TSTP Solution File: SYO222^5 by cocATP---0.2.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SYO222^5 : TPTP v7.5.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n015.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 0s
% DateTime : Tue Mar 29 00:50:56 EDT 2022
% Result : Timeout 300.01s 300.65s
% Output : None
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----No solution output by system
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.10/0.11 % Problem : SYO222^5 : TPTP v7.5.0. Released v4.0.0.
% 0.10/0.12 % Command : python CASC.py /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.12/0.33 % Computer : n015.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPUModel : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % RAMPerCPU : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % DateTime : Fri Mar 11 19:50:18 EST 2022
% 0.12/0.33 % CPUTime :
% 0.12/0.33 ModuleCmd_Load.c(213):ERROR:105: Unable to locate a modulefile for 'python/python27'
% 0.12/0.34 Python 2.7.5
% 1.22/1.44 Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox2/benchmark/', '/export/starexec/sandbox2/benchmark/']
% 1.22/1.44 FOF formula (<kernel.Constant object at 0x1fb75f0>, <kernel.DependentProduct object at 0x1fb7638>) of role type named f
% 1.22/1.44 Using role type
% 1.22/1.44 Declaring f:(fofType->fofType)
% 1.22/1.44 FOF formula (<kernel.Constant object at 0x2b079db931b8>, <kernel.Single object at 0x1fb7b90>) of role type named a
% 1.22/1.44 Using role type
% 1.22/1.44 Declaring a:fofType
% 1.22/1.44 FOF formula (<kernel.Constant object at 0x1fb7950>, <kernel.DependentProduct object at 0x1fb7cf8>) of role type named cP
% 1.22/1.44 Using role type
% 1.22/1.44 Declaring cP:(fofType->Prop)
% 1.22/1.44 FOF formula ((ex (fofType->Prop)) (fun (A:(fofType->Prop))=> ((and (forall (Xx:fofType), ((A (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (A Xz))))))) of role conjecture named cTHM115A
% 1.22/1.44 Conjecture to prove = ((ex (fofType->Prop)) (fun (A:(fofType->Prop))=> ((and (forall (Xx:fofType), ((A (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (A Xz))))))):Prop
% 1.22/1.44 We need to prove ['((ex (fofType->Prop)) (fun (A:(fofType->Prop))=> ((and (forall (Xx:fofType), ((A (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (A Xz)))))))']
% 1.22/1.44 Parameter fofType:Type.
% 1.22/1.44 Parameter f:(fofType->fofType).
% 1.22/1.44 Parameter a:fofType.
% 1.22/1.44 Parameter cP:(fofType->Prop).
% 1.22/1.44 Trying to prove ((ex (fofType->Prop)) (fun (A:(fofType->Prop))=> ((and (forall (Xx:fofType), ((A (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (A Xz)))))))
% 1.22/1.44 Found x00:=(x0 Xx):(cP Xx)
% 1.22/1.44 Found (x0 Xx) as proof of (cP Xx)
% 1.22/1.44 Found (x0 Xx) as proof of (cP Xx)
% 1.22/1.44 Found (fun (x0:(x (f Xx)))=> (x0 Xx)) as proof of (cP Xx)
% 1.22/1.44 Found (fun (Xx:fofType) (x0:(x (f Xx)))=> (x0 Xx)) as proof of ((x (f Xx))->(cP Xx))
% 1.22/1.44 Found (fun (Xx:fofType) (x0:(x (f Xx)))=> (x0 Xx)) as proof of (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))
% 1.22/1.44 Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (A:(fofType->Prop))=> ((and (forall (Xx:fofType), ((A (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (A Xz))))))):(((eq ((fofType->Prop)->Prop)) (fun (A:(fofType->Prop))=> ((and (forall (Xx:fofType), ((A (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (A Xz))))))) (fun (x:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 1.22/1.44 Found (eta_expansion_dep00 (fun (A:(fofType->Prop))=> ((and (forall (Xx:fofType), ((A (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (A Xz))))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (A:(fofType->Prop))=> ((and (forall (Xx:fofType), ((A (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (A Xz))))))) b)
% 1.22/1.44 Found ((eta_expansion_dep0 (fun (x1:(fofType->Prop))=> Prop)) (fun (A:(fofType->Prop))=> ((and (forall (Xx:fofType), ((A (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (A Xz))))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (A:(fofType->Prop))=> ((and (forall (Xx:fofType), ((A (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (A Xz))))))) b)
% 3.48/3.69 Found (((eta_expansion_dep (fofType->Prop)) (fun (x1:(fofType->Prop))=> Prop)) (fun (A:(fofType->Prop))=> ((and (forall (Xx:fofType), ((A (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (A Xz))))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (A:(fofType->Prop))=> ((and (forall (Xx:fofType), ((A (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (A Xz))))))) b)
% 3.48/3.69 Found (((eta_expansion_dep (fofType->Prop)) (fun (x1:(fofType->Prop))=> Prop)) (fun (A:(fofType->Prop))=> ((and (forall (Xx:fofType), ((A (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (A Xz))))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (A:(fofType->Prop))=> ((and (forall (Xx:fofType), ((A (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (A Xz))))))) b)
% 3.48/3.69 Found (((eta_expansion_dep (fofType->Prop)) (fun (x1:(fofType->Prop))=> Prop)) (fun (A:(fofType->Prop))=> ((and (forall (Xx:fofType), ((A (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (A Xz))))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (A:(fofType->Prop))=> ((and (forall (Xx:fofType), ((A (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (A Xz))))))) b)
% 3.48/3.69 Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->Prop)->Prop)) b) b)
% 3.48/3.69 Found (eq_ref0 b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (A:(fofType->Prop))=> ((and (forall (Xx:fofType), ((A (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (A Xz)))))))
% 3.48/3.69 Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (A:(fofType->Prop))=> ((and (forall (Xx:fofType), ((A (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (A Xz)))))))
% 3.48/3.69 Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (A:(fofType->Prop))=> ((and (forall (Xx:fofType), ((A (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (A Xz)))))))
% 3.48/3.69 Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (A:(fofType->Prop))=> ((and (forall (Xx:fofType), ((A (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (A Xz)))))))
% 3.48/3.69 Found eta_expansion000:=(eta_expansion00 a0):(((eq ((fofType->Prop)->Prop)) a0) (fun (x:(fofType->Prop))=> (a0 x)))
% 3.48/3.69 Found (eta_expansion00 a0) as proof of (((eq ((fofType->Prop)->Prop)) a0) b)
% 3.48/3.69 Found ((eta_expansion0 Prop) a0) as proof of (((eq ((fofType->Prop)->Prop)) a0) b)
% 3.48/3.69 Found (((eta_expansion (fofType->Prop)) Prop) a0) as proof of (((eq ((fofType->Prop)->Prop)) a0) b)
% 3.48/3.69 Found (((eta_expansion (fofType->Prop)) Prop) a0) as proof of (((eq ((fofType->Prop)->Prop)) a0) b)
% 3.48/3.69 Found (((eta_expansion (fofType->Prop)) Prop) a0) as proof of (((eq ((fofType->Prop)->Prop)) a0) b)
% 3.48/3.69 Found eq_ref00:=(eq_ref0 a0):(((eq ((fofType->Prop)->Prop)) a0) a0)
% 3.48/3.69 Found (eq_ref0 a0) as proof of (((eq ((fofType->Prop)->Prop)) a0) b)
% 3.48/3.69 Found ((eq_ref ((fofType->Prop)->Prop)) a0) as proof of (((eq ((fofType->Prop)->Prop)) a0) b)
% 3.48/3.69 Found ((eq_ref ((fofType->Prop)->Prop)) a0) as proof of (((eq ((fofType->Prop)->Prop)) a0) b)
% 3.48/3.69 Found ((eq_ref ((fofType->Prop)->Prop)) a0) as proof of (((eq ((fofType->Prop)->Prop)) a0) b)
% 5.71/5.92 Found eq_ref00:=(eq_ref0 (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))):(((eq Prop) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 5.71/5.92 Found (eq_ref0 (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))) as proof of (((eq Prop) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))) b)
% 5.71/5.92 Found ((eq_ref Prop) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))) as proof of (((eq Prop) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))) b)
% 5.71/5.92 Found ((eq_ref Prop) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))) as proof of (((eq Prop) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))) b)
% 5.71/5.92 Found ((eq_ref Prop) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))) as proof of (((eq Prop) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))) b)
% 5.71/5.92 Found eq_ref00:=(eq_ref0 (fun (Xz:fofType)=> (x Xz))):(((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) (fun (Xz:fofType)=> (x Xz)))
% 5.71/5.92 Found (eq_ref0 (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b)
% 5.71/5.92 Found ((eq_ref (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b)
% 5.71/5.92 Found ((eq_ref (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b)
% 5.71/5.92 Found ((eq_ref (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b)
% 5.71/5.92 Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:fofType)=> (x Xz))):(((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) (fun (x0:fofType)=> (x x0)))
% 5.71/5.92 Found (eta_expansion_dep00 (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b)
% 5.71/5.92 Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b)
% 5.71/5.92 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b)
% 5.71/5.92 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b)
% 5.71/5.92 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b)
% 5.71/5.92 Found eq_ref00:=(eq_ref0 (f0 x00)):(((eq Prop) (f0 x00)) (f0 x00))
% 5.71/5.92 Found (eq_ref0 (f0 x00)) as proof of (((eq Prop) (f0 x00)) (x x00))
% 5.71/5.92 Found ((eq_ref Prop) (f0 x00)) as proof of (((eq Prop) (f0 x00)) (x x00))
% 5.71/5.92 Found ((eq_ref Prop) (f0 x00)) as proof of (((eq Prop) (f0 x00)) (x x00))
% 5.71/5.92 Found (fun (x00:fofType)=> ((eq_ref Prop) (f0 x00))) as proof of (((eq Prop) (f0 x00)) (x x00))
% 5.71/5.92 Found (fun (x00:fofType)=> ((eq_ref Prop) (f0 x00))) as proof of (forall (x0:fofType), (((eq Prop) (f0 x0)) (x x0)))
% 5.71/5.92 Found eq_ref00:=(eq_ref0 (f0 x00)):(((eq Prop) (f0 x00)) (f0 x00))
% 5.71/5.92 Found (eq_ref0 (f0 x00)) as proof of (((eq Prop) (f0 x00)) (x x00))
% 8.28/8.48 Found ((eq_ref Prop) (f0 x00)) as proof of (((eq Prop) (f0 x00)) (x x00))
% 8.28/8.48 Found ((eq_ref Prop) (f0 x00)) as proof of (((eq Prop) (f0 x00)) (x x00))
% 8.28/8.48 Found (fun (x00:fofType)=> ((eq_ref Prop) (f0 x00))) as proof of (((eq Prop) (f0 x00)) (x x00))
% 8.28/8.48 Found (fun (x00:fofType)=> ((eq_ref Prop) (f0 x00))) as proof of (forall (x0:fofType), (((eq Prop) (f0 x0)) (x x0)))
% 8.28/8.48 Found eq_ref00:=(eq_ref0 (f0 x00)):(((eq Prop) (f0 x00)) (f0 x00))
% 8.28/8.48 Found (eq_ref0 (f0 x00)) as proof of (((eq Prop) (f0 x00)) (x x00))
% 8.28/8.48 Found ((eq_ref Prop) (f0 x00)) as proof of (((eq Prop) (f0 x00)) (x x00))
% 8.28/8.48 Found ((eq_ref Prop) (f0 x00)) as proof of (((eq Prop) (f0 x00)) (x x00))
% 8.28/8.48 Found (fun (x00:fofType)=> ((eq_ref Prop) (f0 x00))) as proof of (((eq Prop) (f0 x00)) (x x00))
