.74 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of cINDUCTION
% 44.61/45.74 Found x600:=(x60 x5):(P M)
% 44.61/45.74 Found (x60 x5) as proof of (P M)
% 44.61/45.74 Found ((x6 P) x5) as proof of (P M)
% 44.61/45.74 Found (fun (x6:(cNAT M))=> ((x6 P) x5)) as proof of (P M)
% 44.61/45.74 Found (fun (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of ((cNAT M)->(P M))
% 44.61/45.74 Found (fun (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of (forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M)))
% 44.61/45.74 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of (((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))->(forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M))))
% 44.61/45.74 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of cINDUCTION
% 44.61/45.74 Found x600:=(x60 x5):(P M)
% 44.61/45.74 Found (x60 x5) as proof of (P M)
% 44.61/45.74 Found ((x6 P) x5) as proof of (P M)
% 44.61/45.74 Found (fun (x6:(cNAT M))=> ((x6 P) x5)) as proof of (P M)
% 44.61/45.74 Found (fun (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of ((cNAT M)->(P M))
% 44.61/45.74 Found (fun (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of (forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M)))
% 44.61/45.74 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of (((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))->(forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M))))
% 44.61/45.74 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of cINDUCTION
% 44.61/45.74 Found x200:((and cINDUCTION) ((r cZERO) cZERO))
% 44.61/45.74 Found x200 as proof of ((and cINDUCTION) ((r cZERO) cZERO))
% 44.61/45.74 Found x200:((and cINDUCTION) ((r cZERO) cZERO))
% 44.61/45.74 Found x200 as proof of ((and cINDUCTION) ((r cZERO) cZERO))
% 44.61/45.74 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 44.61/45.74 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 44.61/45.74 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 44.61/45.74 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 44.61/45.74 Found x500:=(x50 x4):(P M)
% 44.61/45.74 Found (x50 x4) as proof of (P M)
% 44.61/45.74 Found ((x5 P) x4) as proof of (P M)
% 44.61/45.74 Found (fun (x5:(cNAT M))=> ((x5 P) x4)) as proof of (P M)
% 44.61/45.74 Found (fun (M:((fofType->Prop)->Prop)) (x5:(cNAT M))=> ((x5 P) x4)) as proof of ((cNAT M)->(P M))
% 44.61/45.74 Found (fun (x4:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x5:(cNAT M))=> ((x5 P) x4)) as proof of (forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M)))
% 44.61/45.74 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x4:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x5:(cNAT M))=> ((x5 P) x4)) as proof of (((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))->(forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M))))
% 44.61/45.74 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x4:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x5:(cNAT M))=> ((x5 P) x4)) as proof of cINDUCTION
% 58.42/59.59 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 58.42/59.59 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 58.42/59.59 Found x400:cINDUCTION
% 58.42/59.59 Found x400 as proof of cINDUCTION
% 58.42/59.59 Found x400:cINDUCTION
% 58.42/59.59 Found x400 as proof of cINDUCTION
% 58.42/59.59 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 58.42/59.59 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 58.42/59.59 Found x400:cINDUCTION
% 58.42/59.59 Found x400 as proof of cINDUCTION
% 58.42/59.59 Found x400:cINDUCTION
% 58.42/59.59 Found x400 as proof of cINDUCTION
% 58.42/59.59 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 58.42/59.59 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 58.42/59.59 Found x200:=(x20 x5):(((r Xx0) x5)->((r (cSUCC Xx0)) (cSUCC x5)))
% 58.42/59.59 Found (x20 x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 58.42/59.59 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 58.42/59.59 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 58.42/59.59 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 58.42/59.59 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 58.42/59.59 Found x300:cINDUCTION
% 58.42/59.59 Found x300 as proof of cINDUCTION
% 58.42/59.59 Found x300:cINDUCTION
% 58.42/59.59 Found x300 as proof of cINDUCTION
% 58.42/59.59 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 58.42/59.59 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 58.42/59.59 Found x4:((r cZERO) cZERO)
% 58.42/59.59 Found x4 as proof of ((r cZERO) cZERO)
% 58.42/59.59 Found x500:((r cZERO) cZERO)
% 58.42/59.59 Found x500 as proof of ((r cZERO) cZERO)
% 58.42/59.59 Found x500:((r cZERO) cZERO)
% 58.42/59.59 Found x500 as proof of ((r cZERO) cZERO)
% 58.42/59.59 Found x4:((r cZERO) cZERO)
% 58.42/59.59 Found x4 as proof of ((r cZERO) cZERO)
% 58.42/59.59 Found x400:=(x40 x3):(P M)
% 58.42/59.59 Found (x40 x3) as proof of (P M)
% 58.42/59.59 Found ((x4 P) x3) as proof of (P M)
% 58.42/59.59 Found (fun (x4:(cNAT M))=> ((x4 P) x3)) as proof of (P M)
% 58.42/59.59 Found (fun (M:((fofType->Prop)->Prop)) (x4:(cNAT M))=> ((x4 P) x3)) as proof of ((cNAT M)->(P M))
% 58.42/59.59 Found (fun (x3:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x4:(cNAT M))=> ((x4 P) x3)) as proof of (forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M)))
% 58.42/59.59 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x3:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x4:(cNAT M))=> ((x4 P) x3)) as proof of (((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))->(forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M))))
% 58.42/59.59 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x3:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x4:(cNAT M))=> ((x4 P) x3)) as proof of cINDUCTION
% 58.42/59.59 Found x30:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 58.42/59.59 Found x30 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 58.42/59.59 Found x500:((r cZERO) cZERO)
% 58.42/59.59 Found x500 as proof of ((r cZERO) cZERO)
% 58.42/59.59 Found x500:((r cZERO) cZERO)
% 58.42/59.59 Found x500 as proof of ((r cZERO) cZERO)
% 58.42/59.59 Found x5:((r cZERO) cZERO)
% 58.42/59.59 Found x5 as proof of ((r cZERO) cZERO)
% 58.42/59.59 Found x5:((r cZERO) cZERO)
% 58.42/59.59 Found x5 as proof of ((r cZERO) cZERO)
% 58.42/59.59 Found x400:=(x40 x3):(P M)
% 58.42/59.59 Found (x40 x3) as proof of (P M)
% 58.42/59.59 Found ((x4 P) x3) as proof of (P M)
% 58.42/59.59 Found (fun (x4:(cNAT M))=> ((x4 P) x3)) as proof of (P M)
% 58.42/59.59 Found (fun (M:((fofType->Prop)->Prop)) (x4:(cNAT M))=> ((x4 P) x3)) as proof of ((cNAT M)->(P M))
% 58.42/59.59 Found (fun (x3:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x4:(cNAT M))=> ((x4 P) x3)) as proof of (forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M)))
% 67.42/68.53 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x3:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x4:(cNAT M))=> ((x4 P) x3)) as proof of (((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))->(forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M))))
% 67.42/68.53 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x3:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x4:(cNAT M))=> ((x4 P) x3)) as proof of cINDUCTION
% 67.42/68.53 Found x30:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 67.42/68.53 Found x30 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 67.42/68.53 Found x5:((r cZERO) cZERO)
% 67.42/68.53 Found x5 as proof of ((r cZERO) cZERO)
% 67.42/68.53 Found x5:((r cZERO) cZERO)
% 67.42/68.53 Found x5 as proof of ((r cZERO) cZERO)
% 67.42/68.53 Found x400:((r cZERO) cZERO)
% 67.42/68.53 Found x400 as proof of ((r cZERO) cZERO)
% 67.42/68.53 Found x20:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 67.42/68.53 Found x20 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 67.42/68.53 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 67.42/68.53 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 67.42/68.53 Found x8:(Xp cZERO)
% 67.42/68.53 Instantiate: x5:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 67.42/68.53 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 67.42/68.53 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 67.42/68.53 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 67.42/68.53 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 67.42/68.53 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 67.42/68.53 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 67.42/68.53 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 67.42/68.53 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 67.42/68.53 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 68.42/69.53 Found x8:(Xp cZERO)
% 68.42/69.53 Instantiate: x6:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 68.42/69.53 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 68.42/69.53 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 68.42/69.53 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 68.42/69.53 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 68.42/69.53 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 68.42/69.53 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 68.42/69.53 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 68.42/69.53 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 68.42/69.53 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 68.42/69.53 Found x6:(Xp cZERO)
% 68.42/69.53 Instantiate: x4:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 68.42/69.53 Found (fun (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of (Xp M)
% 68.42/69.53 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 68.42/69.53 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 68.42/69.53 Found (and_rect20 (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 68.42/69.53 Found ((and_rect2 (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 69.11/70.27 Found (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 69.11/70.27 Found (fun (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (Xp M)
% 69.11/70.27 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 69.11/70.27 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (cNAT M)
% 69.11/70.27 Found x6:(Xp cZERO)
% 69.11/70.27 Instantiate: x4:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 69.11/70.27 Found (fun (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of (Xp M)
% 69.11/70.27 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 69.11/70.27 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 69.11/70.27 Found (and_rect20 (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 69.11/70.27 Found ((and_rect2 (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 69.11/70.27 Found (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 69.11/70.27 Found (fun (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (Xp M)
% 69.11/70.27 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 72.22/73.32 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (cNAT M)
% 72.22/73.32 Found x7:(Xp cZERO)
% 72.22/73.32 Instantiate: x5:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 72.22/73.32 Found (fun (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of (Xp M)
% 72.22/73.32 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 72.22/73.32 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 72.22/73.32 Found (and_rect20 (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 72.22/73.32 Found ((and_rect2 (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 72.22/73.32 Found (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 72.22/73.32 Found (fun (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (Xp M)
% 72.22/73.32 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 72.22/73.32 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (cNAT M)
% 72.22/73.32 Found x8:(Xp cZERO)
% 72.22/73.32 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 72.22/73.32 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 72.22/73.32 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 72.22/73.32 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 72.22/73.32 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 72.42/73.53 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 72.42/73.53 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 72.42/73.53 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 72.42/73.53 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 72.42/73.53 Found x7:(Xp cZERO)
% 72.42/73.53 Instantiate: x5:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 72.42/73.53 Found (fun (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of (Xp M)
% 72.42/73.53 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 72.42/73.53 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 72.42/73.53 Found (and_rect20 (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 72.42/73.53 Found ((and_rect2 (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 72.42/73.53 Found (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 72.42/73.53 Found (fun (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (Xp M)
% 72.42/73.53 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 73.92/75.05 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (cNAT M)
% 73.92/75.05 Found x8:(Xp cZERO)
% 73.92/75.05 Instantiate: x6:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 73.92/75.05 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 73.92/75.05 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 73.92/75.05 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 73.92/75.05 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 73.92/75.05 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 73.92/75.05 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 73.92/75.05 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 73.92/75.05 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 73.92/75.05 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 73.92/75.05 Found x8:(Xp cZERO)
% 73.92/75.05 Instantiate: x6:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 73.92/75.05 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 73.92/75.05 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 73.92/75.05 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 73.92/75.05 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 73.92/75.05 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 75.82/76.92 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 75.82/76.92 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 75.82/76.92 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 75.82/76.92 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 75.82/76.92 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 75.82/76.92 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 75.82/76.92 Found x6:(Xp cZERO)
% 75.82/76.92 Found (fun (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of (Xp M)
% 75.82/76.92 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 75.82/76.92 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 75.82/76.92 Found (and_rect20 (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 75.82/76.92 Found ((and_rect2 (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 75.82/76.92 Found (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 75.82/76.92 Found (fun (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (Xp M)
% 75.82/76.92 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 76.32/77.47 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (cNAT M)
% 76.32/77.47 Found x6:(Xp cZERO)
% 76.32/77.47 Instantiate: x1:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 76.32/77.47 Found (fun (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of (Xp M)
% 76.32/77.47 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 76.32/77.47 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 76.32/77.47 Found (and_rect20 (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 76.32/77.47 Found ((and_rect2 (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 76.32/77.47 Found (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 76.32/77.47 Found (fun (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (Xp M)
% 76.32/77.47 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 76.32/77.47 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (cNAT M)
% 76.32/77.47 Found x8:(Xp cZERO)
% 76.32/77.47 Instantiate: x1:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 76.32/77.47 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 76.32/77.47 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 76.32/77.47 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 76.62/77.78 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 76.62/77.78 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 76.62/77.78 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 76.62/77.78 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 76.62/77.78 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 76.62/77.78 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 76.62/77.78 Found x20000:=(x2000 x4):(((r Xx0) x4)->((r (cSUCC Xx0)) (cSUCC x4)))
% 76.62/77.78 Found (x2000 x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 76.62/77.78 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 76.62/77.78 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 76.62/77.78 Found x8:(Xp cZERO)
% 76.62/77.78 Instantiate: x3:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 76.62/77.78 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 76.62/77.78 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 76.62/77.78 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 76.62/77.78 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 76.62/77.78 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 76.62/77.78 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 76.62/77.78 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 77.43/78.50 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 77.43/78.50 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 77.43/78.50 Found x7:(Xp cZERO)
% 77.43/78.50 Instantiate: x1:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 77.43/78.50 Found (fun (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of (Xp M)
% 77.43/78.50 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 77.43/78.50 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 77.43/78.50 Found (and_rect20 (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 77.43/78.50 Found ((and_rect2 (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 77.43/78.50 Found (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 77.43/78.50 Found (fun (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (Xp M)
% 77.43/78.50 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 77.43/78.50 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (cNAT M)
% 77.43/78.50 Found x6:(Xp cZERO)
% 77.43/78.50 Instantiate: x1:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 77.43/78.50 Found (fun (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of (Xp M)
% 77.82/78.91 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 77.82/78.91 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 77.82/78.91 Found (and_rect20 (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 77.82/78.91 Found ((and_rect2 (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 77.82/78.91 Found (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 77.82/78.91 Found (fun (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (Xp M)
% 77.82/78.91 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 77.82/78.91 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (cNAT M)
% 77.82/78.91 Found x6:(Xp cZERO)
% 77.82/78.91 Instantiate: x3:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 77.82/78.91 Found (fun (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of (Xp M)
% 77.82/78.91 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 77.82/78.91 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 77.82/78.91 Found (and_rect20 (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 77.82/78.91 Found ((and_rect2 (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 77.82/78.91 Found (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 77.82/78.91 Found (fun (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (Xp M)
% 78.63/79.74 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 78.63/79.74 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (cNAT M)
% 78.63/79.74 Found x7:(Xp cZERO)
% 78.63/79.74 Instantiate: x3:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 78.63/79.74 Found (fun (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of (Xp M)
% 78.63/79.74 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 78.63/79.74 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 78.63/79.74 Found (and_rect20 (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 78.63/79.74 Found ((and_rect2 (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 78.63/79.74 Found (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 78.63/79.74 Found (fun (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (Xp M)
% 78.63/79.74 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 78.63/79.74 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (cNAT M)
% 78.63/79.74 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 80.73/81.89 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 80.73/81.89 Found x6:(Xp cZERO)
% 80.73/81.89 Instantiate: x2:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 80.73/81.89 Found (fun (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of (Xp M)
% 80.73/81.89 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 80.73/81.89 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 80.73/81.89 Found (and_rect20 (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 80.73/81.89 Found ((and_rect2 (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 80.73/81.89 Found (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 80.73/81.89 Found (fun (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (Xp M)
% 80.73/81.89 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 80.73/81.89 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (cNAT M)
% 80.73/81.89 Found x20:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 80.73/81.89 Found x20 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 80.73/81.89 Found x6:(Xp cZERO)
% 80.73/81.89 Found (fun (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of (Xp M)
% 80.73/81.89 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 80.73/81.89 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 80.73/81.89 Found (and_rect20 (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 80.73/81.89 Found ((and_rect2 (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 80.73/81.89 Found (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 81.02/82.14 Found (fun (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (Xp M)
% 81.02/82.14 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 81.02/82.14 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (cNAT M)
% 81.02/82.14 Found x7:(Xp cZERO)
% 81.02/82.14 Instantiate: x1:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 81.02/82.14 Found (fun (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of (Xp M)
% 81.02/82.14 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 81.02/82.14 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 81.02/82.14 Found (and_rect20 (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 81.02/82.14 Found ((and_rect2 (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 81.02/82.14 Found (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 81.02/82.14 Found (fun (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (Xp M)
% 81.02/82.14 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 81.02/82.14 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (cNAT M)
% 81.43/82.59 Found x7:(Xp cZERO)
% 81.43/82.59 Found (fun (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of (Xp M)
% 81.43/82.59 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 81.43/82.59 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 81.43/82.59 Found (and_rect20 (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 81.43/82.59 Found ((and_rect2 (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 81.43/82.59 Found (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 81.43/82.59 Found (fun (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (Xp M)
% 81.43/82.59 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 81.43/82.59 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (cNAT M)
% 81.43/82.59 Found x6:(Xp cZERO)
% 81.43/82.59 Found (fun (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of (Xp M)
% 81.43/82.59 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 81.43/82.59 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 81.43/82.59 Found (and_rect20 (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 81.43/82.59 Found ((and_rect2 (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 81.43/82.59 Found (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 81.53/82.64 Found (fun (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (Xp M)