% 8.28/8.48 Found (fun (x00:fofType)=> ((eq_ref Prop) (f0 x00))) as proof of (forall (x0:fofType), (((eq Prop) (f0 x0)) (x x0)))
% 8.28/8.48 Found eq_ref00:=(eq_ref0 (f0 x00)):(((eq Prop) (f0 x00)) (f0 x00))
% 8.28/8.48 Found (eq_ref0 (f0 x00)) as proof of (((eq Prop) (f0 x00)) (x x00))
% 8.28/8.48 Found ((eq_ref Prop) (f0 x00)) as proof of (((eq Prop) (f0 x00)) (x x00))
% 8.28/8.48 Found ((eq_ref Prop) (f0 x00)) as proof of (((eq Prop) (f0 x00)) (x x00))
% 8.28/8.48 Found (fun (x00:fofType)=> ((eq_ref Prop) (f0 x00))) as proof of (((eq Prop) (f0 x00)) (x x00))
% 8.28/8.48 Found (fun (x00:fofType)=> ((eq_ref Prop) (f0 x00))) as proof of (forall (x0:fofType), (((eq Prop) (f0 x0)) (x x0)))
% 8.28/8.48 Found eta_expansion000:=(eta_expansion00 (fun (Xz:fofType)=> (x Xz))):(((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) (fun (x0:fofType)=> (x x0)))
% 8.28/8.48 Found (eta_expansion00 (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b)
% 8.28/8.48 Found ((eta_expansion0 Prop) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b)
% 8.28/8.48 Found (((eta_expansion fofType) Prop) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b)
% 8.28/8.48 Found (((eta_expansion fofType) Prop) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b)
% 8.28/8.48 Found (((eta_expansion fofType) Prop) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b)
% 8.28/8.48 Found eta_expansion000:=(eta_expansion00 (fun (Xz:fofType)=> (x Xz))):(((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) (fun (x0:fofType)=> (x x0)))
% 8.28/8.48 Found (eta_expansion00 (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b)
% 8.28/8.48 Found ((eta_expansion0 Prop) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b)
% 8.28/8.48 Found (((eta_expansion fofType) Prop) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b)
% 8.28/8.48 Found (((eta_expansion fofType) Prop) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b)
% 8.28/8.48 Found (((eta_expansion fofType) Prop) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b)
% 8.28/8.48 Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% 8.28/8.48 Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 8.28/8.49 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 8.28/8.49 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 8.28/8.49 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 8.28/8.49 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 8.28/8.49 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 8.28/8.49 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 8.28/8.49 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 8.28/8.55 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 8.28/8.55 Found ((eq_trans0000 ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 8.28/8.55 Found (((eq_trans000 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 8.28/8.55 Found ((((eq_trans00 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) as proof of (((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 8.28/8.55 Found (((((eq_trans0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) as proof of (((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 8.28/8.55 Found ((((((eq_trans Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) as proof of (((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 8.28/8.55 Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% 8.28/8.55 Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 8.28/8.55 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 8.28/8.55 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 8.44/8.69 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 8.44/8.69 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 8.44/8.69 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 8.44/8.69 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 8.44/8.69 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 8.44/8.69 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 8.44/8.69 Found ((eq_trans0000 ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 8.44/8.69 Found (((eq_trans000 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 8.44/8.69 Found ((((eq_trans00 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) as proof of (((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 8.44/8.69 Found (((((eq_trans0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) as proof of (((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 8.44/8.69 Found ((((((eq_trans Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) as proof of (((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 10.94/11.20 Found eq_ref00:=(eq_ref0 (f0 x02)):(((eq Prop) (f0 x02)) (f0 x02))
% 10.94/11.20 Found (eq_ref0 (f0 x02)) as proof of (((eq Prop) (f0 x02)) (x x02))
% 10.94/11.20 Found ((eq_ref Prop) (f0 x02)) as proof of (((eq Prop) (f0 x02)) (x x02))
% 10.94/11.20 Found ((eq_ref Prop) (f0 x02)) as proof of (((eq Prop) (f0 x02)) (x x02))
% 10.94/11.20 Found (fun (x02:fofType)=> ((eq_ref Prop) (f0 x02))) as proof of (((eq Prop) (f0 x02)) (x x02))
% 10.94/11.20 Found (fun (x02:fofType)=> ((eq_ref Prop) (f0 x02))) as proof of (forall (x0:fofType), (((eq Prop) (f0 x0)) (x x0)))
% 10.94/11.20 Found eq_ref00:=(eq_ref0 (f0 x02)):(((eq Prop) (f0 x02)) (f0 x02))
% 10.94/11.20 Found (eq_ref0 (f0 x02)) as proof of (((eq Prop) (f0 x02)) (x x02))
% 10.94/11.20 Found ((eq_ref Prop) (f0 x02)) as proof of (((eq Prop) (f0 x02)) (x x02))
% 10.94/11.20 Found ((eq_ref Prop) (f0 x02)) as proof of (((eq Prop) (f0 x02)) (x x02))
% 10.94/11.20 Found (fun (x02:fofType)=> ((eq_ref Prop) (f0 x02))) as proof of (((eq Prop) (f0 x02)) (x x02))
% 10.94/11.20 Found (fun (x02:fofType)=> ((eq_ref Prop) (f0 x02))) as proof of (forall (x0:fofType), (((eq Prop) (f0 x0)) (x x0)))
% 10.94/11.20 Found eq_ref00:=(eq_ref0 (f0 x02)):(((eq Prop) (f0 x02)) (f0 x02))
% 10.94/11.20 Found (eq_ref0 (f0 x02)) as proof of (((eq Prop) (f0 x02)) (x x02))
% 10.94/11.20 Found ((eq_ref Prop) (f0 x02)) as proof of (((eq Prop) (f0 x02)) (x x02))
% 10.94/11.20 Found ((eq_ref Prop) (f0 x02)) as proof of (((eq Prop) (f0 x02)) (x x02))
% 10.94/11.20 Found (fun (x02:fofType)=> ((eq_ref Prop) (f0 x02))) as proof of (((eq Prop) (f0 x02)) (x x02))
% 10.94/11.20 Found (fun (x02:fofType)=> ((eq_ref Prop) (f0 x02))) as proof of (forall (x0:fofType), (((eq Prop) (f0 x0)) (x x0)))
% 10.94/11.20 Found eq_ref00:=(eq_ref0 (f0 x02)):(((eq Prop) (f0 x02)) (f0 x02))
% 10.94/11.20 Found (eq_ref0 (f0 x02)) as proof of (((eq Prop) (f0 x02)) (x x02))
% 10.94/11.20 Found ((eq_ref Prop) (f0 x02)) as proof of (((eq Prop) (f0 x02)) (x x02))
% 10.94/11.20 Found ((eq_ref Prop) (f0 x02)) as proof of (((eq Prop) (f0 x02)) (x x02))
% 10.94/11.20 Found (fun (x02:fofType)=> ((eq_ref Prop) (f0 x02))) as proof of (((eq Prop) (f0 x02)) (x x02))
% 10.94/11.20 Found (fun (x02:fofType)=> ((eq_ref Prop) (f0 x02))) as proof of (forall (x0:fofType), (((eq Prop) (f0 x0)) (x x0)))
% 10.94/11.20 Found x01:(P0 (f0 x))
% 10.94/11.20 Found (fun (x01:(P0 (f0 x)))=> x01) as proof of (P0 (f0 x))
% 10.94/11.20 Found (fun (x01:(P0 (f0 x)))=> x01) as proof of (P1 (f0 x))
% 10.94/11.20 Found x01:(P0 (f0 x))
% 10.94/11.20 Found (fun (x01:(P0 (f0 x)))=> x01) as proof of (P0 (f0 x))
% 10.94/11.20 Found (fun (x01:(P0 (f0 x)))=> x01) as proof of (P1 (f0 x))
% 10.94/11.20 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 10.94/11.20 Found (eq_ref0 b) as proof of (((eq Prop) b) (f0 x))
% 10.94/11.20 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 10.94/11.20 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 10.94/11.20 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 10.94/11.20 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))):(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 10.94/11.20 Found (eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 11.30/11.51 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 11.30/11.51 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 11.30/11.51 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 11.30/11.51 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 11.30/11.51 Found (eq_ref0 b) as proof of (((eq Prop) b) (f0 x))
% 11.30/11.51 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 11.30/11.51 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 11.30/11.51 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 11.30/11.51 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))):(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 11.30/11.51 Found (eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 11.30/11.51 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 11.30/11.51 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 11.30/11.51 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 16.70/16.91 Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% 16.70/16.91 Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% 16.70/16.91 Found iff_sym as proof of b
% 16.70/16.91 Found x010:=(x01 Xx):(cP Xx)
% 16.70/16.91 Found (x01 Xx) as proof of (cP Xx)
% 16.70/16.91 Found (x01 Xx) as proof of (cP Xx)
% 16.70/16.91 Found (fun (x01:(x (f Xx)))=> (x01 Xx)) as proof of (cP Xx)
% 16.70/16.91 Found (fun (Xx:fofType) (x01:(x (f Xx)))=> (x01 Xx)) as proof of ((x (f Xx))->(cP Xx))
% 16.70/16.91 Found (fun (Xx:fofType) (x01:(x (f Xx)))=> (x01 Xx)) as proof of (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))
% 16.70/16.91 Found ((conj00 (fun (Xx:fofType) (x01:(x (f Xx)))=> (x01 Xx))) iff_sym) as proof of ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) b)
% 16.70/16.91 Found (((conj0 b) (fun (Xx:fofType) (x01:(x (f Xx)))=> (x01 Xx))) iff_sym) as proof of ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) b)
% 16.70/16.91 Found ((((conj (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) b) (fun (Xx:fofType) (x01:(x (f Xx)))=> (x01 Xx))) iff_sym) as proof of ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) b)
% 16.70/16.91 Found ((((conj (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) b) (fun (Xx:fofType) (x01:(x (f Xx)))=> (x01 Xx))) iff_sym) as proof of ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) b)
% 16.70/16.91 Found ((((conj (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) b) (fun (Xx:fofType) (x01:(x (f Xx)))=> (x01 Xx))) iff_sym) as proof of (P b)
% 16.70/16.91 Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% 16.70/16.91 Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 16.70/16.91 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 16.70/16.91 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 16.70/16.91 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 16.70/16.91 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 16.70/16.91 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 16.70/16.91 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 16.70/16.91 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 16.70/16.91 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 16.70/16.91 Found ((eq_trans0000 ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f0 x))->(P ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))))
% 16.70/16.91 Found (((eq_trans000 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f0 x))->(P ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))))
% 16.70/16.91 Found ((((eq_trans00 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) as proof of (forall (P:(Prop->Prop)), ((P (f0 x))->(P ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))))
% 16.70/16.93 Found (((((eq_trans0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) as proof of (forall (P:(Prop->Prop)), ((P (f0 x))->(P ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))))
% 16.70/16.93 Found ((((((eq_trans Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) as proof of (forall (P:(Prop->Prop)), ((P (f0 x))->(P ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))))
% 16.70/16.93 Found ((((((eq_trans Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) as proof of (forall (P:(Prop->Prop)), ((P (f0 x))->(P ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))))
% 16.70/16.93 Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% 16.70/16.93 Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 16.70/16.93 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 16.70/16.93 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 16.70/16.93 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 16.80/17.07 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 16.80/17.07 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 16.80/17.07 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 16.80/17.07 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 16.80/17.07 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 16.80/17.07 Found ((eq_trans0000 ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f0 x))->(P ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))))
% 16.80/17.07 Found (((eq_trans000 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f0 x))->(P ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))))
% 16.80/17.07 Found ((((eq_trans00 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) as proof of (forall (P:(Prop->Prop)), ((P (f0 x))->(P ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))))
% 16.80/17.07 Found (((((eq_trans0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) as proof of (forall (P:(Prop->Prop)), ((P (f0 x))->(P ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))))
% 16.80/17.07 Found ((((((eq_trans Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) as proof of (forall (P:(Prop->Prop)), ((P (f0 x))->(P ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))))
% 21.61/21.82 Found ((((((eq_trans Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) as proof of (forall (P:(Prop->Prop)), ((P (f0 x))->(P ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))))
% 21.61/21.82 Found x010:=(x01 Xx):(cP Xx)
% 21.61/21.82 Found (x01 Xx) as proof of (cP Xx)
% 21.61/21.82 Found (x01 Xx) as proof of (cP Xx)
% 21.61/21.82 Found (fun (x01:(x (f Xx)))=> (x01 Xx)) as proof of (cP Xx)
% 21.61/21.82 Found (fun (Xx:fofType) (x01:(x (f Xx)))=> (x01 Xx)) as proof of ((x (f Xx))->(cP Xx))
% 21.61/21.82 Found (fun (Xx:fofType) (x01:(x (f Xx)))=> (x01 Xx)) as proof of (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))
% 21.61/21.82 Found x01:(P0 (f0 x))
% 21.61/21.82 Found (fun (x01:(P0 (f0 x)))=> x01) as proof of (P0 (f0 x))
% 21.61/21.82 Found (fun (x01:(P0 (f0 x)))=> x01) as proof of (P1 (f0 x))
% 21.61/21.82 Found x01:(P0 (f0 x))
% 21.61/21.82 Found (fun (x01:(P0 (f0 x)))=> x01) as proof of (P0 (f0 x))
% 21.61/21.82 Found (fun (x01:(P0 (f0 x)))=> x01) as proof of (P1 (f0 x))
% 21.61/21.82 Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% 21.61/21.82 Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% 21.61/21.82 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% 21.61/21.82 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% 21.61/21.82 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% 21.61/21.82 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 21.61/21.82 Found (eq_ref0 b) as proof of (((eq Prop) b) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 21.61/21.82 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 21.61/21.82 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 21.61/21.82 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 21.61/21.82 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))):(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 21.93/22.19 Found (eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 21.93/22.19 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 21.93/22.19 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 21.93/22.19 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 21.93/22.19 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 21.93/22.19 Found (eq_ref0 b) as proof of (((eq Prop) b) (f0 x))
% 21.93/22.19 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 21.93/22.19 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 21.93/22.19 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 21.93/22.19 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 21.93/22.19 Found (eq_ref0 b) as proof of (((eq Prop) b) (f0 x))
% 21.93/22.19 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 21.93/22.19 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 21.93/22.19 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 21.93/22.19 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))):(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 21.93/22.19 Found (eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 21.93/22.19 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 21.93/22.19 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 22.52/22.79 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 22.52/22.79 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 22.52/22.79 Found (eq_ref0 b) as proof of (((eq Prop) b) (f0 x))
% 22.52/22.79 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 22.52/22.79 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 22.52/22.79 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 22.52/22.79 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))):(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 22.52/22.79 Found (eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 22.52/22.79 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 22.52/22.79 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 22.52/22.79 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 22.52/22.79 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 22.52/22.79 Found (eq_ref0 b) as proof of (((eq Prop) b) (f0 x))
% 22.52/22.79 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 22.52/22.79 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 22.52/22.79 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 22.52/22.79 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))):(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 23.89/24.13 Found (eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 23.89/24.13 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 23.89/24.13 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 23.89/24.13 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 23.89/24.13 Found eta_expansion000:=(eta_expansion00 b):(((eq ((fofType->Prop)->Prop)) b) (fun (x:(fofType->Prop))=> (b x)))
% 23.89/24.13 Found (eta_expansion00 b) as proof of (((eq ((fofType->Prop)->Prop)) b) b0)
% 23.89/24.13 Found ((eta_expansion0 Prop) b) as proof of (((eq ((fofType->Prop)->Prop)) b) b0)
% 23.89/24.13 Found (((eta_expansion (fofType->Prop)) Prop) b) as proof of (((eq ((fofType->Prop)->Prop)) b) b0)
% 23.89/24.13 Found (((eta_expansion (fofType->Prop)) Prop) b) as proof of (((eq ((fofType->Prop)->Prop)) b) b0)
% 23.89/24.13 Found (((eta_expansion (fofType->Prop)) Prop) b) as proof of (((eq ((fofType->Prop)->Prop)) b) b0)
% 23.89/24.13 Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->Prop)) a0) a0)
% 23.89/24.13 Found (eq_ref0 a0) as proof of (((eq (fofType->Prop)) a0) b)
% 23.89/24.13 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b)
% 23.89/24.13 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b)
% 23.89/24.13 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b)
% 23.89/24.13 Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% 23.89/24.13 Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> (x Xz)))
% 23.89/24.13 Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> (x Xz)))
% 23.89/24.13 Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> (x Xz)))
% 23.89/24.13 Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> (x Xz)))
% 23.89/24.13 Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% 23.89/24.13 Found (eta_expansion_dep00 a0) as proof of (((eq (fofType->Prop)) a0) b)
% 23.89/24.13 Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b)
% 23.89/24.13 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b)
% 23.89/24.13 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b)
% 26.42/26.63 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b)
% 26.42/26.63 Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% 26.42/26.63 Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> (x Xz)))
% 26.42/26.63 Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> (x Xz)))
% 26.42/26.63 Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> (x Xz)))
% 26.42/26.63 Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> (x Xz)))
% 26.42/26.63 Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% 26.42/26.63 Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> (x Xz)))
% 26.42/26.63 Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> (x Xz)))
% 26.42/26.63 Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> (x Xz)))
% 26.42/26.63 Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> (x Xz)))
% 26.42/26.63 Found eq_ref00:=(eq_ref0 x00):(((eq fofType) x00) x00)
% 26.42/26.63 Found (eq_ref0 x00) as proof of (((eq fofType) x00) b)
% 26.42/26.63 Found ((eq_ref fofType) x00) as proof of (((eq fofType) x00) b)
% 26.42/26.63 Found ((eq_ref fofType) x00) as proof of (((eq fofType) x00) b)
% 26.42/26.63 Found ((eq_ref fofType) x00) as proof of (((eq fofType) x00) b)
% 26.42/26.63 Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% 26.42/26.63 Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b)
% 26.42/26.63 Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% 26.42/26.63 Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% 26.42/26.63 Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% 26.42/26.63 Found x0:(P0 (f0 x))
% 26.42/26.63 Instantiate: b:=(f0 x):Prop
% 26.42/26.63 Found x0 as proof of (P1 b)