% 81.53/82.64 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 81.53/82.64 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (cNAT M)
% 81.53/82.64 Found x6:(Xp cZERO)
% 81.53/82.64 Found (fun (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of (Xp M)
% 81.53/82.64 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 81.53/82.64 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 81.53/82.64 Found (and_rect20 (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 81.53/82.64 Found ((and_rect2 (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 81.53/82.64 Found (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 81.53/82.64 Found (fun (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (Xp M)
% 81.53/82.64 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 81.53/82.64 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (cNAT M)
% 81.53/82.69 Found x7:(Xp cZERO)
% 81.53/82.69 Found (fun (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of (Xp M)
% 81.53/82.69 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 81.53/82.69 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 81.53/82.69 Found (and_rect20 (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 81.53/82.69 Found ((and_rect2 (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 81.53/82.69 Found (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 81.53/82.69 Found (fun (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (Xp M)
% 81.53/82.69 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 81.53/82.69 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (cNAT M)
% 81.53/82.69 Found x8:(Xp cZERO)
% 81.53/82.69 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 81.53/82.69 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 81.53/82.69 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 81.53/82.69 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 81.53/82.69 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 81.53/82.69 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 81.53/82.69 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 81.92/83.02 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 81.92/83.02 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 81.92/83.02 Found x8:(Xp cZERO)
% 81.92/83.02 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 81.92/83.02 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 81.92/83.02 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 81.92/83.02 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 81.92/83.02 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 81.92/83.02 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 81.92/83.02 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 81.92/83.02 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 81.92/83.02 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 81.92/83.02 Found x600:=(x60 x5):(P M)
% 81.92/83.02 Found (x60 x5) as proof of (P M)
% 82.02/83.14 Found ((x6 P) x5) as proof of (P M)
% 82.02/83.14 Found (fun (x6:(cNAT M))=> ((x6 P) x5)) as proof of (P M)
% 82.02/83.14 Found (fun (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of ((cNAT M)->(P M))
% 82.02/83.14 Found (fun (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of (forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M)))
% 82.02/83.14 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of (((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))->(forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M))))
% 82.02/83.14 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of cINDUCTION
% 82.02/83.14 Found x600:=(x60 x5):(P M)
% 82.02/83.14 Found (x60 x5) as proof of (P M)
% 82.02/83.14 Found ((x6 P) x5) as proof of (P M)
% 82.02/83.14 Found (fun (x6:(cNAT M))=> ((x6 P) x5)) as proof of (P M)
% 82.02/83.14 Found (fun (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of ((cNAT M)->(P M))
% 82.02/83.14 Found (fun (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of (forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M)))
% 82.02/83.14 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of (((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))->(forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M))))
% 82.02/83.14 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of cINDUCTION
% 82.02/83.14 Found x8:(Xp cZERO)
% 82.02/83.14 Instantiate: x1:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 82.02/83.14 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 82.02/83.14 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 82.02/83.14 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 82.02/83.14 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 82.02/83.14 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 82.02/83.14 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 82.02/83.14 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 82.02/83.14 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 82.73/83.85 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 82.73/83.85 Found x8:(Xp cZERO)
% 82.73/83.85 Instantiate: x1:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 82.73/83.85 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 82.73/83.85 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 82.73/83.85 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 82.73/83.85 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 82.73/83.85 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 82.73/83.85 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 82.73/83.85 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 82.73/83.85 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 82.73/83.85 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 82.73/83.85 Found x8:(Xp cZERO)
% 82.73/83.85 Instantiate: x3:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 82.73/83.85 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 82.73/83.85 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 82.73/83.85 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 82.73/83.85 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 84.53/85.68 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 84.53/85.68 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 84.53/85.68 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 84.53/85.68 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 84.53/85.68 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 84.53/85.68 Found x8:(Xp cZERO)
% 84.53/85.68 Instantiate: x6:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 84.53/85.68 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 84.53/85.68 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 84.53/85.68 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 84.53/85.68 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 84.53/85.68 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 84.53/85.68 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 84.53/85.68 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 84.53/85.68 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 85.03/86.18 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 85.03/86.18 Found x7:(Xp cZERO)
% 85.03/86.18 Found (fun (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of (Xp M)
% 85.03/86.18 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 85.03/86.18 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 85.03/86.18 Found (and_rect20 (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 85.03/86.18 Found ((and_rect2 (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 85.03/86.18 Found (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 85.03/86.18 Found (fun (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (Xp M)
% 85.03/86.18 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 85.03/86.18 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (cNAT M)
% 85.03/86.18 Found x8:(Xp cZERO)
% 85.03/86.18 Instantiate: x6:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 85.03/86.18 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 85.03/86.18 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 85.03/86.18 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 85.03/86.18 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 85.62/86.78 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 85.62/86.78 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 85.62/86.78 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 85.62/86.78 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 85.62/86.78 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 85.62/86.78 Found x8:(Xp cZERO)
% 85.62/86.78 Instantiate: x6:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 85.62/86.78 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 85.62/86.78 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 85.62/86.78 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 85.62/86.78 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 85.62/86.78 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 85.62/86.78 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 85.62/86.78 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 85.62/86.78 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 86.42/87.59 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 86.42/87.59 Found x600:=(x60 x5):(P M)
% 86.42/87.59 Found (x60 x5) as proof of (P M)
% 86.42/87.59 Found ((x6 P) x5) as proof of (P M)
% 86.42/87.59 Found (fun (x6:(cNAT M))=> ((x6 P) x5)) as proof of (P M)
% 86.42/87.59 Found (fun (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of ((cNAT M)->(P M))
% 86.42/87.59 Found (fun (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of (forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M)))
% 86.42/87.59 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of (((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))->(forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M))))
% 86.42/87.59 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of cINDUCTION
% 86.42/87.59 Found x600:=(x60 x5):(P M)
% 86.42/87.59 Found (x60 x5) as proof of (P M)
% 86.42/87.59 Found ((x6 P) x5) as proof of (P M)
% 86.42/87.59 Found (fun (x6:(cNAT M))=> ((x6 P) x5)) as proof of (P M)
% 86.42/87.59 Found (fun (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of ((cNAT M)->(P M))
% 86.42/87.59 Found (fun (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of (forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M)))
% 86.42/87.59 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of (((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))->(forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M))))
% 86.42/87.59 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of cINDUCTION
% 86.42/87.59 Found x8:(Xp cZERO)
% 86.42/87.59 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 86.42/87.59 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 86.42/87.59 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 86.42/87.59 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 86.42/87.59 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 86.42/87.59 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 86.42/87.59 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 86.52/87.61 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 86.52/87.61 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 86.52/87.61 Found x7:(Xp cZERO)
% 86.52/87.61 Instantiate: x5:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 86.52/87.61 Found (fun (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of (Xp M)
% 86.52/87.61 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 86.52/87.61 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 86.52/87.61 Found (and_rect20 (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 86.52/87.61 Found ((and_rect2 (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 86.52/87.61 Found (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 86.52/87.61 Found (fun (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (Xp M)
% 86.52/87.61 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 86.52/87.61 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (cNAT M)
% 87.22/88.33 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 87.22/88.33 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 87.22/88.33 Found x8:(Xp cZERO)
% 87.22/88.33 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 87.22/88.33 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 87.22/88.33 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 87.22/88.33 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 87.22/88.33 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 87.22/88.33 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 87.22/88.33 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 87.22/88.33 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 87.22/88.33 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 87.22/88.33 Found x8:(Xp cZERO)
% 87.22/88.33 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 87.22/88.33 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 87.22/88.33 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 87.22/88.33 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 87.22/88.33 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 87.22/88.33 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 87.22/88.33 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 89.93/91.06 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 89.93/91.06 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 89.93/91.06 Found x6:((r Xx0) x5)
% 89.93/91.06 Instantiate: x5:=X:((fofType->Prop)->Prop)
% 89.93/91.06 Found x6 as proof of ((r Xx0) X)
% 89.93/91.06 Found (x200 x6) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 89.93/91.06 Found ((x20 X) x6) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 89.93/91.06 Found (((x2 Xx0) X) x6) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 89.93/91.06 Found (fun (x7:((r (cSUCC Xx0)) X))=> (((x2 Xx0) X) x6)) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 89.93/91.06 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 89.93/91.06 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 89.93/91.06 Found x8:(Xp cZERO)
% 89.93/91.06 Instantiate: x1:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 89.93/91.06 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 89.93/91.06 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 89.93/91.06 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 89.93/91.06 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 89.93/91.06 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 89.93/91.06 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 89.93/91.06 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 89.93/91.06 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 90.83/91.98 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 90.83/91.98 Found x8:(Xp cZERO)
% 90.83/91.98 Instantiate: x1:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 90.83/91.98 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 90.83/91.98 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 90.83/91.98 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 90.83/91.98 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 90.83/91.98 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 90.83/91.98 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 90.83/91.98 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 90.83/91.98 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 90.83/91.98 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 90.83/91.98 Found x8:(Xp cZERO)
% 90.83/91.98 Instantiate: x1:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 90.83/91.98 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 90.83/91.98 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 90.83/91.98 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 91.23/92.38 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 91.23/92.38 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 91.23/92.38 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 91.23/92.38 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 91.23/92.38 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 91.23/92.38 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 91.23/92.38 Found x8:(Xp cZERO)
% 91.23/92.38 Instantiate: x3:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 91.23/92.38 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 91.23/92.38 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 91.23/92.38 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 91.23/92.38 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 91.23/92.38 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 91.23/92.38 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 91.23/92.38 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 91.23/92.38 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 92.23/93.31 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 92.23/93.31 Found x8:(Xp cZERO)
% 92.23/93.31 Instantiate: x2:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 92.23/93.31 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 92.23/93.31 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 92.23/93.31 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 92.23/93.31 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 92.23/93.31 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 92.23/93.31 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 92.23/93.31 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 92.23/93.31 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 92.23/93.31 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 92.23/93.31 Found x8:(Xp cZERO)
% 92.23/93.31 Instantiate: x4:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 92.23/93.31 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 92.23/93.31 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 92.93/94.03 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 92.93/94.03 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 92.93/94.03 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 92.93/94.03 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 92.93/94.03 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 92.93/94.03 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 92.93/94.03 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 92.93/94.03 Found x8:(Xp cZERO)
% 92.93/94.03 Instantiate: x4:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 92.93/94.03 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 92.93/94.03 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 92.93/94.03 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 92.93/94.03 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 92.93/94.03 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 92.93/94.03 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 92.93/94.03 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 92.93/94.03 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 93.83/94.93 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 93.83/94.93 Found x7:(Xp cZERO)
% 93.83/94.93 Instantiate: x1:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 93.83/94.93 Found (fun (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of (Xp M)
% 93.83/94.93 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 93.83/94.93 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 93.83/94.93 Found (and_rect20 (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 93.83/94.93 Found ((and_rect2 (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 93.83/94.93 Found (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 93.83/94.94 Found (fun (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (Xp M)
% 93.83/94.94 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 93.83/94.94 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (cNAT M)
% 93.83/94.94 Found x8:(Xp cZERO)
% 93.83/94.94 Instantiate: x4:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 93.83/94.94 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 93.83/94.94 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 94.83/95.99 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 94.83/95.99 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 94.83/95.99 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 94.83/95.99 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 94.83/95.99 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 94.83/95.99 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 94.83/95.99 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 94.83/95.99 Found x7:(Xp cZERO)
% 94.83/95.99 Instantiate: x3:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 94.83/95.99 Found (fun (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of (Xp M)
% 94.83/95.99 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 94.83/95.99 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 94.83/95.99 Found (and_rect20 (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 94.83/95.99 Found ((and_rect2 (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 94.83/95.99 Found (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 94.83/95.99 Found (fun (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (Xp M)
% 96.43/97.57 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 96.43/97.57 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (cNAT M)
% 96.43/97.57 Found x8:(Xp cZERO)
% 96.43/97.57 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 96.43/97.57 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 96.43/97.57 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 96.43/97.57 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 96.43/97.57 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 96.43/97.57 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 96.43/97.57 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 96.43/97.57 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 96.43/97.57 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 96.43/97.57 Found x8:(Xp cZERO)
% 96.43/97.57 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 96.43/97.57 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 96.43/97.57 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 96.53/97.61 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 96.53/97.61 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 96.53/97.61 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 96.53/97.61 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 96.53/97.61 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 96.53/97.61 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 96.53/97.61 Found x7:(Xp cZERO)
% 96.53/97.61 Found (fun (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of (Xp M)
% 96.53/97.61 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 96.53/97.61 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 96.53/97.61 Found (and_rect20 (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 96.53/97.61 Found ((and_rect2 (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 96.53/97.61 Found (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 96.53/97.61 Found (fun (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (Xp M)
% 96.53/97.61 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 96.63/97.70 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (cNAT M)
% 96.63/97.70 Found x8:(Xp cZERO)
% 96.63/97.70 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 96.63/97.70 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 96.63/97.70 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 96.63/97.70 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 96.63/97.70 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 96.63/97.70 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 96.63/97.70 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 96.63/97.70 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 96.63/97.70 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 96.63/97.70 Found x8:(Xp cZERO)
% 96.63/97.70 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 96.63/97.70 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 96.63/97.70 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 96.63/97.70 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 98.44/99.52 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 98.44/99.52 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 98.44/99.52 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 98.44/99.52 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 98.44/99.52 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 98.44/99.52 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 98.44/99.52 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 98.44/99.52 Found x6:((r Xx0) x1)
% 98.44/99.52 Instantiate: x1:=X:((fofType->Prop)->Prop)
% 98.44/99.52 Found x6 as proof of ((r Xx0) X)
% 98.44/99.52 Found (x300 x6) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 98.44/99.52 Found ((x30 X) x6) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 98.44/99.52 Found (((x3 Xx0) X) x6) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 98.44/99.52 Found (fun (x7:((r (cSUCC Xx0)) X))=> (((x3 Xx0) X) x6)) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 98.44/99.52 Found x6:((r Xx0) x3)
% 98.44/99.52 Instantiate: x3:=X:((fofType->Prop)->Prop)
% 98.44/99.52 Found x6 as proof of ((r Xx0) X)
% 98.44/99.52 Found (x200 x6) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 98.44/99.52 Found ((x20 X) x6) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 98.44/99.52 Found (((x2 Xx0) X) x6) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 98.44/99.52 Found (fun (x7:((r (cSUCC Xx0)) X))=> (((x2 Xx0) X) x6)) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 98.44/99.52 Found x500:=(x50 x4):(P M)
% 98.44/99.52 Found (x50 x4) as proof of (P M)
% 98.44/99.52 Found ((x5 P) x4) as proof of (P M)
% 98.44/99.52 Found (fun (x5:(cNAT M))=> ((x5 P) x4)) as proof of (P M)
% 98.44/99.52 Found (fun (M:((fofType->Prop)->Prop)) (x5:(cNAT M))=> ((x5 P) x4)) as proof of ((cNAT M)->(P M))
% 98.44/99.52 Found (fun (x4:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x5:(cNAT M))=> ((x5 P) x4)) as proof of (forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M)))
% 98.44/99.52 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x4:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x5:(cNAT M))=> ((x5 P) x4)) as proof of (((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))->(forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M))))
% 98.44/99.52 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x4:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x5:(cNAT M))=> ((x5 P) x4)) as proof of cINDUCTION
% 105.54/106.63 Found x200:=(x20 x5):(((r Xx0) x5)->((r (cSUCC Xx0)) (cSUCC x5)))
% 105.54/106.63 Found (x20 x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 105.54/106.63 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 105.54/106.63 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 105.54/106.63 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 105.54/106.63 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 105.54/106.63 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 105.54/106.63 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 105.54/106.63 Found x6:((r Xx0) x1)
% 105.54/106.63 Instantiate: x1:=X:((fofType->Prop)->Prop)
% 105.54/106.63 Found x6 as proof of ((r Xx0) X)
% 105.54/106.63 Found (x300 x6) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 105.54/106.63 Found ((x30 X) x6) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 105.54/106.63 Found (((x3 Xx0) X) x6) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 105.54/106.63 Found (fun (x7:((r (cSUCC Xx0)) X))=> (((x3 Xx0) X) x6)) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 105.54/106.63 Found x6:((r Xx0) x1)
% 105.54/106.63 Instantiate: x1:=X:((fofType->Prop)->Prop)
% 105.54/106.63 Found x6 as proof of ((r Xx0) X)
% 105.54/106.63 Found (x300 x6) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 105.54/106.63 Found ((x30 X) x6) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 105.54/106.63 Found (((x3 Xx0) X) x6) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 105.54/106.63 Found (fun (x7:((r (cSUCC Xx0)) X))=> (((x3 Xx0) X) x6)) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 105.54/106.63 Found x6:((r Xx0) x3)
% 105.54/106.63 Instantiate: x3:=X:((fofType->Prop)->Prop)
% 105.54/106.63 Found x6 as proof of ((r Xx0) X)
% 105.54/106.63 Found (x200 x6) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 105.54/106.63 Found ((x20 X) x6) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 105.54/106.63 Found (((x2 Xx0) X) x6) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 105.54/106.63 Found (fun (x7:((r (cSUCC Xx0)) X))=> (((x2 Xx0) X) x6)) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 105.54/106.63 Found x400:cINDUCTION
% 105.54/106.63 Found x400 as proof of cINDUCTION
% 105.54/106.63 Found x400:cINDUCTION
% 105.54/106.63 Found x400 as proof of cINDUCTION
% 105.54/106.63 Found x400:cINDUCTION
% 105.54/106.63 Found x400 as proof of cINDUCTION
% 105.54/106.63 Found x400:cINDUCTION
% 105.54/106.63 Found x400 as proof of cINDUCTION
% 105.54/106.63 Found x20:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 105.54/106.63 Found x20 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 105.54/106.63 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 105.54/106.63 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 105.54/106.63 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 105.54/106.63 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 105.54/106.63 Found x400:cINDUCTION
% 105.54/106.63 Found x400 as proof of cINDUCTION
% 105.54/106.63 Found x400:cINDUCTION
% 105.54/106.63 Found x400 as proof of cINDUCTION
% 105.54/106.63 Found x400:cINDUCTION
% 105.54/106.63 Found x400 as proof of cINDUCTION
% 105.54/106.63 Found x400:cINDUCTION
% 105.54/106.63 Found x400 as proof of cINDUCTION
% 105.54/106.63 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 105.54/106.63 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 105.54/106.63 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 105.54/106.63 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 105.54/106.63 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 105.54/106.63 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 105.54/106.63 Found x2:((r Xx0) x1)
% 105.54/106.63 Instantiate: x1:=X:((fofType->Prop)->Prop)
% 105.54/106.63 Found x2 as proof of ((r Xx0) X)
% 105.54/106.63 Found (x400 x2) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 111.74/112.89 Found ((x40 X) x2) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 111.74/112.89 Found (((x4 Xx0) X) x2) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 111.74/112.89 Found (fun (x7:((r (cSUCC Xx0)) X))=> (((x4 Xx0) X) x2)) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 111.74/112.89 Found x4:((r Xx0) x1)
% 111.74/112.89 Instantiate: x1:=X:((fofType->Prop)->Prop)
% 111.74/112.89 Found x4 as proof of ((r Xx0) X)
% 111.74/112.89 Found (x300 x4) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 111.74/112.89 Found ((x30 X) x4) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 111.74/112.89 Found (((x3 Xx0) X) x4) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 111.74/112.89 Found (fun (x7:((r (cSUCC Xx0)) X))=> (((x3 Xx0) X) x4)) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 111.74/112.89 Found x4:((r Xx0) x1)
% 111.74/112.89 Instantiate: x1:=X:((fofType->Prop)->Prop)
% 111.74/112.89 Found x4 as proof of ((r Xx0) X)
% 111.74/112.89 Found (x300 x4) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 111.74/112.89 Found ((x30 X) x4) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 111.74/112.89 Found (((x3 Xx0) X) x4) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 111.74/112.89 Found (fun (x7:((r (cSUCC Xx0)) X))=> (((x3 Xx0) X) x4)) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 111.74/112.89 Found x4:((r Xx0) x3)
% 111.74/112.89 Instantiate: x3:=X:((fofType->Prop)->Prop)
% 111.74/112.89 Found x4 as proof of ((r Xx0) X)
% 111.74/112.89 Found (x200 x4) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 111.74/112.89 Found ((x20 X) x4) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 111.74/112.89 Found (((x2 Xx0) X) x4) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 111.74/112.89 Found (fun (x7:((r (cSUCC Xx0)) X))=> (((x2 Xx0) X) x4)) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 111.74/112.89 Found x51:=(x5 x50):((r Xx0) x1)
% 111.74/112.89 Instantiate: x1:=X:((fofType->Prop)->Prop)
% 111.74/112.89 Found (x5 x50) as proof of ((r Xx0) X)
% 111.74/112.89 Found (x5 x50) as proof of ((r Xx0) X)
% 111.74/112.89 Found (x300 (x5 x50)) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 111.74/112.89 Found ((x30 X) (x5 x50)) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 111.74/112.89 Found (((x3 Xx0) X) (x5 x50)) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 111.74/112.89 Found (fun (x6:((r (cSUCC Xx0)) X))=> (((x3 Xx0) X) (x5 x50))) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 111.74/112.89 Found x51:=(x5 x50):((r Xx0) x3)
% 111.74/112.89 Instantiate: x3:=X:((fofType->Prop)->Prop)
% 111.74/112.89 Found (x5 x50) as proof of ((r Xx0) X)
% 111.74/112.89 Found (x5 x50) as proof of ((r Xx0) X)
% 111.74/112.89 Found (x200 (x5 x50)) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 111.74/112.89 Found ((x20 X) (x5 x50)) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 111.74/112.89 Found (((x2 Xx0) X) (x5 x50)) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 111.74/112.89 Found (fun (x6:((r (cSUCC Xx0)) X))=> (((x2 Xx0) X) (x5 x50))) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 111.74/112.89 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 111.74/112.89 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 111.74/112.89 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 111.74/112.89 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 111.74/112.89 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 111.74/112.89 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 111.74/112.89 Found x50:=(x5 x30):((r Xx0) x1)
% 111.74/112.89 Instantiate: x1:=X:((fofType->Prop)->Prop)
% 111.74/112.89 Found (x5 x30) as proof of ((r Xx0) X)
% 111.74/112.89 Found (x5 x30) as proof of ((r Xx0) X)
% 111.74/112.89 Found (x3000 (x5 x30)) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 111.74/112.89 Found ((x300 X) (x5 x30)) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 111.74/112.89 Found (((x30 Xx0) X) (x5 x30)) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 111.74/112.89 Found (fun (x6:((r (cSUCC Xx0)) X))=> (((x30 Xx0) X) (x5 x30))) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 111.74/112.89 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 111.74/112.89 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 111.74/112.89 Found x10:(Xp cZERO)
% 111.74/112.89 Instantiate: x8:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 111.74/112.89 Found (fun (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of (Xp M)
% 111.74/112.89 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 115.55/116.64 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 115.55/116.64 Found (and_rect20 (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 115.55/116.64 Found ((and_rect2 (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 115.55/116.64 Found (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 115.55/116.64 Found (fun (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (Xp M)
% 115.55/116.64 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 115.55/116.64 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (cNAT M)
% 115.55/116.64 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 115.55/116.64 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 115.55/116.64 Found x200:=(x20 x5):(((r Xx0) x5)->((r (cSUCC Xx0)) (cSUCC x5)))
% 115.55/116.64 Found (x20 x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 115.55/116.64 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 115.55/116.64 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 115.55/116.64 Found x200:=(x20 x5):(((r Xx0) x5)->((r (cSUCC Xx0)) (cSUCC x5)))
% 115.55/116.64 Found (x20 x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 115.55/116.64 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 115.55/116.64 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 115.55/116.64 Found x300:cINDUCTION
% 115.55/116.64 Found x300 as proof of cINDUCTION
% 115.55/116.64 Found x300:cINDUCTION
% 115.55/116.64 Found x300 as proof of cINDUCTION
% 115.55/116.64 Found x300:cINDUCTION
% 115.55/116.64 Found x300 as proof of cINDUCTION
% 115.55/116.64 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 115.55/116.64 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 115.55/116.64 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 115.55/116.64 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 115.55/116.64 Found x10:(Xp cZERO)
% 115.55/116.64 Instantiate: x8:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 116.14/117.21 Found (fun (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of (Xp M)
% 116.14/117.21 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 116.14/117.21 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 116.14/117.21 Found (and_rect20 (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 116.14/117.21 Found ((and_rect2 (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 116.14/117.21 Found (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 116.14/117.21 Found (fun (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (Xp M)
% 116.14/117.21 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 116.14/117.21 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (cNAT M)
% 116.14/117.21 Found x10:(Xp cZERO)
% 116.14/117.21 Instantiate: x8:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 116.14/117.21 Found (fun (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of (Xp M)
% 116.14/117.21 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 116.14/117.21 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 116.14/117.21 Found (and_rect20 (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 116.14/117.21 Found ((and_rect2 (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 116.14/117.21 Found (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 116.14/117.21 Found (fun (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (Xp M)
% 118.64/119.78 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 118.64/119.78 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (cNAT M)
% 118.64/119.78 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 118.64/119.78 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 118.64/119.78 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 118.64/119.78 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 118.64/119.78 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 118.64/119.78 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 118.64/119.78 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 118.64/119.78 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 118.64/119.78 Found x210:=(x21 x30):((r Xx0) x1)
% 118.64/119.78 Instantiate: x1:=X:((fofType->Prop)->Prop)
% 118.64/119.78 Found (x21 x30) as proof of ((r Xx0) X)
% 118.64/119.78 Found ((x2 x20) x30) as proof of ((r Xx0) X)
% 118.64/119.78 Found ((x2 x20) x30) as proof of ((r Xx0) X)
% 118.64/119.78 Found (x3000 ((x2 x20) x30)) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 118.64/119.78 Found ((x300 X) ((x2 x20) x30)) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 118.64/119.78 Found (((x30 Xx0) X) ((x2 x20) x30)) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 118.64/119.78 Found (fun (x5:((r (cSUCC Xx0)) X))=> (((x30 Xx0) X) ((x2 x20) x30))) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 118.64/119.78 Found x31:=(x3 x30):((r Xx0) x1)
% 118.64/119.78 Instantiate: x1:=X:((fofType->Prop)->Prop)
% 118.64/119.78 Found (x3 x30) as proof of ((r Xx0) X)
% 118.64/119.78 Found (x3 x30) as proof of ((r Xx0) X)
% 118.64/119.78 Found (x3000 (x3 x30)) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 118.64/119.78 Found ((x300 X) (x3 x30)) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 118.64/119.78 Found (((x30 Xx0) X) (x3 x30)) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 118.64/119.78 Found (fun (x6:((r (cSUCC Xx0)) X))=> (((x30 Xx0) X) (x3 x30))) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 118.64/119.78 Found x410:=(x41 x50):((r Xx0) x1)
% 118.64/119.78 Instantiate: x1:=X:((fofType->Prop)->Prop)
% 118.64/119.78 Found (x41 x50) as proof of ((r Xx0) X)
% 118.64/119.78 Found ((x4 x40) x50) as proof of ((r Xx0) X)
% 118.64/119.78 Found ((x4 x40) x50) as proof of ((r Xx0) X)
% 118.64/119.78 Found (x300 ((x4 x40) x50)) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 118.64/119.78 Found ((x30 X) ((x4 x40) x50)) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 118.64/119.78 Found (((x3 Xx0) X) ((x4 x40) x50)) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 118.64/119.78 Found (fun (x5:((r (cSUCC Xx0)) X))=> (((x3 Xx0) X) ((x4 x40) x50))) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 118.64/119.78 Found x410:=(x41 x50):((r Xx0) x3)
% 118.64/119.78 Instantiate: x3:=X:((fofType->Prop)->Prop)
% 122.26/123.33 Found (x41 x50) as proof of ((r Xx0) X)
% 122.26/123.33 Found ((x4 x40) x50) as proof of ((r Xx0) X)
% 122.26/123.33 Found ((x4 x40) x50) as proof of ((r Xx0) X)
% 122.26/123.33 Found (x200 ((x4 x40) x50)) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 122.26/123.33 Found ((x20 X) ((x4 x40) x50)) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 122.26/123.33 Found (((x2 Xx0) X) ((x4 x40) x50)) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 122.26/123.33 Found (fun (x5:((r (cSUCC Xx0)) X))=> (((x2 Xx0) X) ((x4 x40) x50))) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 122.26/123.33 Found x10:(Xp cZERO)
% 122.26/123.33 Instantiate: x6:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 122.26/123.33 Found (fun (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of (Xp M)
% 122.26/123.33 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 122.26/123.33 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 122.26/123.33 Found (and_rect20 (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 122.26/123.33 Found ((and_rect2 (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 122.26/123.33 Found (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 122.26/123.33 Found (fun (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (Xp M)
% 122.26/123.33 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 122.26/123.33 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (cNAT M)
% 122.26/123.33 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 122.26/123.33 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 122.26/123.33 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 122.26/123.33 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 122.26/123.33 Found x2000:=(x200 x6):(((r Xx0) x6)->((r (cSUCC Xx0)) (cSUCC x6)))
% 122.26/123.33 Found (x200 x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 122.26/123.33 Found ((x20 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 122.26/123.33 Found ((x20 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 122.26/123.33 Found x10:(Xp cZERO)
% 122.84/123.91 Instantiate: x6:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 122.84/123.91 Found (fun (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of (Xp M)
% 122.84/123.91 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 122.84/123.91 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 122.84/123.91 Found (and_rect20 (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 122.84/123.91 Found ((and_rect2 (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 122.84/123.91 Found (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 122.84/123.91 Found (fun (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (Xp M)
% 122.84/123.91 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 122.84/123.91 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (cNAT M)
% 122.84/123.91 Found x10:(Xp cZERO)
% 122.84/123.91 Instantiate: x6:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 122.84/123.91 Found (fun (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of (Xp M)
% 122.84/123.91 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 122.84/123.91 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 122.84/123.91 Found (and_rect20 (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 122.84/123.91 Found ((and_rect2 (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 122.84/123.91 Found (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 122.84/123.91 Found (fun (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (Xp M)
% 130.34/131.46 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 130.34/131.46 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (cNAT M)
% 130.34/131.46 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 130.34/131.46 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 130.34/131.46 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 130.34/131.46 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 130.34/131.46 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 130.34/131.46 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 130.34/131.46 Found x10:(Xp cZERO)
% 130.34/131.46 Instantiate: x8:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 130.34/131.46 Found (fun (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of (Xp M)
% 130.34/131.46 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 130.34/131.46 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 130.34/131.46 Found (and_rect20 (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 130.34/131.46 Found ((and_rect2 (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 130.34/131.46 Found (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 130.34/131.46 Found (fun (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (Xp M)
% 130.34/131.46 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 130.96/132.09 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (cNAT M)
% 130.96/132.09 Found x10:(Xp cZERO)
% 130.96/132.09 Instantiate: x8:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 130.96/132.09 Found (fun (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of (Xp M)
% 130.96/132.09 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 130.96/132.09 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 130.96/132.09 Found (and_rect20 (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 130.96/132.09 Found ((and_rect2 (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 130.96/132.09 Found (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 130.96/132.09 Found (fun (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (Xp M)
% 130.96/132.09 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 130.96/132.09 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (cNAT M)
% 130.96/132.09 Found x10:(Xp cZERO)
% 130.96/132.09 Instantiate: x8:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 130.96/132.09 Found (fun (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of (Xp M)
% 130.96/132.09 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 131.45/132.57 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 131.45/132.57 Found (and_rect20 (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 131.45/132.57 Found ((and_rect2 (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 131.45/132.57 Found (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 131.45/132.57 Found (fun (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (Xp M)
% 131.45/132.57 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 131.45/132.57 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (cNAT M)
% 131.45/132.57 Found x10:(Xp cZERO)
% 131.45/132.57 Instantiate: x8:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 131.45/132.57 Found (fun (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of (Xp M)
% 131.45/132.57 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 131.45/132.57 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 131.45/132.57 Found (and_rect20 (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 131.45/132.57 Found ((and_rect2 (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 131.45/132.57 Found (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 131.45/132.57 Found (fun (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (Xp M)
% 132.35/133.48 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 132.35/133.48 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (cNAT M)
% 132.35/133.48 Found x10:(Xp cZERO)
% 132.35/133.48 Instantiate: x8:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 132.35/133.48 Found (fun (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of (Xp M)
% 132.35/133.48 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 132.35/133.48 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 132.35/133.48 Found (and_rect20 (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 132.35/133.48 Found ((and_rect2 (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 132.35/133.48 Found (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 132.35/133.48 Found (fun (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (Xp M)
% 132.35/133.48 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 132.35/133.48 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (cNAT M)
% 132.35/133.48 Found x500:((r cZERO) cZERO)
% 132.35/133.48 Found x500 as proof of ((r cZERO) cZERO)
% 138.66/139.74 Found x500:((r cZERO) cZERO)
% 138.66/139.74 Found x500 as proof of ((r cZERO) cZERO)
% 138.66/139.74 Found x30:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 138.66/139.74 Found x30 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 138.66/139.74 Found x500:((r cZERO) cZERO)
% 138.66/139.74 Found x500 as proof of ((r cZERO) cZERO)
% 138.66/139.74 Found x500:((r cZERO) cZERO)
% 138.66/139.74 Found x500 as proof of ((r cZERO) cZERO)