% 26.42/26.63 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))):(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 26.42/26.63 Found (eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 26.42/26.63 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 26.42/26.63 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 26.42/26.63 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 27.10/27.33 Found x0:(P0 (f0 x))
% 27.10/27.33 Instantiate: b:=(f0 x):Prop
% 27.10/27.33 Found x0 as proof of (P1 b)
% 27.10/27.33 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))):(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 27.10/27.33 Found (eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 27.10/27.33 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 27.10/27.33 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 27.10/27.33 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 27.10/27.33 Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% 27.10/27.33 Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 27.10/27.33 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 27.10/27.33 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 27.10/27.33 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 27.10/27.33 Found (eq_sym010 ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) b) (f0 x))
% 27.10/27.33 Found ((eq_sym01 b) ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) b) (f0 x))
% 27.10/27.33 Found (((eq_sym0 (f0 x)) b) ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) b) (f0 x))
% 27.10/27.33 Found (((eq_sym0 (f0 x)) b) ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) b) (f0 x))
% 27.10/27.33 Found ((eq_trans0000 ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (((eq_sym0 (f0 x)) b) ((eq_ref Prop) (f0 x)))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x))
% 27.10/27.33 Found (((eq_trans000 (f0 x)) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (((eq_sym0 (f0 x)) b) ((eq_ref Prop) (f0 x)))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x))
% 27.10/27.33 Found ((((eq_trans00 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((eq_ref Prop) (f0 x)))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x))
% 27.10/27.33 Found (((((eq_trans0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((eq_ref Prop) (f0 x)))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x))
% 27.10/27.33 Found ((((((eq_trans Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((eq_ref Prop) (f0 x)))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x))
% 27.10/27.33 Found ((((((eq_trans Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((eq_ref Prop) (f0 x)))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x))
% 27.10/27.34 Found (eq_sym000 ((((((eq_trans Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((eq_ref Prop) (f0 x))))) as proof of (forall (P:(Prop->Prop)), ((P (f0 x))->(P ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))))
% 27.10/27.34 Found ((eq_sym00 (f0 x)) ((((((eq_trans Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((eq_ref Prop) (f0 x))))) as proof of (forall (P:(Prop->Prop)), ((P (f0 x))->(P ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))))
% 27.10/27.34 Found (((eq_sym0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) ((((((eq_trans Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((eq_ref Prop) (f0 x))))) as proof of (forall (P:(Prop->Prop)), ((P (f0 x))->(P ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))))
% 28.08/28.31 Found ((((eq_sym Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) ((((((eq_trans Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) ((((eq_sym Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((eq_ref Prop) (f0 x))))) as proof of (forall (P:(Prop->Prop)), ((P (f0 x))->(P ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))))
% 28.08/28.31 Found ((((eq_sym Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) ((((((eq_trans Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) ((((eq_sym Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((eq_ref Prop) (f0 x))))) as proof of (forall (P:(Prop->Prop)), ((P (f0 x))->(P ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))))
% 28.08/28.31 Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->Prop)) a0) a0)
% 28.08/28.31 Found (eq_ref0 a0) as proof of (((eq (fofType->Prop)) a0) b)
% 28.08/28.31 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b)
% 28.08/28.31 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b)
% 28.08/28.31 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b)
% 28.08/28.31 Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% 28.08/28.31 Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> (x Xz)))
% 28.08/28.31 Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> (x Xz)))
% 28.08/28.31 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> (x Xz)))
% 30.02/30.29 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> (x Xz)))
% 30.02/30.29 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> (x Xz)))
% 30.02/30.29 Found eta_expansion000:=(eta_expansion00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% 30.02/30.29 Found (eta_expansion00 a0) as proof of (((eq (fofType->Prop)) a0) b)
% 30.02/30.29 Found ((eta_expansion0 Prop) a0) as proof of (((eq (fofType->Prop)) a0) b)
% 30.02/30.29 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) b)
% 30.02/30.29 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) b)
% 30.02/30.29 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) b)
% 30.02/30.29 Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% 30.02/30.29 Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> (x Xz)))
% 30.02/30.29 Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> (x Xz)))
% 30.02/30.29 Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> (x Xz)))
% 30.02/30.29 Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> (x Xz)))
% 30.02/30.29 Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% 30.02/30.29 Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> (x Xz)))
% 30.02/30.29 Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> (x Xz)))
% 30.02/30.29 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> (x Xz)))
% 30.02/30.29 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> (x Xz)))
% 30.02/30.29 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> (x Xz)))
% 30.02/30.29 Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% 30.02/30.29 Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 30.02/30.29 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 30.02/30.29 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 30.02/30.29 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 30.02/30.29 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 30.02/30.29 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 30.02/30.29 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 30.02/30.29 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 30.02/30.29 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 30.02/30.29 Found ((eq_trans00000 ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 30.02/30.29 Found ((eq_trans00000 ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 30.02/30.29 Found (((fun (x0:(((eq Prop) (f0 x)) b)) (x00:(((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((eq_trans0000 x0) x00) P0)) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 30.02/30.30 Found (((fun (x0:(((eq Prop) (f0 x)) b)) (x00:(((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_trans000 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) P0)) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 30.02/30.30 Found (((fun (x0:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (x00:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((((eq_trans00 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) P0)) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 30.02/30.30 Found (((fun (x0:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (x00:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((((eq_trans0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) P0)) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 30.20/30.43 Found (((fun (x0:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (x00:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((((((eq_trans Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) P0)) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 30.20/30.43 Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (x00:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((((((eq_trans Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) P0)) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 30.20/30.43 Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% 30.20/30.43 Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 30.20/30.43 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 30.20/30.43 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 30.20/30.43 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 30.20/30.43 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 30.20/30.43 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 30.20/30.43 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 30.20/30.43 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 30.20/30.43 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 30.20/30.43 Found ((eq_trans00000 ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 30.20/30.43 Found ((eq_trans00000 ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 30.20/30.43 Found (((fun (x0:(((eq Prop) (f0 x)) b)) (x00:(((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((eq_trans0000 x0) x00) P0)) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 30.20/30.43 Found (((fun (x0:(((eq Prop) (f0 x)) b)) (x00:(((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_trans000 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) P0)) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 30.20/30.43 Found (((fun (x0:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (x00:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((((eq_trans00 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) P0)) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 30.20/30.44 Found (((fun (x0:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (x00:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((((eq_trans0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) P0)) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 30.20/30.44 Found (((fun (x0:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (x00:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((((((eq_trans Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) P0)) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 30.20/30.44 Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (x00:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((((((eq_trans Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) P0)) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 32.29/32.51 Found x00:(cP a)