% 138.66/139.74 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 138.66/139.74 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 138.66/139.74 Found x30:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 138.66/139.74 Found x30 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 138.66/139.74 Found x10:(Xp cZERO)
% 138.66/139.74 Instantiate: x4:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 138.66/139.74 Found (fun (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of (Xp M)
% 138.66/139.74 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 138.66/139.74 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 138.66/139.74 Found (and_rect20 (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 138.66/139.74 Found ((and_rect2 (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 138.66/139.74 Found (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 138.66/139.74 Found (fun (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (Xp M)
% 138.66/139.74 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 138.66/139.74 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (cNAT M)
% 138.66/139.74 Found x10:(Xp cZERO)
% 138.66/139.74 Instantiate: x6:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 138.66/139.74 Found (fun (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of (Xp M)
% 138.66/139.74 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 138.95/140.07 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 138.95/140.07 Found (and_rect20 (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 138.95/140.07 Found ((and_rect2 (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 138.95/140.07 Found (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 138.95/140.07 Found (fun (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (Xp M)
% 138.95/140.07 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 138.95/140.07 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (cNAT M)
% 138.95/140.07 Found x10:(Xp cZERO)
% 138.95/140.07 Instantiate: x6:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 138.95/140.07 Found (fun (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of (Xp M)
% 138.95/140.07 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 138.95/140.07 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 138.95/140.07 Found (and_rect20 (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 138.95/140.07 Found ((and_rect2 (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 138.95/140.07 Found (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 138.95/140.07 Found (fun (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (Xp M)
% 140.15/141.24 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 140.15/141.24 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (cNAT M)
% 140.15/141.24 Found x10:(Xp cZERO)
% 140.15/141.24 Instantiate: x6:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 140.15/141.24 Found (fun (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of (Xp M)
% 140.15/141.24 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 140.15/141.24 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 140.15/141.24 Found (and_rect20 (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 140.15/141.24 Found ((and_rect2 (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 140.15/141.24 Found (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 140.15/141.24 Found (fun (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (Xp M)
% 140.15/141.24 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 140.15/141.24 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (cNAT M)
% 140.15/141.24 Found x10:(Xp cZERO)
% 140.15/141.24 Instantiate: x6:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 140.87/141.91 Found (fun (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of (Xp M)
% 140.87/141.91 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 140.87/141.91 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 140.87/141.91 Found (and_rect20 (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 140.87/141.91 Found ((and_rect2 (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 140.87/141.91 Found (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 140.87/141.91 Found (fun (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (Xp M)
% 140.87/141.91 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 140.87/141.91 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (cNAT M)
% 140.87/141.91 Found x10:(Xp cZERO)
% 140.87/141.91 Instantiate: x6:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 140.87/141.91 Found (fun (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of (Xp M)
% 140.87/141.91 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 140.87/141.91 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 140.87/141.91 Found (and_rect20 (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 140.87/141.91 Found ((and_rect2 (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 140.87/141.91 Found (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 140.87/141.91 Found (fun (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (Xp M)
% 141.76/142.81 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 141.76/142.81 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (cNAT M)
% 141.76/142.81 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 141.76/142.81 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 141.76/142.81 Found x10:(Xp cZERO)
% 141.76/142.81 Instantiate: x8:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 141.76/142.81 Found (fun (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of (Xp M)
% 141.76/142.81 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 141.76/142.81 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 141.76/142.81 Found (and_rect20 (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 141.76/142.81 Found ((and_rect2 (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 141.76/142.81 Found (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 141.76/142.81 Found (fun (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (Xp M)
% 141.76/142.81 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 141.76/142.81 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (cNAT M)
% 147.76/148.81 Found x10:(Xp cZERO)
% 147.76/148.81 Instantiate: x8:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 147.76/148.81 Found (fun (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of (Xp M)
% 147.76/148.81 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 147.76/148.81 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 147.76/148.81 Found (and_rect20 (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 147.76/148.81 Found ((and_rect2 (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 147.76/148.81 Found (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 147.76/148.81 Found (fun (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (Xp M)
% 147.76/148.81 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 147.76/148.81 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (cNAT M)
% 147.76/148.81 Found x10:(Xp cZERO)
% 147.76/148.81 Instantiate: x2:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 147.76/148.81 Found (fun (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of (Xp M)
% 147.76/148.81 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 147.76/148.81 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 147.76/148.81 Found (and_rect20 (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 147.76/148.81 Found ((and_rect2 (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 148.36/149.41 Found (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 148.36/149.41 Found (fun (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (Xp M)
% 148.36/149.41 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 148.36/149.41 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (cNAT M)
% 148.36/149.41 Found x10:(Xp cZERO)
% 148.36/149.41 Instantiate: x4:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 148.36/149.41 Found (fun (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of (Xp M)
% 148.36/149.41 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 148.36/149.41 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 148.36/149.41 Found (and_rect20 (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 148.36/149.41 Found ((and_rect2 (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 148.36/149.41 Found (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 148.36/149.41 Found (fun (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (Xp M)
% 148.36/149.41 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 149.26/150.35 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (cNAT M)
% 149.26/150.35 Found x10:(Xp cZERO)
% 149.26/150.35 Instantiate: x4:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 149.26/150.35 Found (fun (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of (Xp M)
% 149.26/150.35 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 149.26/150.35 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 149.26/150.35 Found (and_rect20 (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 149.26/150.35 Found ((and_rect2 (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 149.26/150.35 Found (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 149.26/150.35 Found (fun (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (Xp M)
% 149.26/150.35 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 149.26/150.35 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (cNAT M)
% 149.26/150.35 Found x10:(Xp cZERO)
% 149.26/150.35 Instantiate: x4:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 149.26/150.35 Found (fun (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of (Xp M)
% 149.26/150.35 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 149.26/150.35 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 149.76/150.84 Found (and_rect20 (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 149.76/150.84 Found ((and_rect2 (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 149.76/150.84 Found (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 149.76/150.84 Found (fun (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (Xp M)
% 149.76/150.84 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 149.76/150.84 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (cNAT M)
% 149.76/150.84 Found x10:(Xp cZERO)
% 149.76/150.84 Instantiate: x4:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 149.76/150.84 Found (fun (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of (Xp M)
% 149.76/150.84 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 149.76/150.84 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 149.76/150.84 Found (and_rect20 (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 149.76/150.84 Found ((and_rect2 (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 149.76/150.84 Found (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 149.76/150.84 Found (fun (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (Xp M)
% 149.76/150.84 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 151.66/152.72 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (cNAT M)
% 151.66/152.72 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 151.66/152.72 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 151.66/152.72 Found x6:(Xp cZERO)
% 151.66/152.72 Instantiate: x4:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 151.66/152.72 Found (fun (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of (Xp M)
% 151.66/152.72 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 151.66/152.72 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 151.66/152.72 Found (and_rect20 (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 151.66/152.72 Found ((and_rect2 (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 151.66/152.72 Found (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 151.66/152.72 Found (fun (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (Xp M)
% 151.66/152.72 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 151.66/152.72 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (cNAT M)
% 151.66/152.72 Found x10:(Xp cZERO)
% 151.66/152.72 Instantiate: x6:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 151.66/152.72 Found (fun (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of (Xp M)
% 152.06/153.17 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 152.06/153.17 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 152.06/153.17 Found (and_rect20 (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 152.06/153.17 Found ((and_rect2 (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 152.06/153.17 Found (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 152.06/153.17 Found (fun (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (Xp M)
% 152.06/153.17 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 152.06/153.17 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (cNAT M)
% 152.06/153.17 Found x10:(Xp cZERO)
% 152.06/153.17 Instantiate: x6:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 152.06/153.17 Found (fun (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of (Xp M)
% 152.06/153.17 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 152.06/153.17 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 152.06/153.17 Found (and_rect20 (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 152.06/153.17 Found ((and_rect2 (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 152.06/153.17 Found (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 152.06/153.17 Found (fun (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (Xp M)
% 156.66/157.78 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 156.66/157.78 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (cNAT M)
% 156.66/157.78 Found x20:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 156.66/157.78 Found x20 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 156.66/157.78 Found x20:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 156.66/157.78 Found x20 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 156.66/157.78 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 156.66/157.78 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 156.66/157.78 Found x6:(Xp cZERO)
% 156.66/157.78 Instantiate: x1:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 156.66/157.78 Found (fun (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of (Xp M)
% 156.66/157.78 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 156.66/157.78 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 156.66/157.78 Found (and_rect20 (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 156.66/157.78 Found ((and_rect2 (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 156.66/157.78 Found (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 156.66/157.78 Found (fun (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (Xp M)
% 156.66/157.78 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 158.76/159.82 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (cNAT M)
% 158.76/159.82 Found x6:(Xp cZERO)
% 158.76/159.82 Instantiate: x3:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 158.76/159.82 Found (fun (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of (Xp M)
% 158.76/159.82 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 158.76/159.82 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 158.76/159.82 Found (and_rect20 (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 158.76/159.82 Found ((and_rect2 (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 158.76/159.82 Found (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 158.76/159.82 Found (fun (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (Xp M)
% 158.76/159.82 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 158.76/159.82 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (cNAT M)
% 158.76/159.82 Found x600:=(x60 x5):(P M)
% 158.76/159.82 Found (x60 x5) as proof of (P M)
% 158.76/159.82 Found ((x6 P) x5) as proof of (P M)
% 158.76/159.82 Found (fun (x6:(cNAT M))=> ((x6 P) x5)) as proof of (P M)
% 158.76/159.82 Found (fun (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of ((cNAT M)->(P M))
% 158.76/159.82 Found (fun (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of (forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M)))
% 158.76/159.82 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of (((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))->(forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M))))
% 159.46/160.57 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of cINDUCTION
% 159.46/160.57 Found x600:=(x60 x5):(P M)
% 159.46/160.57 Found (x60 x5) as proof of (P M)
% 159.46/160.57 Found ((x6 P) x5) as proof of (P M)
% 159.46/160.57 Found (fun (x6:(cNAT M))=> ((x6 P) x5)) as proof of (P M)
% 159.46/160.57 Found (fun (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of ((cNAT M)->(P M))
% 159.46/160.57 Found (fun (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of (forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M)))
% 159.46/160.57 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of (((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))->(forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M))))
% 159.46/160.57 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of cINDUCTION
% 159.46/160.57 Found x500:((r cZERO) cZERO)
% 159.46/160.57 Found x500 as proof of ((r cZERO) cZERO)
% 159.46/160.57 Found x500:((r cZERO) cZERO)
% 159.46/160.57 Found x500 as proof of ((r cZERO) cZERO)
% 159.46/160.57 Found x10:(Xp cZERO)
% 159.46/160.57 Instantiate: x2:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 159.46/160.57 Found (fun (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of (Xp M)
% 159.46/160.57 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 159.46/160.57 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 159.46/160.57 Found (and_rect20 (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 159.46/160.57 Found ((and_rect2 (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 159.46/160.57 Found (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 159.46/160.57 Found (fun (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (Xp M)
% 159.46/160.57 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 159.46/160.57 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (cNAT M)
% 160.26/161.39 Found x9:(Xp cZERO)
% 160.26/161.39 Instantiate: x7:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 160.26/161.39 Found (fun (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of (Xp M)
% 160.26/161.39 Found (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 160.26/161.39 Found (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 160.26/161.39 Found (and_rect20 (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 160.26/161.39 Found ((and_rect2 (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 160.26/161.39 Found (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 160.26/161.39 Found (fun (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (Xp M)
% 160.26/161.39 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 160.26/161.39 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (cNAT M)
% 160.26/161.39 Found x10:(Xp cZERO)
% 160.26/161.39 Instantiate: x4:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 160.26/161.39 Found (fun (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of (Xp M)
% 160.26/161.39 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 160.26/161.39 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 160.26/161.39 Found (and_rect20 (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 160.26/161.39 Found ((and_rect2 (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 160.26/161.39 Found (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 160.66/161.77 Found (fun (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (Xp M)
% 160.66/161.77 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 160.66/161.77 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (cNAT M)
% 160.66/161.77 Found x10:(Xp cZERO)
% 160.66/161.77 Instantiate: x2:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 160.66/161.77 Found (fun (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of (Xp M)
% 160.66/161.77 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 160.66/161.77 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 160.66/161.77 Found (and_rect20 (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 160.66/161.77 Found ((and_rect2 (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 160.66/161.77 Found (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 160.66/161.78 Found (fun (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (Xp M)
% 160.66/161.78 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 162.16/163.29 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (cNAT M)
% 162.16/163.29 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 162.16/163.29 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 162.16/163.29 Found x10:(Xp cZERO)
% 162.16/163.29 Instantiate: x4:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 162.16/163.29 Found (fun (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of (Xp M)
% 162.16/163.29 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 162.16/163.29 Found (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 162.16/163.29 Found (and_rect20 (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 162.16/163.29 Found ((and_rect2 (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 162.16/163.29 Found (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10)) as proof of (Xp M)
% 162.16/163.29 Found (fun (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (Xp M)
% 162.16/163.29 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 162.16/163.29 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x9:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x10:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x10) x9)) (Xp M)) (fun (x10:(Xp cZERO)) (x11:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x10))) as proof of (cNAT M)
% 162.16/163.29 Found x6:(Xp cZERO)
% 162.16/163.29 Found (fun (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of (Xp M)
% 162.16/163.29 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 162.16/163.29 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 162.26/163.36 Found (and_rect20 (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 162.26/163.36 Found ((and_rect2 (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 162.26/163.36 Found (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 162.26/163.36 Found (fun (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (Xp M)
% 162.26/163.36 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 162.26/163.36 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (cNAT M)
% 162.26/163.36 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 162.26/163.36 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 162.26/163.36 Found x6:(Xp cZERO)
% 162.26/163.36 Found (fun (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of (Xp M)
% 162.26/163.36 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 162.26/163.36 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 162.26/163.36 Found (and_rect20 (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 162.26/163.36 Found ((and_rect2 (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 162.26/163.36 Found (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 162.26/163.36 Found (fun (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (Xp M)
% 162.26/163.36 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 163.36/164.47 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (cNAT M)
% 163.36/164.47 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 163.36/164.47 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 163.36/164.47 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 163.36/164.47 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 163.36/164.47 Found x600:=(x60 x5):(P M)
% 163.36/164.47 Found (x60 x5) as proof of (P M)
% 163.36/164.47 Found ((x6 P) x5) as proof of (P M)
% 163.36/164.47 Found (fun (x6:(cNAT M))=> ((x6 P) x5)) as proof of (P M)
% 163.36/164.47 Found (fun (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of ((cNAT M)->(P M))
% 163.36/164.47 Found (fun (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of (forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M)))
% 163.36/164.47 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of (((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))->(forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M))))
% 163.36/164.47 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of cINDUCTION
% 163.36/164.47 Found x30:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 163.36/164.47 Found x30 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 163.36/164.47 Found x600:=(x60 x5):(P M)
% 163.36/164.47 Found (x60 x5) as proof of (P M)
% 163.36/164.47 Found ((x6 P) x5) as proof of (P M)
% 163.36/164.47 Found (fun (x6:(cNAT M))=> ((x6 P) x5)) as proof of (P M)
% 163.36/164.47 Found (fun (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of ((cNAT M)->(P M))
% 163.36/164.47 Found (fun (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of (forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M)))
% 163.36/164.47 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of (((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))->(forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M))))
% 163.36/164.47 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of cINDUCTION
% 163.36/164.47 Found x500:((r cZERO) cZERO)
% 163.36/164.47 Found x500 as proof of ((r cZERO) cZERO)
% 163.36/164.47 Found x500:((r cZERO) cZERO)
% 163.36/164.47 Found x500 as proof of ((r cZERO) cZERO)
% 163.36/164.47 Found x20000:=(x2000 x4):(((r Xx0) x4)->((r (cSUCC Xx0)) (cSUCC x4)))
% 163.36/164.47 Found (x2000 x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 163.36/164.47 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 166.16/167.28 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 166.16/167.28 Found x20000:=(x2000 x4):(((r Xx0) x4)->((r (cSUCC Xx0)) (cSUCC x4)))
% 166.16/167.28 Found (x2000 x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 166.16/167.28 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 166.16/167.28 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 166.16/167.28 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 166.16/167.28 Found x9:(Xp cZERO)
% 166.16/167.28 Instantiate: x7:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 166.16/167.28 Found (fun (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of (Xp M)
% 166.16/167.28 Found (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 166.16/167.28 Found (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 166.16/167.28 Found (and_rect20 (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 166.16/167.28 Found ((and_rect2 (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 166.16/167.28 Found (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 166.16/167.28 Found (fun (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (Xp M)
% 166.16/167.28 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 166.16/167.28 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (cNAT M)
% 166.16/167.28 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 166.16/167.28 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 166.16/167.28 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 166.16/167.28 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 166.16/167.28 Found x20:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 166.16/167.28 Found x20 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 166.16/167.28 Found x30:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 169.16/170.20 Found x30 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 169.16/170.20 Found x20000:=(x2000 x4):(((r Xx0) x4)->((r (cSUCC Xx0)) (cSUCC x4)))
% 169.16/170.20 Found (x2000 x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 169.16/170.20 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 169.16/170.20 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 169.16/170.20 Found x20000:=(x2000 x4):(((r Xx0) x4)->((r (cSUCC Xx0)) (cSUCC x4)))
% 169.16/170.20 Found (x2000 x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 169.16/170.20 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 169.16/170.20 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 169.16/170.20 Found x20000:=(x2000 x4):(((r Xx0) x4)->((r (cSUCC Xx0)) (cSUCC x4)))
% 169.16/170.20 Found (x2000 x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 169.16/170.20 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 169.16/170.20 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 169.16/170.20 Found x20000:=(x2000 x6):(((r Xx0) x6)->((r (cSUCC Xx0)) (cSUCC x6)))
% 169.16/170.20 Found (x2000 x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 169.16/170.20 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 169.16/170.20 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 169.16/170.20 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 169.16/170.20 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 169.16/170.20 Found x9:(Xp cZERO)
% 169.16/170.20 Instantiate: x5:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 169.16/170.20 Found (fun (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of (Xp M)
% 169.16/170.20 Found (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 169.16/170.20 Found (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 169.16/170.20 Found (and_rect20 (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 169.16/170.20 Found ((and_rect2 (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 169.16/170.20 Found (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 169.16/170.20 Found (fun (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (Xp M)
% 169.16/170.20 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 169.16/170.20 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (cNAT M)
% 171.77/172.81 Found x8:(Xp cZERO)
% 171.77/172.81 Instantiate: x6:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 171.77/172.81 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 171.77/172.81 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 171.77/172.81 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 171.77/172.81 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 171.77/172.81 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 171.77/172.81 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 171.77/172.81 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 171.77/172.81 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 171.77/172.81 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 171.77/172.81 Found x8:(Xp cZERO)
% 171.77/172.81 Instantiate: x6:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 171.77/172.81 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 171.77/172.81 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 171.77/172.81 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 171.77/172.81 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 171.77/172.81 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 171.77/172.81 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 172.55/173.65 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 172.55/173.65 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 172.55/173.65 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 172.55/173.65 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 172.55/173.65 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 172.55/173.65 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 172.55/173.65 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 172.55/173.65 Found x9:(Xp cZERO)
% 172.55/173.65 Instantiate: x5:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 172.55/173.65 Found (fun (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of (Xp M)
% 172.55/173.65 Found (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 172.55/173.65 Found (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 172.55/173.65 Found (and_rect20 (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 172.55/173.65 Found ((and_rect2 (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 172.55/173.65 Found (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 172.55/173.65 Found (fun (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (Xp M)
% 172.55/173.65 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 174.46/175.56 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (cNAT M)
% 174.46/175.56 Found x8:(Xp cZERO)
% 174.46/175.56 Instantiate: x6:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 174.46/175.56 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 174.46/175.56 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 174.46/175.56 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 174.46/175.56 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 174.46/175.56 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 174.46/175.56 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 174.46/175.56 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 174.46/175.56 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 174.46/175.56 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 174.46/175.56 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 174.46/175.56 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 174.46/175.56 Found x8:(Xp cZERO)
% 174.46/175.56 Instantiate: x6:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 174.86/176.00 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 174.86/176.00 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 174.86/176.00 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 174.86/176.00 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 174.86/176.00 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 174.86/176.00 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 174.86/176.00 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 174.86/176.00 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 174.86/176.00 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 174.86/176.00 Found x9:(Xp cZERO)
% 174.86/176.00 Instantiate: x7:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 174.86/176.00 Found (fun (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of (Xp M)
% 174.86/176.00 Found (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 174.86/176.00 Found (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 174.86/176.00 Found (and_rect20 (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 174.86/176.00 Found ((and_rect2 (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 174.86/176.00 Found (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 174.86/176.00 Found (fun (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (Xp M)
% 176.96/178.06 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 176.96/178.06 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (cNAT M)
% 176.96/178.06 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 176.96/178.06 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 176.96/178.06 Found x9:(Xp cZERO)
% 176.96/178.06 Instantiate: x7:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 176.96/178.06 Found (fun (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of (Xp M)
% 176.96/178.06 Found (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 176.96/178.06 Found (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 176.96/178.06 Found (and_rect20 (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 176.96/178.06 Found ((and_rect2 (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 176.96/178.06 Found (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 176.96/178.06 Found (fun (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (Xp M)
% 176.96/178.06 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 176.96/178.06 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (cNAT M)
% 178.75/179.83 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 178.75/179.83 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 178.75/179.83 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 178.75/179.83 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 178.75/179.83 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 178.75/179.83 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 178.75/179.83 Found x20000:=(x2000 x6):(((r Xx0) x6)->((r (cSUCC Xx0)) (cSUCC x6)))
% 178.75/179.83 Found (x2000 x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x3))
% 178.75/179.83 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x3))
% 178.75/179.83 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x3))
% 178.75/179.83 Found x500:=(x50 x4):(P M)
% 178.75/179.83 Found (x50 x4) as proof of (P M)
% 178.75/179.83 Found ((x5 P) x4) as proof of (P M)
% 178.75/179.83 Found (fun (x5:(cNAT M))=> ((x5 P) x4)) as proof of (P M)
% 178.75/179.83 Found (fun (M:((fofType->Prop)->Prop)) (x5:(cNAT M))=> ((x5 P) x4)) as proof of ((cNAT M)->(P M))
% 178.75/179.83 Found (fun (x4:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x5:(cNAT M))=> ((x5 P) x4)) as proof of (forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M)))
% 178.75/179.83 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x4:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x5:(cNAT M))=> ((x5 P) x4)) as proof of (((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))->(forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M))))
% 178.75/179.83 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x4:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x5:(cNAT M))=> ((x5 P) x4)) as proof of cINDUCTION
% 178.75/179.83 Found x9:(Xp cZERO)
% 178.75/179.83 Instantiate: x7:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 178.75/179.83 Found (fun (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of (Xp M)
% 178.75/179.83 Found (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 178.75/179.83 Found (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 178.75/179.83 Found (and_rect20 (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 178.75/179.83 Found ((and_rect2 (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 178.75/179.83 Found (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 178.75/179.83 Found (fun (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (Xp M)
% 181.46/182.58 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 181.46/182.58 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (cNAT M)
% 181.46/182.58 Found x8:(Xp cZERO)
% 181.46/182.58 Instantiate: x6:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 181.46/182.58 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 181.46/182.58 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 181.46/182.58 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 181.46/182.58 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 181.46/182.58 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 181.46/182.58 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 181.46/182.58 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 181.46/182.58 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 181.46/182.58 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 181.46/182.58 Found x9:(Xp cZERO)
% 181.46/182.58 Instantiate: x7:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 181.46/182.58 Found (fun (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of (Xp M)
% 184.76/185.86 Found (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 184.76/185.86 Found (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 184.76/185.86 Found (and_rect20 (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 184.76/185.86 Found ((and_rect2 (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 184.76/185.86 Found (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 184.76/185.86 Found (fun (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (Xp M)
% 184.76/185.86 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 184.76/185.86 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (cNAT M)
% 184.76/185.86 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 184.76/185.86 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 184.76/185.86 Found x400:cINDUCTION
% 184.76/185.86 Found x400 as proof of cINDUCTION
% 184.76/185.86 Found x400:cINDUCTION
% 184.76/185.86 Found x400 as proof of cINDUCTION
% 184.76/185.86 Found x400:cINDUCTION
% 184.76/185.86 Found x400 as proof of cINDUCTION
% 184.76/185.86 Found x400:cINDUCTION
% 184.76/185.86 Found x400 as proof of cINDUCTION
% 184.76/185.86 Found x400:cINDUCTION
% 184.76/185.86 Found x400 as proof of cINDUCTION
% 184.76/185.86 Found x400:cINDUCTION
% 184.76/185.86 Found x400 as proof of cINDUCTION
% 184.76/185.86 Found x200:=(x20 x5):(((r Xx0) x5)->((r (cSUCC Xx0)) (cSUCC x5)))
% 184.76/185.86 Found (x20 x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 184.76/185.86 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 184.76/185.86 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 184.76/185.86 Found x200:=(x20 x5):(((r Xx0) x5)->((r (cSUCC Xx0)) (cSUCC x5)))
% 184.76/185.86 Found (x20 x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 184.76/185.86 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 184.76/185.86 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 184.76/185.86 Found x200:=(x20 x5):(((r Xx0) x5)->((r (cSUCC Xx0)) (cSUCC x5)))
% 184.76/185.86 Found (x20 x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 184.76/185.86 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 184.76/185.86 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 184.76/185.86 Found x200:=(x20 x5):(((r Xx0) x5)->((r (cSUCC Xx0)) (cSUCC x5)))
% 184.76/185.86 Found (x20 x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 184.76/185.86 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 185.46/186.59 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 185.46/186.59 Found x8:(Xp cZERO)
% 185.46/186.59 Instantiate: x4:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 185.46/186.59 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 185.46/186.59 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 185.46/186.59 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 185.46/186.59 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 185.46/186.59 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 185.46/186.59 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 185.46/186.59 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 185.46/186.59 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 185.46/186.59 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 185.46/186.59 Found x8:(Xp cZERO)
% 185.46/186.59 Instantiate: x4:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 185.46/186.59 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 185.46/186.59 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 185.46/186.59 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 185.46/186.59 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 185.46/186.59 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 185.46/186.59 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 187.86/188.93 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 187.86/188.93 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 187.86/188.93 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 187.86/188.93 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 187.86/188.93 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 187.86/188.93 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 187.86/188.93 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 187.86/188.93 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 187.86/188.93 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 187.86/188.93 Found x400:cINDUCTION
% 187.86/188.93 Found x400 as proof of cINDUCTION
% 187.86/188.93 Found x400:cINDUCTION
% 187.86/188.93 Found x400 as proof of cINDUCTION
% 187.86/188.93 Found x400:cINDUCTION
% 187.86/188.93 Found x400 as proof of cINDUCTION
% 187.86/188.93 Found x400:cINDUCTION
% 187.86/188.93 Found x400 as proof of cINDUCTION
% 187.86/188.93 Found x400:cINDUCTION
% 187.86/188.93 Found x400 as proof of cINDUCTION
% 187.86/188.93 Found x400:cINDUCTION
% 187.86/188.93 Found x400 as proof of cINDUCTION
% 187.86/188.93 Found x8:(Xp cZERO)
% 187.86/188.93 Instantiate: x4:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 187.86/188.93 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 187.86/188.93 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 187.86/188.93 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 187.86/188.93 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 187.86/188.93 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 187.86/188.93 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 187.86/188.93 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 188.56/189.62 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 188.56/189.62 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 188.56/189.62 Found x9:(Xp cZERO)
% 188.56/189.62 Instantiate: x5:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 188.56/189.62 Found (fun (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of (Xp M)
% 188.56/189.62 Found (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 188.56/189.62 Found (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 188.56/189.62 Found (and_rect20 (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 188.56/189.62 Found ((and_rect2 (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 188.56/189.62 Found (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 188.56/189.62 Found (fun (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (Xp M)
% 188.56/189.62 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 188.56/189.62 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (cNAT M)
% 190.56/191.68 Found x9:(Xp cZERO)
% 190.56/191.68 Instantiate: x5:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 190.56/191.68 Found (fun (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of (Xp M)
% 190.56/191.68 Found (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 190.56/191.68 Found (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 190.56/191.68 Found (and_rect20 (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 190.56/191.68 Found ((and_rect2 (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 190.56/191.68 Found (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 190.56/191.68 Found (fun (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (Xp M)
% 190.56/191.68 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 190.56/191.68 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (cNAT M)
% 190.56/191.68 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 190.56/191.68 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 190.56/191.68 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 190.56/191.68 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 190.56/191.68 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 190.56/191.68 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 190.56/191.68 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 190.56/191.68 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 190.56/191.68 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 190.56/191.68 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 192.76/193.82 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 192.76/193.82 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 192.76/193.82 Found x8:(Xp cZERO)
% 192.76/193.82 Instantiate: x2:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 192.76/193.82 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 192.76/193.82 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 192.76/193.82 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 192.76/193.82 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 192.76/193.82 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 192.76/193.82 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 192.76/193.82 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 192.76/193.82 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 192.76/193.82 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 192.76/193.82 Found x8:(Xp cZERO)
% 192.76/193.82 Instantiate: x2:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 192.76/193.82 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 192.76/193.82 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 192.76/193.82 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 192.76/193.82 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 192.76/193.82 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 192.76/193.82 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 194.77/195.84 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 194.77/195.84 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 194.77/195.84 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 194.77/195.84 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 194.77/195.84 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 194.77/195.84 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 194.77/195.84 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 194.77/195.84 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 194.77/195.84 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 194.77/195.84 Found x9:(Xp cZERO)
% 194.77/195.84 Instantiate: x3:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 194.77/195.84 Found (fun (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of (Xp M)
% 194.77/195.84 Found (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 194.77/195.84 Found (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 194.77/195.84 Found (and_rect20 (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 194.77/195.84 Found ((and_rect2 (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 194.77/195.84 Found (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 194.77/195.84 Found (fun (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (Xp M)
% 195.97/197.07 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 195.97/197.07 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (cNAT M)
% 195.97/197.07 Found x8:(Xp cZERO)