% 32.29/32.51 Instantiate: Xx0:=a:fofType
% 32.29/32.51 Found x00 as proof of (P Xx0)
% 32.29/32.51 Found eq_ref00:=(eq_ref0 (f Xx0)):(((eq fofType) (f Xx0)) (f Xx0))
% 32.29/32.51 Found (eq_ref0 (f Xx0)) as proof of (((eq fofType) (f Xx0)) (f Xx))
% 32.29/32.51 Found ((eq_ref fofType) (f Xx0)) as proof of (((eq fofType) (f Xx0)) (f Xx))
% 32.29/32.51 Found ((eq_ref fofType) (f Xx0)) as proof of (((eq fofType) (f Xx0)) (f Xx))
% 32.29/32.51 Found ((eq_ref fofType) (f Xx0)) as proof of (((eq fofType) (f Xx0)) (f Xx))
% 32.29/32.51 Found eq_ref00:=(eq_ref0 (f Xx)):(((eq fofType) (f Xx)) (f Xx))
% 32.29/32.51 Found (eq_ref0 (f Xx)) as proof of (((eq fofType) (f Xx)) (f x02))
% 32.29/32.51 Found ((eq_ref fofType) (f Xx)) as proof of (((eq fofType) (f Xx)) (f x02))
% 32.29/32.51 Found ((eq_ref fofType) (f Xx)) as proof of (((eq fofType) (f Xx)) (f x02))
% 32.29/32.51 Found ((eq_ref fofType) (f Xx)) as proof of (((eq fofType) (f Xx)) (f x02))
% 32.29/32.51 Found x00:(cP a)
% 32.29/32.51 Instantiate: b:=a:fofType
% 32.29/32.51 Found x00 as proof of (P b)
% 32.29/32.51 Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% 32.29/32.51 Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b)
% 32.29/32.51 Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% 32.29/32.51 Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% 32.29/32.51 Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% 32.29/32.51 Found eq_ref00:=(eq_ref0 x02):(((eq fofType) x02) x02)
% 32.29/32.51 Found (eq_ref0 x02) as proof of (((eq fofType) x02) b)
% 32.29/32.51 Found ((eq_ref fofType) x02) as proof of (((eq fofType) x02) b)
% 32.29/32.51 Found ((eq_ref fofType) x02) as proof of (((eq fofType) x02) b)
% 32.29/32.51 Found ((eq_ref fofType) x02) as proof of (((eq fofType) x02) b)
% 32.29/32.51 Found x01:(cP a)
% 32.29/32.51 Instantiate: Xx0:=a:fofType
% 32.29/32.51 Found x01 as proof of (P Xx0)
% 32.29/32.51 Found eq_ref00:=(eq_ref0 (f Xx0)):(((eq fofType) (f Xx0)) (f Xx0))
% 32.29/32.51 Found (eq_ref0 (f Xx0)) as proof of (((eq fofType) (f Xx0)) (f Xx))
% 32.29/32.51 Found ((eq_ref fofType) (f Xx0)) as proof of (((eq fofType) (f Xx0)) (f Xx))
% 32.29/32.51 Found ((eq_ref fofType) (f Xx0)) as proof of (((eq fofType) (f Xx0)) (f Xx))
% 32.29/32.51 Found ((eq_ref fofType) (f Xx0)) as proof of (((eq fofType) (f Xx0)) (f Xx))
% 32.29/32.51 Found x1:(cP a)
% 32.29/32.51 Instantiate: Xx0:=a:fofType
% 32.29/32.51 Found x1 as proof of (P Xx0)
% 32.29/32.51 Found eq_ref00:=(eq_ref0 (f Xx0)):(((eq fofType) (f Xx0)) (f Xx0))
% 32.29/32.51 Found (eq_ref0 (f Xx0)) as proof of (((eq fofType) (f Xx0)) (f Xx))
% 32.29/32.51 Found ((eq_ref fofType) (f Xx0)) as proof of (((eq fofType) (f Xx0)) (f Xx))
% 32.29/32.51 Found ((eq_ref fofType) (f Xx0)) as proof of (((eq fofType) (f Xx0)) (f Xx))
% 32.29/32.51 Found ((eq_ref fofType) (f Xx0)) as proof of (((eq fofType) (f Xx0)) (f Xx))
% 32.29/32.51 Found eq_ref00:=(eq_ref0 (f Xx)):(((eq fofType) (f Xx)) (f Xx))
% 32.29/32.51 Found (eq_ref0 (f Xx)) as proof of (((eq fofType) (f Xx)) (f x00))
% 32.29/32.51 Found ((eq_ref fofType) (f Xx)) as proof of (((eq fofType) (f Xx)) (f x00))
% 32.29/32.51 Found ((eq_ref fofType) (f Xx)) as proof of (((eq fofType) (f Xx)) (f x00))
% 32.29/32.51 Found ((eq_ref fofType) (f Xx)) as proof of (((eq fofType) (f Xx)) (f x00))
% 32.29/32.51 Found x01:(cP a)
% 32.29/32.51 Instantiate: b:=a:fofType
% 32.29/32.51 Found x01 as proof of (P b)
% 32.29/32.51 Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% 32.73/32.94 Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b)
% 32.73/32.94 Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% 32.73/32.94 Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% 32.73/32.94 Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% 32.73/32.94 Found x1:(cP a)
% 32.73/32.94 Instantiate: b:=a:fofType
% 32.73/32.94 Found x1 as proof of (P b)
% 32.73/32.94 Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% 32.73/32.94 Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b)
% 32.73/32.94 Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% 32.73/32.94 Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% 32.73/32.94 Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% 32.73/32.94 Found x02:(P0 (f0 x))
% 32.73/32.94 Found (fun (x02:(P0 (f0 x)))=> x02) as proof of (P0 (f0 x))
% 32.73/32.94 Found (fun (x02:(P0 (f0 x)))=> x02) as proof of (P1 (f0 x))
% 32.73/32.94 Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% 32.73/32.94 Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 32.73/32.94 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 32.73/32.94 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 32.73/32.94 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 32.73/32.94 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 32.73/32.94 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 32.73/32.94 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 32.73/32.94 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 32.73/32.94 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 32.73/32.94 Found (((eq_trans00000 ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f0 x)))=> x02)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 32.73/32.94 Found (((eq_trans00000 ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f0 x)))=> x02)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 32.73/32.94 Found ((((fun (x0:(((eq Prop) (f0 x)) b)) (x00:(((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((eq_trans0000 x0) x00) (fun (x1:Prop)=> ((P0 (f0 x))->(P0 x1))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f0 x)))=> x02)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 32.73/32.94 Found ((((fun (x0:(((eq Prop) (f0 x)) b)) (x00:(((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_trans000 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) (fun (x1:Prop)=> ((P0 (f0 x))->(P0 x1))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f0 x)))=> x02)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 32.73/32.95 Found ((((fun (x0:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (x00:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((((eq_trans00 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) (fun (x1:Prop)=> ((P0 (f0 x))->(P0 x1))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (fun (x02:(P0 (f0 x)))=> x02)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 32.73/32.95 Found ((((fun (x0:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (x00:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((((eq_trans0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) (fun (x1:Prop)=> ((P0 (f0 x))->(P0 x1))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (fun (x02:(P0 (f0 x)))=> x02)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 32.73/32.95 Found ((((fun (x0:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (x00:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((((((eq_trans Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) (fun (x1:Prop)=> ((P0 (f0 x))->(P0 x1))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (fun (x02:(P0 (f0 x)))=> x02)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 32.88/33.08 Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (x00:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((((((eq_trans Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) (fun (x1:Prop)=> ((P0 (f0 x))->(P0 x1))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (fun (x02:(P0 (f0 x)))=> x02))) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 32.88/33.08 Found x02:(P0 (f0 x))
% 32.88/33.08 Found (fun (x02:(P0 (f0 x)))=> x02) as proof of (P0 (f0 x))
% 32.88/33.08 Found (fun (x02:(P0 (f0 x)))=> x02) as proof of (P1 (f0 x))
% 32.88/33.08 Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% 32.88/33.08 Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 32.88/33.08 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 32.88/33.08 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 32.88/33.08 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 32.88/33.08 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 32.88/33.08 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 32.88/33.08 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 32.88/33.08 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 32.88/33.09 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 32.88/33.09 Found (((eq_trans00000 ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f0 x)))=> x02)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 32.88/33.09 Found (((eq_trans00000 ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f0 x)))=> x02)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 32.88/33.09 Found ((((fun (x0:(((eq Prop) (f0 x)) b)) (x00:(((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((eq_trans0000 x0) x00) (fun (x1:Prop)=> ((P0 (f0 x))->(P0 x1))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f0 x)))=> x02)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 32.88/33.09 Found ((((fun (x0:(((eq Prop) (f0 x)) b)) (x00:(((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_trans000 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) (fun (x1:Prop)=> ((P0 (f0 x))->(P0 x1))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f0 x)))=> x02)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 32.88/33.09 Found ((((fun (x0:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (x00:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((((eq_trans00 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) (fun (x1:Prop)=> ((P0 (f0 x))->(P0 x1))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (fun (x02:(P0 (f0 x)))=> x02)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 32.88/33.10 Found ((((fun (x0:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (x00:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((((eq_trans0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) (fun (x1:Prop)=> ((P0 (f0 x))->(P0 x1))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (fun (x02:(P0 (f0 x)))=> x02)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 32.88/33.10 Found ((((fun (x0:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (x00:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((((((eq_trans Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) (fun (x1:Prop)=> ((P0 (f0 x))->(P0 x1))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (fun (x02:(P0 (f0 x)))=> x02)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 32.88/33.10 Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (x00:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((((((eq_trans Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) (fun (x1:Prop)=> ((P0 (f0 x))->(P0 x1))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (fun (x02:(P0 (f0 x)))=> x02))) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 33.91/34.11 Found eq_ref00:=(eq_ref0 x00):(((eq fofType) x00) x00)
% 33.91/34.11 Found (eq_ref0 x00) as proof of (((eq fofType) x00) b)
% 33.91/34.11 Found ((eq_ref fofType) x00) as proof of (((eq fofType) x00) b)
% 33.91/34.11 Found ((eq_ref fofType) x00) as proof of (((eq fofType) x00) b)
% 33.91/34.11 Found ((eq_ref fofType) x00) as proof of (((eq fofType) x00) b)
% 33.91/34.11 Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% 33.91/34.11 Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% 33.91/34.11 Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% 33.91/34.11 Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% 33.91/34.11 Found (fun (x:(fofType->Prop))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% 33.91/34.11 Found (fun (x:(fofType->Prop))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(fofType->Prop)), (((eq Prop) (f1 x)) (f0 x)))
% 33.91/34.11 Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% 33.91/34.11 Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% 33.91/34.11 Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% 33.91/34.11 Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% 33.91/34.11 Found (fun (x:(fofType->Prop))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% 33.91/34.11 Found (fun (x:(fofType->Prop))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(fofType->Prop)), (((eq Prop) (f1 x)) (f0 x)))
% 33.91/34.11 Found x1:(cP a)
% 33.91/34.11 Instantiate: b:=a:fofType
% 33.91/34.11 Found (fun (x2:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))=> x1) as proof of (P b)
% 33.91/34.11 Found (fun (x1:(cP a)) (x2:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))=> x1) as proof of ((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy)))->(P b))
% 33.91/34.11 Found (fun (x1:(cP a)) (x2:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))=> x1) as proof of ((cP a)->((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy)))->(P b)))
% 33.91/34.11 Found (and_rect00 (fun (x1:(cP a)) (x2:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))=> x1)) as proof of (P b)
% 33.91/34.11 Found ((and_rect0 (P b)) (fun (x1:(cP a)) (x2:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))=> x1)) as proof of (P b)
% 33.91/34.11 Found (((fun (P0:Type) (x1:((cP a)->((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy)))->P0)))=> (((((and_rect (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy)))) P0) x1) x0)) (P b)) (fun (x1:(cP a)) (x2:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))=> x1)) as proof of (P b)
% 33.91/34.11 Found (((fun (P0:Type) (x1:((cP a)->((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy)))->P0)))=> (((((and_rect (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy)))) P0) x1) x0)) (P b)) (fun (x1:(cP a)) (x2:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))=> x1)) as proof of (P b)
% 34.69/34.94 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))):(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 34.69/34.94 Found (eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 34.69/34.94 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 34.69/34.94 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 34.69/34.94 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 34.69/34.94 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))):(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 34.69/34.94 Found (eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 34.69/34.94 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 35.86/36.10 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 35.86/36.10 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 35.86/36.10 Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% 35.86/36.10 Found (eq_ref0 a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 35.86/36.10 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 35.86/36.10 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 35.86/36.10 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 35.86/36.10 Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% 35.86/36.10 Found (eq_ref0 a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 35.86/36.10 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 35.86/36.10 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 35.86/36.10 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 35.86/36.10 Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% 35.86/36.10 Found (eq_ref0 a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 35.86/36.10 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 35.86/36.10 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 35.86/36.10 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 35.86/36.10 Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% 35.86/36.10 Found (eq_ref0 a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 35.86/36.10 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 36.80/37.01 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 36.80/37.01 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 36.80/37.01 Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% 36.80/37.01 Found (eq_ref0 a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 36.80/37.01 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 36.80/37.01 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 36.80/37.01 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 36.80/37.01 Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% 36.80/37.01 Found (eq_ref0 a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 36.80/37.01 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 36.80/37.01 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 36.80/37.01 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 36.80/37.01 Found x0:(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 36.80/37.01 Instantiate: b:=((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))):Prop
% 36.80/37.01 Found x0 as proof of (P1 b)
% 36.80/37.01 Found x0:(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 36.80/37.01 Instantiate: b:=((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))):Prop
% 36.80/37.01 Found x0 as proof of (P1 b)
% 36.80/37.01 Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% 36.80/37.01 Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 36.80/37.01 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 36.80/37.01 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 36.80/37.01 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 36.80/37.01 Found ((eq_sym0100 ((eq_ref Prop) (f0 x))) x0) as proof of (P0 (f0 x))
% 36.80/37.01 Found ((eq_sym0100 ((eq_ref Prop) (f0 x))) x0) as proof of (P0 (f0 x))
% 36.80/37.01 Found (((fun (x00:(((eq Prop) (f0 x)) b))=> ((eq_sym010 x00) P0)) ((eq_ref Prop) (f0 x))) x0) as proof of (P0 (f0 x))
% 36.80/37.01 Found (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((eq_sym01 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P0)) ((eq_ref Prop) (f0 x))) x0) as proof of (P0 (f0 x))
% 36.81/37.02 Found (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P0)) ((eq_ref Prop) (f0 x))) x0) as proof of (P0 (f0 x))
% 36.81/37.02 Found (fun (x0:(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P0)) ((eq_ref Prop) (f0 x))) x0)) as proof of (P0 (f0 x))
% 36.81/37.02 Found (fun (P0:(Prop->Prop)) (x0:(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P0)) ((eq_ref Prop) (f0 x))) x0)) as proof of ((P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))->(P0 (f0 x)))
% 36.81/37.02 Found (fun (P0:(Prop->Prop)) (x0:(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P0)) ((eq_ref Prop) (f0 x))) x0)) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x))
% 36.81/37.02 Found (eq_sym000 (fun (P0:(Prop->Prop)) (x0:(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P0)) ((eq_ref Prop) (f0 x))) x0))) as proof of (((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 36.81/37.04 Found ((eq_sym00 (f0 x)) (fun (P0:(Prop->Prop)) (x0:(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P0)) ((eq_ref Prop) (f0 x))) x0))) as proof of (((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 36.81/37.04 Found (((eq_sym0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) (fun (P0:(Prop->Prop)) (x0:(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P0)) ((eq_ref Prop) (f0 x))) x0))) as proof of (((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 36.81/37.04 Found ((((eq_sym Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) (fun (P0:(Prop->Prop)) (x0:(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((((eq_sym Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P0)) ((eq_ref Prop) (f0 x))) x0))) as proof of (((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 36.81/37.04 Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% 36.81/37.04 Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 36.81/37.04 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 36.81/37.05 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 36.81/37.05 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 36.81/37.05 Found ((eq_sym0100 ((eq_ref Prop) (f0 x))) x0) as proof of (P0 (f0 x))
% 36.81/37.05 Found ((eq_sym0100 ((eq_ref Prop) (f0 x))) x0) as proof of (P0 (f0 x))
% 36.81/37.05 Found (((fun (x00:(((eq Prop) (f0 x)) b))=> ((eq_sym010 x00) P0)) ((eq_ref Prop) (f0 x))) x0) as proof of (P0 (f0 x))
% 36.81/37.05 Found (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((eq_sym01 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P0)) ((eq_ref Prop) (f0 x))) x0) as proof of (P0 (f0 x))
% 36.81/37.05 Found (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P0)) ((eq_ref Prop) (f0 x))) x0) as proof of (P0 (f0 x))
% 36.81/37.05 Found (fun (x0:(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P0)) ((eq_ref Prop) (f0 x))) x0)) as proof of (P0 (f0 x))
% 36.81/37.05 Found (fun (P0:(Prop->Prop)) (x0:(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P0)) ((eq_ref Prop) (f0 x))) x0)) as proof of ((P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))->(P0 (f0 x)))
% 36.81/37.05 Found (fun (P0:(Prop->Prop)) (x0:(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P0)) ((eq_ref Prop) (f0 x))) x0)) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x))