% 195.97/197.07 Instantiate: x2:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 195.97/197.07 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 195.97/197.07 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 195.97/197.07 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 195.97/197.07 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 195.97/197.07 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 195.97/197.07 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 195.97/197.07 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 195.97/197.07 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 195.97/197.07 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 195.97/197.07 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 198.46/199.50 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 198.46/199.50 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 198.46/199.50 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 198.46/199.50 Found x9:(Xp cZERO)
% 198.46/199.50 Instantiate: x3:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 198.46/199.50 Found (fun (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of (Xp M)
% 198.46/199.50 Found (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 198.46/199.50 Found (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 198.46/199.50 Found (and_rect20 (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 198.46/199.50 Found ((and_rect2 (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 198.46/199.50 Found (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 198.46/199.50 Found (fun (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (Xp M)
% 198.46/199.50 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 198.46/199.50 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (cNAT M)
% 198.46/199.50 Found x8:(Xp cZERO)
% 198.46/199.50 Instantiate: x2:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 198.46/199.50 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 198.46/199.50 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 198.46/199.50 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 198.46/199.50 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 198.46/199.50 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 199.17/200.20 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 199.17/200.20 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 199.17/200.20 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 199.17/200.20 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 199.17/200.20 Found x9:(Xp cZERO)
% 199.17/200.20 Instantiate: x3:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 199.17/200.20 Found (fun (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of (Xp M)
% 199.17/200.20 Found (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 199.17/200.20 Found (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 199.17/200.20 Found (and_rect20 (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 199.17/200.20 Found ((and_rect2 (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 199.17/200.20 Found (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 199.17/200.20 Found (fun (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (Xp M)
% 199.17/200.20 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 202.76/203.84 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (cNAT M)
% 202.76/203.84 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 202.76/203.84 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 202.76/203.84 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 202.76/203.84 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 202.76/203.84 Found x9:(Xp cZERO)
% 202.76/203.84 Instantiate: x3:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 202.76/203.84 Found (fun (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of (Xp M)
% 202.76/203.84 Found (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 202.76/203.84 Found (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 202.76/203.84 Found (and_rect20 (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 202.76/203.84 Found ((and_rect2 (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 202.76/203.84 Found (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9)) as proof of (Xp M)
% 202.76/203.84 Found (fun (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (Xp M)
% 202.76/203.84 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 202.76/203.84 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x8:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x9:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x9) x8)) (Xp M)) (fun (x9:(Xp cZERO)) (x10:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x9))) as proof of (cNAT M)
% 202.76/203.84 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 202.76/203.84 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 209.37/210.46 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 209.37/210.46 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 209.37/210.46 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 209.37/210.46 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 209.37/210.46 Found x300:cINDUCTION
% 209.37/210.46 Found x300 as proof of cINDUCTION
% 209.37/210.46 Found x300:cINDUCTION
% 209.37/210.46 Found x300 as proof of cINDUCTION
% 209.37/210.46 Found x300:cINDUCTION
% 209.37/210.46 Found x300 as proof of cINDUCTION
% 209.37/210.46 Found x300:cINDUCTION
% 209.37/210.46 Found x300 as proof of cINDUCTION
% 209.37/210.46 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 209.37/210.46 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 209.37/210.46 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 209.37/210.46 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 209.37/210.46 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 209.37/210.46 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 209.37/210.46 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 209.37/210.46 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 209.37/210.46 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 209.37/210.46 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 209.37/210.47 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 209.37/210.47 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 209.37/210.47 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 209.37/210.47 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 209.37/210.47 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 209.37/210.47 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 209.37/210.47 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 209.37/210.47 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 209.37/210.47 Found x20:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 209.37/210.47 Found x20 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 209.37/210.47 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 209.37/210.47 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 209.37/210.47 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 209.37/210.47 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 209.37/210.47 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 213.47/214.59 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 213.47/214.59 Found x20:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 213.47/214.59 Found x20 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 213.47/214.59 Found x200:=(x20 x5):(((r Xx0) x5)->((r (cSUCC Xx0)) (cSUCC x5)))
% 213.47/214.59 Found (x20 x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 213.47/214.59 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 213.47/214.59 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 213.47/214.59 Found x200:=(x20 x5):(((r Xx0) x5)->((r (cSUCC Xx0)) (cSUCC x5)))
% 213.47/214.59 Found (x20 x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 213.47/214.59 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 213.47/214.59 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 213.47/214.59 Found x200:=(x20 x5):(((r Xx0) x5)->((r (cSUCC Xx0)) (cSUCC x5)))
% 213.47/214.59 Found (x20 x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 213.47/214.59 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 213.47/214.59 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 213.47/214.59 Found x20:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 213.47/214.59 Found x20 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 213.47/214.59 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 213.47/214.59 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 213.47/214.59 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 213.47/214.59 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 213.47/214.59 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 213.47/214.59 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 213.47/214.59 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 213.47/214.59 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 213.47/214.59 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 213.47/214.59 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 213.47/214.59 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 213.47/214.59 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 213.47/214.59 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 213.47/214.59 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 213.47/214.59 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 213.47/214.59 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 213.47/214.59 Found x410:=(x41 x400):((r Xx0) x1)
% 213.47/214.59 Instantiate: x1:=X:((fofType->Prop)->Prop)
% 213.47/214.59 Found (x41 x400) as proof of ((r Xx0) X)
% 213.47/214.59 Found ((fun (x401:cINDUCTION)=> ((x4 x401) x50)) x400) as proof of ((r Xx0) X)
% 213.47/214.59 Found ((fun (x401:cINDUCTION)=> ((x4 x401) x50)) x400) as proof of ((r Xx0) X)
% 213.47/214.59 Found (x300 ((fun (x401:cINDUCTION)=> ((x4 x401) x50)) x400)) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 213.47/214.59 Found ((x30 X) ((fun (x401:cINDUCTION)=> ((x4 x401) x50)) x400)) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 218.87/219.99 Found (((x3 Xx0) X) ((fun (x401:cINDUCTION)=> ((x4 x401) x50)) x400)) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 218.87/219.99 Found (fun (x5:((r (cSUCC Xx0)) X))=> (((x3 Xx0) X) ((fun (x401:cINDUCTION)=> ((x4 x401) x50)) x400))) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 218.87/219.99 Found x410:=(x41 x400):((r Xx0) x3)
% 218.87/219.99 Instantiate: x3:=X:((fofType->Prop)->Prop)
% 218.87/219.99 Found (x41 x400) as proof of ((r Xx0) X)
% 218.87/219.99 Found ((fun (x401:cINDUCTION)=> ((x4 x401) x50)) x400) as proof of ((r Xx0) X)
% 218.87/219.99 Found ((fun (x401:cINDUCTION)=> ((x4 x401) x50)) x400) as proof of ((r Xx0) X)
% 218.87/219.99 Found (x200 ((fun (x401:cINDUCTION)=> ((x4 x401) x50)) x400)) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 218.87/219.99 Found ((x20 X) ((fun (x401:cINDUCTION)=> ((x4 x401) x50)) x400)) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 218.87/219.99 Found (((x2 Xx0) X) ((fun (x401:cINDUCTION)=> ((x4 x401) x50)) x400)) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 218.87/219.99 Found (fun (x5:((r (cSUCC Xx0)) X))=> (((x2 Xx0) X) ((fun (x401:cINDUCTION)=> ((x4 x401) x50)) x400))) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 218.87/219.99 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 218.87/219.99 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 218.87/219.99 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 218.87/219.99 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 218.87/219.99 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 218.87/219.99 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 218.87/219.99 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 218.87/219.99 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 218.87/219.99 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 218.87/219.99 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 218.87/219.99 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 218.87/219.99 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 218.87/219.99 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 218.87/219.99 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 218.87/219.99 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 218.87/219.99 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 218.87/219.99 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 218.87/219.99 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 218.87/219.99 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 218.87/219.99 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 218.87/219.99 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 218.87/219.99 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 218.87/219.99 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 224.27/225.38 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 224.27/225.38 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 224.27/225.38 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 224.27/225.38 Found x500:((r cZERO) cZERO)
% 224.27/225.38 Found x500 as proof of ((r cZERO) cZERO)
% 224.27/225.38 Found x500:((r cZERO) cZERO)
% 224.27/225.38 Found x500 as proof of ((r cZERO) cZERO)
% 224.27/225.38 Found x500:((r cZERO) cZERO)
% 224.27/225.38 Found x500 as proof of ((r cZERO) cZERO)
% 224.27/225.38 Found x500:((r cZERO) cZERO)
% 224.27/225.38 Found x500 as proof of ((r cZERO) cZERO)
% 224.27/225.38 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 224.27/225.38 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 224.27/225.38 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 224.27/225.38 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 224.27/225.38 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 224.27/225.38 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 224.27/225.38 Found x20:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 224.27/225.38 Found x20 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 224.27/225.38 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 224.27/225.38 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 224.27/225.38 Found x500:((r cZERO) cZERO)
% 224.27/225.38 Found x500 as proof of ((r cZERO) cZERO)
% 224.27/225.38 Found x30:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 224.27/225.38 Found x30 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 224.27/225.38 Found x500:((r cZERO) cZERO)
% 224.27/225.38 Found x500 as proof of ((r cZERO) cZERO)
% 224.27/225.38 Found x30:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 224.27/225.38 Found x30 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 224.27/225.38 Found x2000:=(x200 x6):(((r Xx0) x6)->((r (cSUCC Xx0)) (cSUCC x6)))
% 224.27/225.38 Found (x200 x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 224.27/225.38 Found ((x20 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 224.27/225.38 Found ((x20 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 224.27/225.38 Found ((x20 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 224.27/225.38 Found x2000:=(x200 x6):(((r Xx0) x6)->((r (cSUCC Xx0)) (cSUCC x6)))
% 224.27/225.38 Found (x200 x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 224.27/225.38 Found ((x20 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 224.27/225.38 Found ((x20 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 224.27/225.38 Found x500:((r cZERO) cZERO)
% 224.27/225.38 Found x500 as proof of ((r cZERO) cZERO)
% 224.27/225.38 Found x500:((r cZERO) cZERO)
% 224.27/225.38 Found x500 as proof of ((r cZERO) cZERO)
% 224.27/225.38 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 224.27/225.38 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 224.27/225.38 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 224.27/225.38 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 224.27/225.38 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 226.78/227.84 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 226.78/227.84 Found x30:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 226.78/227.84 Found x30 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 226.78/227.84 Found x30:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 226.78/227.84 Found x30 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 226.78/227.84 Found x2000:=(x200 x6):(((r Xx0) x6)->((r (cSUCC Xx0)) (cSUCC x6)))
% 226.78/227.84 Found (x200 x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 226.78/227.84 Found ((x20 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 226.78/227.84 Found ((x20 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 226.78/227.84 Found x2000:=(x200 x6):(((r Xx0) x6)->((r (cSUCC Xx0)) (cSUCC x6)))
% 226.78/227.84 Found (x200 x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 226.78/227.84 Found ((x20 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 226.78/227.84 Found ((x20 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 226.78/227.84 Found x2000:=(x200 x6):(((r Xx0) x6)->((r (cSUCC Xx0)) (cSUCC x6)))
% 226.78/227.84 Found (x200 x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 226.78/227.84 Found ((x20 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 226.78/227.84 Found ((x20 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 226.78/227.84 Found x7:(Xp cZERO)
% 226.78/227.84 Found (fun (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of (Xp M)
% 226.78/227.84 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 226.78/227.84 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 226.78/227.84 Found (and_rect20 (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 226.78/227.84 Found ((and_rect2 (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 226.78/227.84 Found (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 226.78/227.84 Found (fun (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (Xp M)
% 226.78/227.84 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 226.78/227.84 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (cNAT M)
% 227.97/229.00 Found x7:(Xp cZERO)
% 227.97/229.00 Found (fun (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of (Xp M)
% 227.97/229.00 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 227.97/229.00 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 227.97/229.00 Found (and_rect20 (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 227.97/229.00 Found ((and_rect2 (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 227.97/229.00 Found (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 227.97/229.00 Found (fun (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (Xp M)
% 227.97/229.00 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 227.97/229.00 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (cNAT M)
% 227.97/229.00 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 227.97/229.00 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 227.97/229.00 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 227.97/229.00 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 227.97/229.00 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 227.97/229.00 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 227.97/229.00 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 227.97/229.00 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 227.97/229.00 Found x20:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 227.97/229.00 Found x20 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 229.57/230.60 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 229.57/230.60 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 229.57/230.60 Found x7:(Xp cZERO)
% 229.57/230.60 Found (fun (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of (Xp M)
% 229.57/230.60 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 229.57/230.60 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 229.57/230.60 Found (and_rect20 (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 229.57/230.60 Found ((and_rect2 (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 229.57/230.60 Found (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 229.57/230.60 Found (fun (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (Xp M)
% 229.57/230.60 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 229.57/230.60 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (cNAT M)
% 229.57/230.60 Found x7:(Xp cZERO)
% 229.57/230.60 Found (fun (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of (Xp M)
% 229.57/230.60 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 229.57/230.60 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 229.57/230.60 Found (and_rect20 (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 229.57/230.60 Found ((and_rect2 (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 229.57/230.60 Found (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 229.57/230.60 Found (fun (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (Xp M)
% 236.18/237.29 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 236.18/237.29 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (cNAT M)
% 236.18/237.29 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 236.18/237.29 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 236.18/237.29 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 236.18/237.29 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 236.18/237.29 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 236.18/237.29 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 236.18/237.29 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 236.18/237.29 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 236.18/237.29 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 236.18/237.29 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 236.18/237.29 Found x20:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 236.18/237.29 Found x20 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 236.18/237.29 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 236.18/237.29 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 236.18/237.29 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 236.18/237.29 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 236.18/237.29 Found x30:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 236.18/237.29 Found x30 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 236.18/237.29 Found x20:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 236.18/237.29 Found x20 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 241.47/242.51 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 241.47/242.51 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 241.47/242.51 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 241.47/242.51 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 241.47/242.51 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 241.47/242.51 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 241.47/242.51 Found x20:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 241.47/242.51 Found x20 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 241.47/242.52 Found x30:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 241.47/242.52 Found x30 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 241.47/242.52 Found x20000:=(x2000 x4):(((r Xx0) x4)->((r (cSUCC Xx0)) (cSUCC x4)))
% 241.47/242.52 Found (x2000 x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 241.47/242.52 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 241.47/242.52 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 241.47/242.52 Found x20000:=(x2000 x4):(((r Xx0) x4)->((r (cSUCC Xx0)) (cSUCC x4)))
% 241.47/242.52 Found (x2000 x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 241.47/242.52 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 241.47/242.52 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 241.47/242.52 Found x30:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 241.47/242.52 Found x30 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 241.47/242.52 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 241.47/242.52 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 241.47/242.52 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 241.47/242.52 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 241.47/242.52 Found x30:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 241.47/242.52 Found x30 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 241.47/242.52 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 241.47/242.52 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 241.47/242.52 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 241.47/242.52 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 241.47/242.52 Found x7:(Xp cZERO)
% 241.47/242.52 Instantiate: x1:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 241.47/242.52 Found (fun (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of (Xp M)