% 36.81/37.07 Found (eq_sym000 (fun (P0:(Prop->Prop)) (x0:(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P0)) ((eq_ref Prop) (f0 x))) x0))) as proof of (((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 36.81/37.07 Found ((eq_sym00 (f0 x)) (fun (P0:(Prop->Prop)) (x0:(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P0)) ((eq_ref Prop) (f0 x))) x0))) as proof of (((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 36.81/37.07 Found (((eq_sym0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) (fun (P0:(Prop->Prop)) (x0:(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P0)) ((eq_ref Prop) (f0 x))) x0))) as proof of (((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 36.81/37.07 Found ((((eq_sym Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) (fun (P0:(Prop->Prop)) (x0:(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((((eq_sym Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P0)) ((eq_ref Prop) (f0 x))) x0))) as proof of (((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 40.71/40.95 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 40.71/40.95 Found (eq_ref0 b) as proof of (((eq Prop) b) (f0 x))
% 40.71/40.95 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 40.71/40.95 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 40.71/40.95 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 40.71/40.95 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))):(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 40.71/40.95 Found (eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 40.71/40.95 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 40.71/40.95 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 40.71/40.95 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 40.71/40.95 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 40.71/40.95 Found (eq_ref0 b) as proof of (((eq Prop) b) (f0 x))
% 40.71/40.95 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 40.71/40.95 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 40.71/40.95 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 40.71/40.95 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))):(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 40.99/41.28 Found (eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 40.99/41.28 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 40.99/41.28 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 40.99/41.28 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 40.99/41.28 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 40.99/41.28 Found (eq_ref0 b) as proof of (((eq Prop) b) (f0 x))
% 40.99/41.28 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 40.99/41.28 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 40.99/41.28 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 40.99/41.28 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))):(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 40.99/41.28 Found (eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 40.99/41.28 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 40.99/41.28 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 50.70/50.95 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 50.70/50.95 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 50.70/50.95 Found (eq_ref0 b) as proof of (((eq Prop) b) (f0 x))
% 50.70/50.95 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 50.70/50.95 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 50.70/50.95 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 50.70/50.95 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))):(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 50.70/50.95 Found (eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 50.70/50.95 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 50.70/50.95 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 50.70/50.95 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 50.70/50.95 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))):(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 59.59/59.82 Found (eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 59.59/59.82 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 59.59/59.82 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 59.59/59.82 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 59.59/59.82 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 59.59/59.82 Found (eq_ref0 b) as proof of (((eq Prop) b) (f0 x))
% 59.59/59.82 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 59.59/59.82 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 59.59/59.82 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 59.59/59.82 Found iff_refl:=(fun (A:Prop)=> ((((conj (A->A)) (A->A)) (fun (H:A)=> H)) (fun (H:A)=> H))):(forall (P:Prop), ((iff P) P))
% 59.59/59.82 Instantiate: a0:=(forall (P:Prop), ((iff P) P)):Prop
% 59.59/59.82 Found iff_refl as proof of a0
% 59.59/59.82 Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->Prop)->Prop)) b) b)
% 59.59/59.82 Found (eq_ref0 b) as proof of (((eq ((fofType->Prop)->Prop)) b) b0)
% 59.59/59.82 Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) b0)
% 59.59/59.82 Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) b0)
% 59.59/59.82 Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) b0)
% 59.59/59.82 Found True:Prop
% 59.59/59.82 Found True as proof of Prop
% 59.59/59.82 Found (x020 True) as proof of (cP Xx)
% 59.59/59.82 Found ((x02 Xx) True) as proof of (cP Xx)
% 59.59/59.82 Found ((x02 Xx) True) as proof of (cP Xx)
% 59.59/59.82 Found (fun (x02:(x (f Xx)))=> ((x02 Xx) True)) as proof of (cP Xx)
% 59.59/59.82 Found (fun (Xx:fofType) (x02:(x (f Xx)))=> ((x02 Xx) True)) as proof of ((x (f Xx))->(cP Xx))
% 59.59/59.82 Found (fun (Xx:fofType) (x02:(x (f Xx)))=> ((x02 Xx) True)) as proof of (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))
% 59.59/59.82 Found ((conj00 (fun (Xx:fofType) (x02:(x (f Xx)))=> ((x02 Xx) True))) iff_refl) as proof of ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) a0)
% 59.59/59.82 Found (((conj0 a0) (fun (Xx:fofType) (x02:(x (f Xx)))=> ((x02 Xx) True))) iff_refl) as proof of ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) a0)
% 59.59/59.82 Found ((((conj (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) a0) (fun (Xx:fofType) (x02:(x (f Xx)))=> ((x02 Xx) True))) iff_refl) as proof of ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) a0)
% 59.59/59.82 Found ((((conj (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) a0) (fun (Xx:fofType) (x02:(x (f Xx)))=> ((x02 Xx) True))) iff_refl) as proof of ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) a0)
% 59.59/59.82 Found ((((conj (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) a0) (fun (Xx:fofType) (x02:(x (f Xx)))=> ((x02 Xx) True))) iff_refl) as proof of (P a0)
% 59.59/59.82 Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% 61.36/61.58 Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b)
% 61.36/61.58 Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% 61.36/61.58 Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% 61.36/61.58 Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% 61.36/61.58 Found x020:=(x02 Xx):(cP Xx)
% 61.36/61.58 Found (x02 Xx) as proof of (cP Xx)
% 61.36/61.58 Found (x02 Xx) as proof of (cP Xx)
% 61.36/61.58 Found (fun (x02:(x (f Xx)))=> (x02 Xx)) as proof of (cP Xx)
% 61.36/61.58 Found (fun (Xx:fofType) (x02:(x (f Xx)))=> (x02 Xx)) as proof of ((x (f Xx))->(cP Xx))
% 61.36/61.58 Found (fun (Xx:fofType) (x02:(x (f Xx)))=> (x02 Xx)) as proof of (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))
% 61.36/61.58 Found x0:(P0 (f0 x))
% 61.36/61.58 Instantiate: b:=(f0 x):Prop
% 61.36/61.58 Found x0 as proof of (P1 b)
% 61.36/61.58 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))):(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 61.36/61.58 Found (eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 61.36/61.58 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 61.36/61.58 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 61.36/61.58 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 61.36/61.58 Found x0:(P0 (f0 x))
% 61.36/61.58 Instantiate: b:=(f0 x):Prop
% 61.36/61.58 Found x0 as proof of (P1 b)
% 61.36/61.58 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))):(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 61.36/61.58 Found (eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 62.19/62.47 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 62.19/62.47 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 62.19/62.47 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 62.19/62.47 Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% 62.19/62.47 Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 62.19/62.47 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 62.19/62.47 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 62.19/62.47 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 62.19/62.47 Found (eq_sym010 ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) b) (f0 x))
% 62.19/62.47 Found ((eq_sym01 b) ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) b) (f0 x))
% 62.19/62.47 Found (((eq_sym0 (f0 x)) b) ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) b) (f0 x))
% 62.19/62.47 Found (((eq_sym0 (f0 x)) b) ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) b) (f0 x))
% 62.19/62.47 Found ((eq_trans0000 ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (((eq_sym0 (f0 x)) b) ((eq_ref Prop) (f0 x)))) as proof of (forall (P:(Prop->Prop)), ((P ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))->(P (f0 x))))
% 62.19/62.47 Found (((eq_trans000 (f0 x)) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (((eq_sym0 (f0 x)) b) ((eq_ref Prop) (f0 x)))) as proof of (forall (P:(Prop->Prop)), ((P ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))->(P (f0 x))))
% 62.19/62.47 Found ((((eq_trans00 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((eq_ref Prop) (f0 x)))) as proof of (forall (P:(Prop->Prop)), ((P ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))->(P (f0 x))))
% 62.25/62.47 Found (((((eq_trans0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((eq_ref Prop) (f0 x)))) as proof of (forall (P:(Prop->Prop)), ((P ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))->(P (f0 x))))
% 62.25/62.47 Found ((((((eq_trans Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((eq_ref Prop) (f0 x)))) as proof of (forall (P:(Prop->Prop)), ((P ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))->(P (f0 x))))
% 62.25/62.47 Found ((((((eq_trans Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((eq_ref Prop) (f0 x)))) as proof of (forall (P:(Prop->Prop)), ((P ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))->(P (f0 x))))
% 62.25/62.47 Found ((((((eq_trans Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((eq_ref Prop) (f0 x)))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x))
% 62.25/62.48 Found (eq_sym000 ((((((eq_trans Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((eq_ref Prop) (f0 x))))) as proof of (forall (P:(Prop->Prop)), ((P (f0 x))->(P ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))))