% 241.47/242.52 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 241.47/242.52 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 242.47/243.54 Found (and_rect20 (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 242.47/243.54 Found ((and_rect2 (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 242.47/243.54 Found (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 242.47/243.54 Found (fun (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (Xp M)
% 242.47/243.54 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 242.47/243.54 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (cNAT M)
% 242.47/243.54 Found x30:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 242.47/243.54 Found x30 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 242.47/243.54 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 242.47/243.54 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 242.47/243.54 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 242.47/243.54 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 242.47/243.54 Found x6:(Xp cZERO)
% 242.47/243.54 Instantiate: x4:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 242.47/243.54 Found (fun (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of (Xp M)
% 242.47/243.54 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 242.47/243.54 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 242.47/243.54 Found (and_rect20 (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 242.47/243.54 Found ((and_rect2 (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 242.47/243.54 Found (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 243.18/244.26 Found (fun (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (Xp M)
% 243.18/244.26 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 243.18/244.26 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (cNAT M)
% 243.18/244.26 Found x6:(Xp cZERO)
% 243.18/244.26 Instantiate: x4:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 243.18/244.26 Found (fun (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of (Xp M)
% 243.18/244.26 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 243.18/244.26 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 243.18/244.26 Found (and_rect20 (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 243.18/244.26 Found ((and_rect2 (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 243.18/244.26 Found (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 243.18/244.26 Found (fun (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (Xp M)
% 243.18/244.26 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 243.18/244.26 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (cNAT M)
% 247.68/248.79 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 247.68/248.79 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 247.68/248.79 Found x7:(Xp cZERO)
% 247.68/248.79 Found (fun (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of (Xp M)
% 247.68/248.79 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 247.68/248.79 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 247.68/248.79 Found (and_rect20 (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 247.68/248.79 Found ((and_rect2 (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 247.68/248.79 Found (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 247.68/248.79 Found (fun (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (Xp M)
% 247.68/248.79 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 247.68/248.79 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (cNAT M)
% 247.68/248.79 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 247.68/248.79 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 247.68/248.79 Found x600:=(x60 x5):(P M)
% 247.68/248.79 Found (x60 x5) as proof of (P M)
% 247.68/248.79 Found ((x6 P) x5) as proof of (P M)
% 247.68/248.79 Found (fun (x6:(cNAT M))=> ((x6 P) x5)) as proof of (P M)
% 247.68/248.79 Found (fun (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of ((cNAT M)->(P M))
% 247.68/248.79 Found (fun (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of (forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M)))
% 247.68/248.79 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of (((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))->(forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M))))
% 248.38/249.43 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of cINDUCTION
% 248.38/249.43 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 248.38/249.43 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 248.38/249.43 Found x600:=(x60 x5):(P M)
% 248.38/249.43 Found (x60 x5) as proof of (P M)
% 248.38/249.43 Found ((x6 P) x5) as proof of (P M)
% 248.38/249.43 Found (fun (x6:(cNAT M))=> ((x6 P) x5)) as proof of (P M)
% 248.38/249.43 Found (fun (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of ((cNAT M)->(P M))
% 248.38/249.43 Found (fun (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of (forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M)))
% 248.38/249.43 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of (((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))->(forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M))))
% 248.38/249.43 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of cINDUCTION
% 248.38/249.43 Found x6:(Xp cZERO)
% 248.38/249.43 Instantiate: x1:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 248.38/249.43 Found (fun (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of (Xp M)
% 248.38/249.43 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 248.38/249.43 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 248.38/249.43 Found (and_rect20 (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 248.38/249.43 Found ((and_rect2 (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 248.38/249.43 Found (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 248.38/249.43 Found (fun (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (Xp M)
% 248.38/249.43 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 248.38/249.43 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (cNAT M)
% 249.38/250.43 Found x6:(Xp cZERO)
% 249.38/250.43 Instantiate: x1:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 249.38/250.43 Found (fun (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of (Xp M)
% 249.38/250.43 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 249.38/250.43 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 249.38/250.43 Found (and_rect20 (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 249.38/250.43 Found ((and_rect2 (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 249.38/250.43 Found (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 249.38/250.43 Found (fun (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (Xp M)
% 249.38/250.43 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 249.38/250.43 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (cNAT M)
% 249.38/250.43 Found x6:(Xp cZERO)
% 249.38/250.43 Instantiate: x3:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 249.38/250.43 Found (fun (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of (Xp M)
% 249.38/250.43 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 249.38/250.43 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 249.38/250.43 Found (and_rect20 (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 249.38/250.43 Found ((and_rect2 (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 249.79/250.88 Found (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 249.79/250.88 Found (fun (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (Xp M)
% 249.79/250.88 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 249.79/250.88 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (cNAT M)
% 249.79/250.88 Found x6:(Xp cZERO)
% 249.79/250.88 Instantiate: x3:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 249.79/250.88 Found (fun (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of (Xp M)
% 249.79/250.88 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 249.79/250.88 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 249.79/250.88 Found (and_rect20 (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 249.79/250.88 Found ((and_rect2 (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 249.79/250.88 Found (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 249.79/250.88 Found (fun (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (Xp M)
% 249.79/250.88 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 251.88/252.96 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (cNAT M)
% 251.88/252.96 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 251.88/252.96 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 251.88/252.96 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 251.88/252.96 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 251.88/252.96 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 251.88/252.96 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 251.88/252.96 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 251.88/252.96 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 251.88/252.96 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 251.88/252.96 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 251.88/252.96 Found x600:=(x60 x5):(P M)
% 251.88/252.96 Found (x60 x5) as proof of (P M)
% 251.88/252.96 Found ((x6 P) x5) as proof of (P M)
% 251.88/252.96 Found (fun (x6:(cNAT M))=> ((x6 P) x5)) as proof of (P M)
% 251.88/252.96 Found (fun (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of ((cNAT M)->(P M))
% 251.88/252.96 Found (fun (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of (forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M)))
% 251.88/252.96 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of (((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))->(forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M))))
% 251.88/252.96 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of cINDUCTION
% 251.88/252.96 Found x600:=(x60 x5):(P M)
% 251.88/252.96 Found (x60 x5) as proof of (P M)
% 251.88/252.96 Found ((x6 P) x5) as proof of (P M)
% 251.88/252.96 Found (fun (x6:(cNAT M))=> ((x6 P) x5)) as proof of (P M)
% 251.88/252.96 Found (fun (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of ((cNAT M)->(P M))
% 251.88/252.96 Found (fun (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of (forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M)))
% 251.88/252.96 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of (((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))->(forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M))))
% 251.88/252.96 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x5:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x6:(cNAT M))=> ((x6 P) x5)) as proof of cINDUCTION
% 251.88/252.96 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 251.88/252.96 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 252.68/253.70 Found x6:(Xp cZERO)
% 252.68/253.70 Found (fun (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of (Xp M)
% 252.68/253.70 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 252.68/253.70 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 252.68/253.70 Found (and_rect20 (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 252.68/253.70 Found ((and_rect2 (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 252.68/253.70 Found (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 252.68/253.70 Found (fun (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (Xp M)
% 252.68/253.70 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 252.68/253.70 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (cNAT M)
% 252.68/253.70 Found x20:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 252.68/253.70 Found x20 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 252.68/253.70 Found x20:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 252.68/253.70 Found x20 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 252.68/253.70 Found x6:(Xp cZERO)
% 252.68/253.70 Found (fun (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of (Xp M)
% 252.68/253.70 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 252.68/253.70 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 252.68/253.70 Found (and_rect20 (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 252.68/253.70 Found ((and_rect2 (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 252.68/253.70 Found (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 252.68/253.72 Found (fun (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (Xp M)
% 252.68/253.72 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 252.68/253.72 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (cNAT M)
% 252.68/253.72 Found x6:(Xp cZERO)
% 252.68/253.72 Found (fun (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of (Xp M)
% 252.68/253.72 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 252.68/253.72 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 252.68/253.72 Found (and_rect20 (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 252.68/253.72 Found ((and_rect2 (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 252.68/253.72 Found (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 252.68/253.72 Found (fun (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (Xp M)
% 252.68/253.72 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 252.68/253.72 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (cNAT M)
% 258.19/259.27 Found x20:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 258.19/259.27 Found x20 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 258.19/259.27 Found x6:(Xp cZERO)
% 258.19/259.27 Found (fun (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of (Xp M)
% 258.19/259.27 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 258.19/259.27 Found (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 258.19/259.27 Found (and_rect20 (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 258.19/259.27 Found ((and_rect2 (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 258.19/259.27 Found (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6)) as proof of (Xp M)
% 258.19/259.27 Found (fun (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (Xp M)
% 258.19/259.27 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 258.19/259.27 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x5:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x6:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x6) x5)) (Xp M)) (fun (x6:(Xp cZERO)) (x7:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x6))) as proof of (cNAT M)
% 258.19/259.27 Found x20:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 258.19/259.27 Found x20 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 258.19/259.27 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 258.19/259.27 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 258.19/259.27 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 258.19/259.27 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 258.19/259.27 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 260.18/261.25 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 260.18/261.25 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 260.18/261.25 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 260.18/261.25 Found x7:(Xp cZERO)
% 260.18/261.25 Instantiate: x5:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 260.18/261.25 Found (fun (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of (Xp M)
% 260.18/261.25 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 260.18/261.25 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 260.18/261.25 Found (and_rect20 (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 260.18/261.25 Found ((and_rect2 (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 260.18/261.25 Found (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 260.18/261.25 Found (fun (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (Xp M)
% 260.18/261.25 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 260.18/261.25 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (cNAT M)
% 260.18/261.25 Found x20:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 260.18/261.25 Found x20 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 260.18/261.25 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 260.18/261.25 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 260.18/261.25 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 260.18/261.25 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 260.18/261.25 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 265.19/266.23 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 265.19/266.23 Found x20000:=(x2000 x4):(((r Xx0) x4)->((r (cSUCC Xx0)) (cSUCC x4)))
% 265.19/266.23 Found (x2000 x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 265.19/266.23 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 265.19/266.23 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 265.19/266.23 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 265.19/266.23 Found x20000:=(x2000 x4):(((r Xx0) x4)->((r (cSUCC Xx0)) (cSUCC x4)))
% 265.19/266.23 Found (x2000 x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 265.19/266.23 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 265.19/266.23 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 265.19/266.23 Found x20000:=(x2000 x4):(((r Xx0) x4)->((r (cSUCC Xx0)) (cSUCC x4)))
% 265.19/266.23 Found (x2000 x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 265.19/266.23 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 265.19/266.23 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 265.19/266.23 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 265.19/266.23 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 265.19/266.23 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 265.19/266.23 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 265.19/266.23 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 265.19/266.23 Found x20:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 265.19/266.23 Found x20 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 265.19/266.23 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 265.19/266.23 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 265.19/266.23 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 265.19/266.23 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 265.19/266.23 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 265.19/266.23 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 265.19/266.23 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 265.19/266.23 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 265.19/266.23 Found x20000:=(x2000 x4):(((r Xx0) x4)->((r (cSUCC Xx0)) (cSUCC x4)))
% 265.19/266.23 Found (x2000 x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 265.19/266.23 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 265.19/266.23 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 265.19/266.23 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 265.19/266.23 Found x20000:=(x2000 x4):(((r Xx0) x4)->((r (cSUCC Xx0)) (cSUCC x4)))
% 265.19/266.23 Found (x2000 x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 265.19/266.23 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 265.19/266.23 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 265.19/266.23 Found x500:=(x50 x4):(P M)
% 265.19/266.23 Found (x50 x4) as proof of (P M)
% 265.19/266.23 Found ((x5 P) x4) as proof of (P M)
% 265.19/266.23 Found (fun (x5:(cNAT M))=> ((x5 P) x4)) as proof of (P M)
% 265.19/266.23 Found (fun (M:((fofType->Prop)->Prop)) (x5:(cNAT M))=> ((x5 P) x4)) as proof of ((cNAT M)->(P M))
% 265.19/266.23 Found (fun (x4:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x5:(cNAT M))=> ((x5 P) x4)) as proof of (forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M)))
% 267.19/268.23 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x4:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x5:(cNAT M))=> ((x5 P) x4)) as proof of (((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))->(forall (M:((fofType->Prop)->Prop)), ((cNAT M)->(P M))))
% 267.19/268.23 Found (fun (P:(((fofType->Prop)->Prop)->Prop)) (x4:((and (P cZERO)) (forall (X:((fofType->Prop)->Prop)), ((P X)->(P (cSUCC X)))))) (M:((fofType->Prop)->Prop)) (x5:(cNAT M))=> ((x5 P) x4)) as proof of cINDUCTION
% 267.19/268.23 Found x20000:=(x2000 x4):(((r Xx0) x4)->((r (cSUCC Xx0)) (cSUCC x4)))
% 267.19/268.23 Found (x2000 x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 267.19/268.23 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 267.19/268.23 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 267.19/268.23 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 267.19/268.23 Found x20000:=(x2000 x4):(((r Xx0) x4)->((r (cSUCC Xx0)) (cSUCC x4)))
% 267.19/268.23 Found (x2000 x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 267.19/268.23 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 267.19/268.23 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 267.19/268.23 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 267.19/268.23 Found x20000:=(x2000 x4):(((r Xx0) x4)->((r (cSUCC Xx0)) (cSUCC x4)))
% 267.19/268.23 Found (x2000 x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 267.19/268.23 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 267.19/268.23 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 267.19/268.23 Found x20000:=(x2000 x4):(((r Xx0) x4)->((r (cSUCC Xx0)) (cSUCC x4)))
% 267.19/268.23 Found (x2000 x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 267.19/268.23 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 267.19/268.23 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 267.19/268.23 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 267.19/268.23 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 267.19/268.23 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 267.19/268.23 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 267.19/268.23 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 267.19/268.23 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 267.19/268.23 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 267.19/268.23 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 267.19/268.23 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 267.19/268.23 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 267.19/268.23 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 267.19/268.23 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 267.19/268.23 Found x20000:=(x2000 x6):(((r Xx0) x6)->((r (cSUCC Xx0)) (cSUCC x6)))
% 267.19/268.23 Found (x2000 x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 267.19/268.23 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 267.19/268.23 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 267.19/268.23 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 267.19/268.23 Found x20000:=(x2000 x6):(((r Xx0) x6)->((r (cSUCC Xx0)) (cSUCC x6)))
% 267.19/268.23 Found (x2000 x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 267.19/268.23 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 267.19/268.23 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 271.49/272.59 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 271.49/272.59 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 271.49/272.59 Found x20000:=(x2000 x4):(((r Xx0) x4)->((r (cSUCC Xx0)) (cSUCC x4)))
% 271.49/272.59 Found (x2000 x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 271.49/272.59 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 271.49/272.59 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 271.49/272.59 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 271.49/272.59 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 271.49/272.59 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 271.49/272.59 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 271.49/272.59 Found x20000:=(x2000 x4):(((r Xx0) x4)->((r (cSUCC Xx0)) (cSUCC x4)))
% 271.49/272.59 Found (x2000 x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 271.49/272.59 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 271.49/272.59 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 271.49/272.59 Found x20000:=(x2000 x4):(((r Xx0) x4)->((r (cSUCC Xx0)) (cSUCC x4)))
% 271.49/272.59 Found (x2000 x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 271.49/272.59 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 271.49/272.59 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 271.49/272.59 Found x20000:=(x2000 x4):(((r Xx0) x4)->((r (cSUCC Xx0)) (cSUCC x4)))
% 271.49/272.59 Found (x2000 x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 271.49/272.59 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 271.49/272.59 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 271.49/272.59 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 271.49/272.59 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 271.49/272.59 Found x20000:=(x2000 x4):(((r Xx0) x4)->((r (cSUCC Xx0)) (cSUCC x4)))
% 271.49/272.59 Found (x2000 x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 271.49/272.59 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 271.49/272.59 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 271.49/272.59 Found x20000:=(x2000 x4):(((r Xx0) x4)->((r (cSUCC Xx0)) (cSUCC x4)))
% 271.49/272.59 Found (x2000 x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 271.49/272.59 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 271.49/272.59 Found ((x200 Xx0) x4) as proof of (((r Xx0) x4)->((r (cSUCC Xx0)) x3))
% 271.49/272.59 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 271.49/272.59 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 271.49/272.59 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 271.49/272.59 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 271.49/272.59 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 271.49/272.59 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 271.49/272.59 Found x400:cINDUCTION
% 271.49/272.59 Found x400 as proof of cINDUCTION
% 271.49/272.59 Found x400:cINDUCTION
% 271.49/272.59 Found x400 as proof of cINDUCTION
% 271.49/272.59 Found x400:cINDUCTION
% 271.49/272.59 Found x400 as proof of cINDUCTION
% 271.49/272.59 Found x400:cINDUCTION
% 271.49/272.59 Found x400 as proof of cINDUCTION
% 271.49/272.59 Found x400:cINDUCTION
% 271.49/272.59 Found x400 as proof of cINDUCTION
% 271.49/272.59 Found x400:cINDUCTION
% 271.49/272.59 Found x400 as proof of cINDUCTION
% 271.49/272.59 Found x400:cINDUCTION
% 271.49/272.59 Found x400 as proof of cINDUCTION
% 271.49/272.59 Found x400:cINDUCTION
% 271.49/272.59 Found x400 as proof of cINDUCTION
% 271.49/272.59 Found x20000:=(x2000 x6):(((r Xx0) x6)->((r (cSUCC Xx0)) (cSUCC x6)))
% 272.49/273.58 Found (x2000 x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 272.49/273.58 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 272.49/273.58 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 272.49/273.58 Found x20000:=(x2000 x6):(((r Xx0) x6)->((r (cSUCC Xx0)) (cSUCC x6)))
% 272.49/273.58 Found (x2000 x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 272.49/273.58 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 272.49/273.58 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 272.49/273.58 Found x20000:=(x2000 x6):(((r Xx0) x6)->((r (cSUCC Xx0)) (cSUCC x6)))
% 272.49/273.58 Found (x2000 x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 272.49/273.58 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 272.49/273.58 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 272.49/273.58 Found x20000:=(x2000 x6):(((r Xx0) x6)->((r (cSUCC Xx0)) (cSUCC x6)))
% 272.49/273.58 Found (x2000 x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 272.49/273.58 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 272.49/273.58 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 272.49/273.58 Found x20000:=(x2000 x6):(((r Xx0) x6)->((r (cSUCC Xx0)) (cSUCC x6)))
% 272.49/273.58 Found (x2000 x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 272.49/273.58 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 272.49/273.58 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 272.49/273.58 Found x20000:=(x2000 x6):(((r Xx0) x6)->((r (cSUCC Xx0)) (cSUCC x6)))
% 272.49/273.58 Found (x2000 x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 272.49/273.58 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 272.49/273.58 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 272.49/273.58 Found x20000:=(x2000 x6):(((r Xx0) x6)->((r (cSUCC Xx0)) (cSUCC x6)))
% 272.49/273.58 Found (x2000 x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 272.49/273.58 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 272.49/273.58 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 272.49/273.58 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 272.49/273.58 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 272.49/273.58 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 272.49/273.58 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 272.49/273.58 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 272.49/273.58 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 272.49/273.58 Found x8:(Xp cZERO)
% 272.49/273.58 Instantiate: x6:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 272.49/273.58 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 272.49/273.58 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 272.49/273.58 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 272.49/273.58 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 272.49/273.58 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 272.49/273.58 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 272.49/273.58 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 276.29/277.31 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 276.29/277.31 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 276.29/277.31 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 276.29/277.31 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 276.29/277.31 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 276.29/277.31 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 276.29/277.31 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 276.29/277.31 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 276.29/277.31 Found x30:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 276.29/277.31 Found x30 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 276.29/277.31 Found x400:cINDUCTION
% 276.29/277.31 Found x400 as proof of cINDUCTION
% 276.29/277.31 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 276.29/277.31 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 276.29/277.31 Found x400:cINDUCTION
% 276.29/277.31 Found x400 as proof of cINDUCTION
% 276.29/277.31 Found x400:cINDUCTION
% 276.29/277.31 Found x400 as proof of cINDUCTION
% 276.29/277.31 Found x50:((r Xx0) x1)
% 276.29/277.31 Instantiate: x1:=X:((fofType->Prop)->Prop)
% 276.29/277.31 Found x50 as proof of ((r Xx0) X)
% 276.29/277.31 Found (x3000 x50) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 276.29/277.31 Found ((x300 X) x50) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 276.29/277.31 Found (((x30 Xx0) X) x50) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 276.29/277.31 Found (fun (x6:((r (cSUCC Xx0)) X))=> (((x30 Xx0) X) x50)) as proof of ((r (cSUCC Xx0)) (cSUCC X))
% 276.29/277.31 Found x400:cINDUCTION
% 276.29/277.31 Found x400 as proof of cINDUCTION
% 276.29/277.31 Found x400:cINDUCTION
% 276.29/277.31 Found x400 as proof of cINDUCTION
% 276.29/277.31 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 276.29/277.31 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 276.29/277.31 Found x400:cINDUCTION
% 276.29/277.31 Found x400 as proof of cINDUCTION
% 276.29/277.31 Found x400:cINDUCTION
% 276.29/277.31 Found x400 as proof of cINDUCTION
% 276.29/277.31 Found x400:cINDUCTION
% 276.29/277.31 Found x400 as proof of cINDUCTION
% 276.29/277.31 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 276.29/277.31 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 276.29/277.31 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 278.99/280.09 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 278.99/280.09 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 278.99/280.09 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 278.99/280.09 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 278.99/280.09 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 278.99/280.09 Found x20000:=(x2000 x6):(((r Xx0) x6)->((r (cSUCC Xx0)) (cSUCC x6)))
% 278.99/280.09 Found (x2000 x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 278.99/280.09 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 278.99/280.09 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 278.99/280.09 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 278.99/280.09 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 278.99/280.09 Found x20000:=(x2000 x6):(((r Xx0) x6)->((r (cSUCC Xx0)) (cSUCC x6)))
% 278.99/280.09 Found (x2000 x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 278.99/280.09 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 278.99/280.09 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x5))
% 278.99/280.09 Found x30:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 278.99/280.09 Found x30 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 278.99/280.09 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 278.99/280.09 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 278.99/280.09 Found x20000:=(x2000 x6):(((r Xx0) x6)->((r (cSUCC Xx0)) (cSUCC x6)))
% 278.99/280.09 Found (x2000 x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x3))
% 278.99/280.09 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x3))
% 278.99/280.09 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x3))
% 278.99/280.09 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x3))
% 278.99/280.09 Found x200:=(x20 x5):(((r Xx0) x5)->((r (cSUCC Xx0)) (cSUCC x5)))
% 278.99/280.09 Found (x20 x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 278.99/280.09 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 278.99/280.09 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 278.99/280.09 Found x200:=(x20 x5):(((r Xx0) x5)->((r (cSUCC Xx0)) (cSUCC x5)))
% 278.99/280.09 Found (x20 x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 278.99/280.09 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 278.99/280.09 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 278.99/280.09 Found x200:=(x20 x5):(((r Xx0) x5)->((r (cSUCC Xx0)) (cSUCC x5)))
% 278.99/280.09 Found (x20 x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 278.99/280.09 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 278.99/280.09 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 278.99/280.09 Found x200:=(x20 x5):(((r Xx0) x5)->((r (cSUCC Xx0)) (cSUCC x5)))
% 278.99/280.09 Found (x20 x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 278.99/280.09 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 278.99/280.09 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 278.99/280.09 Found x200:=(x20 x5):(((r Xx0) x5)->((r (cSUCC Xx0)) (cSUCC x5)))
% 278.99/280.09 Found (x20 x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 278.99/280.09 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 278.99/280.09 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 278.99/280.09 Found x200:=(x20 x5):(((r Xx0) x5)->((r (cSUCC Xx0)) (cSUCC x5)))
% 278.99/280.09 Found (x20 x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 278.99/280.09 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 278.99/280.09 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 278.99/280.09 Found x200:=(x20 x5):(((r Xx0) x5)->((r (cSUCC Xx0)) (cSUCC x5)))
% 283.19/284.26 Found (x20 x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 283.19/284.26 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 283.19/284.26 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 283.19/284.26 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 283.19/284.26 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 283.19/284.26 Found x200:=(x20 x5):(((r Xx0) x5)->((r (cSUCC Xx0)) (cSUCC x5)))
% 283.19/284.26 Found (x20 x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 283.19/284.26 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 283.19/284.26 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 283.19/284.26 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 283.19/284.26 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 283.19/284.26 Found x200:=(x20 x5):(((r Xx0) x5)->((r (cSUCC Xx0)) (cSUCC x5)))
% 283.19/284.26 Found (x20 x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 283.19/284.26 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 283.19/284.26 Found ((x2 Xx0) x5) as proof of (((r Xx0) x5)->((r (cSUCC Xx0)) x4))
% 283.19/284.26 Found x20000:=(x2000 x6):(((r Xx0) x6)->((r (cSUCC Xx0)) (cSUCC x6)))
% 283.19/284.26 Found (x2000 x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x3))
% 283.19/284.26 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x3))
% 283.19/284.26 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x3))
% 283.19/284.26 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 283.19/284.26 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 283.19/284.26 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 283.19/284.26 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 283.19/284.26 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 283.19/284.26 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 283.19/284.26 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 283.19/284.26 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 283.19/284.26 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 283.19/284.26 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 283.19/284.26 Found x20000:=(x2000 x6):(((r Xx0) x6)->((r (cSUCC Xx0)) (cSUCC x6)))
% 283.19/284.26 Found (x2000 x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x3))
% 283.19/284.26 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x3))
% 283.19/284.26 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x3))
% 283.19/284.26 Found x20000:=(x2000 x6):(((r Xx0) x6)->((r (cSUCC Xx0)) (cSUCC x6)))
% 283.19/284.26 Found (x2000 x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x3))
% 283.19/284.26 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x3))
% 283.19/284.26 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x3))
% 283.19/284.26 Found x20000:=(x2000 x6):(((r Xx0) x6)->((r (cSUCC Xx0)) (cSUCC x6)))
% 283.19/284.26 Found (x2000 x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x3))
% 283.19/284.26 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x3))
% 283.19/284.26 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x3))
% 283.19/284.26 Found x8:(Xp cZERO)
% 283.19/284.26 Instantiate: x4:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 283.19/284.26 Found (fun (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of (Xp M)
% 283.19/284.26 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 285.49/286.58 Found (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 285.49/286.58 Found (and_rect20 (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 285.49/286.58 Found ((and_rect2 (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 285.49/286.58 Found (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8)) as proof of (Xp M)
% 285.49/286.58 Found (fun (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (Xp M)
% 285.49/286.58 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 285.49/286.58 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x7:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x8:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x8) x7)) (Xp M)) (fun (x8:(Xp cZERO)) (x9:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x8))) as proof of (cNAT M)
% 285.49/286.58 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 285.49/286.58 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 285.49/286.58 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 285.49/286.58 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 285.49/286.58 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 285.49/286.58 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 285.49/286.58 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 285.49/286.58 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 285.49/286.58 Found x20000:=(x2000 x6):(((r Xx0) x6)->((r (cSUCC Xx0)) (cSUCC x6)))
% 285.49/286.58 Found (x2000 x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x3))
% 285.49/286.58 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x3))
% 285.49/286.58 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x3))
% 285.49/286.58 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 285.49/286.58 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 285.49/286.58 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 288.59/289.66 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 288.59/289.66 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 288.59/289.66 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 288.59/289.66 Found x20000:=(x2000 x6):(((r Xx0) x6)->((r (cSUCC Xx0)) (cSUCC x6)))
% 288.59/289.66 Found (x2000 x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x3))
% 288.59/289.66 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x3))
% 288.59/289.66 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x3))
% 288.59/289.66 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 288.59/289.66 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 288.59/289.66 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 288.59/289.66 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 288.59/289.66 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 288.59/289.66 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 288.59/289.66 Found x20000:=(x2000 x6):(((r Xx0) x6)->((r (cSUCC Xx0)) (cSUCC x6)))
% 288.59/289.66 Found (x2000 x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x3))
% 288.59/289.66 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x3))
% 288.59/289.66 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x3))
% 288.59/289.66 Found x20000:=(x2000 x6):(((r Xx0) x6)->((r (cSUCC Xx0)) (cSUCC x6)))
% 288.59/289.66 Found (x2000 x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x3))
% 288.59/289.66 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x3))
% 288.59/289.66 Found ((x200 Xx0) x6) as proof of (((r Xx0) x6)->((r (cSUCC Xx0)) x3))
% 288.59/289.66 Found x30:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 288.59/289.66 Found x30 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 288.59/289.66 Found x30:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 288.59/289.66 Found x30 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 288.59/289.66 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 288.59/289.66 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 288.59/289.66 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 288.59/289.66 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 288.59/289.66 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 288.59/289.66 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 288.59/289.66 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 288.59/289.66 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 288.59/289.66 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 288.59/289.66 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 288.59/289.66 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 290.90/291.95 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 290.90/291.95 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 290.90/291.95 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 290.90/291.95 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 290.90/291.95 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 290.90/291.95 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 290.90/291.95 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 290.90/291.95 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 290.90/291.96 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 290.90/291.96 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 290.90/291.96 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 290.90/291.96 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 290.90/291.96 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 290.90/291.96 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 290.90/291.96 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 290.90/291.96 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 290.90/291.96 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 290.90/291.96 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 290.90/291.96 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 290.90/291.96 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 290.90/291.96 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 290.90/291.96 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 290.90/291.96 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 290.90/291.96 Found x30:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 290.90/291.96 Found x30 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 290.90/291.96 Found x30:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 290.90/291.96 Found x30 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 290.90/291.96 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 290.90/291.96 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 290.90/291.96 Found x300:cINDUCTION
% 290.90/291.96 Found x300 as proof of cINDUCTION
% 290.90/291.96 Found x300:cINDUCTION
% 290.90/291.96 Found x300 as proof of cINDUCTION
% 290.90/291.96 Found x300:cINDUCTION
% 290.90/291.96 Found x300 as proof of cINDUCTION
% 293.10/294.11 Found x300:cINDUCTION
% 293.10/294.11 Found x300 as proof of cINDUCTION
% 293.10/294.11 Found x300:cINDUCTION
% 293.10/294.11 Found x300 as proof of cINDUCTION
% 293.10/294.11 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 293.10/294.11 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 293.10/294.11 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 293.10/294.11 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 293.10/294.11 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 293.10/294.11 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 293.10/294.11 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 293.10/294.11 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 293.10/294.11 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 293.10/294.11 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 293.10/294.11 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 293.10/294.11 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 293.10/294.11 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 293.10/294.11 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 293.10/294.11 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 293.10/294.11 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 293.10/294.11 Found x7:(Xp cZERO)
% 293.10/294.11 Instantiate: x1:=(fun (Xp0:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp0 Xx)))->False)):((fofType->Prop)->Prop)
% 293.10/294.11 Found (fun (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of (Xp M)
% 293.10/294.11 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 293.10/294.11 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 293.10/294.11 Found (and_rect20 (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 293.10/294.11 Found ((and_rect2 (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 293.10/294.11 Found (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 293.10/294.11 Found (fun (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (Xp M)
% 293.10/294.11 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))->(Xp M))
% 296.70/297.72 Found (fun (Xp:(((fofType->Prop)->Prop)->Prop)) (x6:((and (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))))=> (((fun (P:Type) (x7:((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->P)))=> (((((and_rect (Xp cZERO)) (forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))) P) x7) x6)) (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7))) as proof of (cNAT M)
% 296.70/297.72 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 296.70/297.72 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 296.70/297.72 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 296.70/297.72 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 296.70/297.72 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 296.70/297.72 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 296.70/297.72 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 296.70/297.72 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 296.70/297.72 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 296.70/297.72 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 296.70/297.72 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 296.70/297.72 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 296.70/297.72 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 296.70/297.72 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 296.70/297.72 Found x200:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 296.70/297.72 Found x200 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 296.70/297.72 Found x300:(forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 296.70/297.72 Found x300 as proof of (forall (Xx00:((fofType->Prop)->Prop)) (Xy:((fofType->Prop)->Prop)), (((r Xx00) Xy)->((r (cSUCC Xx00)) (cSUCC Xy))))
% 296.70/297.72 Found x7:(Xp cZERO)
% 296.70/297.72 Found (fun (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of (Xp M)
% 296.70/297.72 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M))
% 296.70/297.72 Found (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7) as proof of ((Xp cZERO)->((forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx))))->(Xp M)))
% 296.70/297.72 Found (and_rect20 (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Prop)->Prop)), ((Xp Xx)->(Xp (cSUCC Xx)))))=> x7)) as proof of (Xp M)
% 296.70/297.72 Found ((and_rect2 (Xp M)) (fun (x7:(Xp cZERO)) (x8:(forall (Xx:((fofType->Pr
%------------------------------------------------------------------------------