% 62.25/62.48 Found ((eq_sym00 (f0 x)) ((((((eq_trans Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((eq_ref Prop) (f0 x))))) as proof of (forall (P:(Prop->Prop)), ((P (f0 x))->(P ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))))
% 62.25/62.48 Found (((eq_sym0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) ((((((eq_trans Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((eq_ref Prop) (f0 x))))) as proof of (forall (P:(Prop->Prop)), ((P (f0 x))->(P ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))))
% 67.36/67.63 Found ((((eq_sym Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) ((((((eq_trans Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) ((((eq_sym Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((eq_ref Prop) (f0 x))))) as proof of (forall (P:(Prop->Prop)), ((P (f0 x))->(P ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))))
% 67.36/67.63 Found ((((eq_sym Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) ((((((eq_trans Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) ((((eq_sym Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((eq_ref Prop) (f0 x))))) as proof of (forall (P:(Prop->Prop)), ((P (f0 x))->(P ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))))
% 67.36/67.63 Found x00:(cP a)
% 67.36/67.63 Instantiate: Xx0:=a:fofType
% 67.36/67.63 Found x00 as proof of (P Xx0)
% 67.36/67.63 Found eq_ref00:=(eq_ref0 (f Xx0)):(((eq fofType) (f Xx0)) (f Xx0))
% 67.36/67.63 Found (eq_ref0 (f Xx0)) as proof of (((eq fofType) (f Xx0)) (f Xx))
% 67.36/67.63 Found ((eq_ref fofType) (f Xx0)) as proof of (((eq fofType) (f Xx0)) (f Xx))
% 67.36/67.63 Found ((eq_ref fofType) (f Xx0)) as proof of (((eq fofType) (f Xx0)) (f Xx))
% 67.36/67.63 Found ((eq_ref fofType) (f Xx0)) as proof of (((eq fofType) (f Xx0)) (f Xx))
% 67.36/67.63 Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% 67.36/67.63 Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 67.47/67.70 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 67.47/67.70 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 67.47/67.70 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 67.47/67.70 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 67.47/67.70 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 67.47/67.70 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 67.47/67.70 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 67.47/67.70 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 67.47/67.70 Found ((eq_trans00000 ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 67.47/67.70 Found ((eq_trans00000 ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 67.47/67.70 Found (((fun (x0:(((eq Prop) (f0 x)) b)) (x00:(((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((eq_trans0000 x0) x00) P0)) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 67.47/67.70 Found (((fun (x0:(((eq Prop) (f0 x)) b)) (x00:(((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_trans000 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) P0)) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 67.47/67.70 Found (((fun (x0:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (x00:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((((eq_trans00 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) P0)) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 67.47/67.71 Found (((fun (x0:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (x00:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((((eq_trans0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) P0)) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 67.47/67.71 Found (((fun (x0:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (x00:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((((((eq_trans Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) P0)) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 67.47/67.71 Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (x00:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((((((eq_trans Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) P0)) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 67.53/67.84 Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (x00:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((((((eq_trans Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) P0)) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))) as proof of (forall (P:(Prop->Prop)), ((P (f0 x))->(P ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))))
% 67.53/67.84 Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% 67.53/67.84 Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 67.53/67.84 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 67.53/67.84 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 67.53/67.84 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 67.53/67.84 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 67.53/67.84 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 67.53/67.84 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 67.53/67.85 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 67.53/67.85 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 67.53/67.85 Found ((eq_trans00000 ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 67.53/67.85 Found ((eq_trans00000 ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 67.53/67.85 Found (((fun (x0:(((eq Prop) (f0 x)) b)) (x00:(((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((eq_trans0000 x0) x00) P0)) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 67.53/67.85 Found (((fun (x0:(((eq Prop) (f0 x)) b)) (x00:(((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_trans000 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) P0)) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 67.53/67.85 Found (((fun (x0:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (x00:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((((eq_trans00 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) P0)) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 67.53/67.85 Found (((fun (x0:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (x00:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((((eq_trans0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) P0)) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 67.53/67.85 Found (((fun (x0:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (x00:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((((((eq_trans Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) P0)) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 67.53/67.85 Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (x00:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((((((eq_trans Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) P0)) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 71.79/72.06 Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (x00:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((((((eq_trans Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) P0)) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))) as proof of (forall (P:(Prop->Prop)), ((P (f0 x))->(P ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))))
% 71.79/72.06 Found x00:(cP a)
% 71.79/72.06 Instantiate: b:=a:fofType
% 71.79/72.06 Found x00 as proof of (P b)
% 71.79/72.06 Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% 71.79/72.06 Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b)
% 71.79/72.06 Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% 71.79/72.06 Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% 71.79/72.06 Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% 71.79/72.06 Found x01:(cP a)
% 71.79/72.06 Instantiate: Xx0:=a:fofType
% 71.79/72.06 Found x01 as proof of (P Xx0)
% 71.79/72.06 Found eq_ref00:=(eq_ref0 (f Xx0)):(((eq fofType) (f Xx0)) (f Xx0))
% 71.79/72.06 Found (eq_ref0 (f Xx0)) as proof of (((eq fofType) (f Xx0)) (f Xx))
% 71.79/72.06 Found ((eq_ref fofType) (f Xx0)) as proof of (((eq fofType) (f Xx0)) (f Xx))
% 71.79/72.06 Found ((eq_ref fofType) (f Xx0)) as proof of (((eq fofType) (f Xx0)) (f Xx))
% 71.79/72.06 Found ((eq_ref fofType) (f Xx0)) as proof of (((eq fofType) (f Xx0)) (f Xx))
% 71.79/72.06 Found x1:(cP a)
% 71.79/72.06 Instantiate: Xx0:=a:fofType
% 71.79/72.06 Found x1 as proof of (P Xx0)
% 71.79/72.06 Found eq_ref00:=(eq_ref0 (f Xx0)):(((eq fofType) (f Xx0)) (f Xx0))
% 71.79/72.06 Found (eq_ref0 (f Xx0)) as proof of (((eq fofType) (f Xx0)) (f Xx))
% 71.79/72.06 Found ((eq_ref fofType) (f Xx0)) as proof of (((eq fofType) (f Xx0)) (f Xx))
% 71.79/72.06 Found ((eq_ref fofType) (f Xx0)) as proof of (((eq fofType) (f Xx0)) (f Xx))
% 71.79/72.06 Found ((eq_ref fofType) (f Xx0)) as proof of (((eq fofType) (f Xx0)) (f Xx))
% 71.79/72.06 Found True:Prop
% 71.79/72.06 Found True as proof of Prop
% 71.79/72.06 Found (x020 True) as proof of (cP Xx)
% 71.79/72.06 Found ((x02 Xx) True) as proof of (cP Xx)
% 71.79/72.06 Found ((x02 Xx) True) as proof of (cP Xx)
% 71.79/72.06 Found (fun (x02:(x (f Xx)))=> ((x02 Xx) True)) as proof of (cP Xx)
% 71.79/72.06 Found (fun (Xx:fofType) (x02:(x (f Xx)))=> ((x02 Xx) True)) as proof of ((x (f Xx))->(cP Xx))
% 74.20/74.48 Found (fun (Xx:fofType) (x02:(x (f Xx)))=> ((x02 Xx) True)) as proof of (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))
% 74.20/74.48 Found x01:(cP a)
% 74.20/74.48 Instantiate: b:=a:fofType
% 74.20/74.48 Found x01 as proof of (P b)
% 74.20/74.48 Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% 74.20/74.48 Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b)
% 74.20/74.48 Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% 74.20/74.48 Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% 74.20/74.48 Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% 74.20/74.48 Found x1:(cP a)
% 74.20/74.48 Instantiate: b:=a:fofType
% 74.20/74.48 Found x1 as proof of (P b)
% 74.20/74.48 Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% 74.20/74.48 Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b)
% 74.20/74.48 Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% 74.20/74.48 Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% 74.20/74.48 Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% 74.20/74.48 Found x02:(P0 (f0 x))
% 74.20/74.48 Found (fun (x02:(P0 (f0 x)))=> x02) as proof of (P0 (f0 x))
% 74.20/74.48 Found (fun (x02:(P0 (f0 x)))=> x02) as proof of (P1 (f0 x))
% 74.20/74.48 Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% 74.20/74.48 Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 74.20/74.48 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 74.20/74.48 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 74.20/74.48 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 74.20/74.48 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 74.20/74.48 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 74.20/74.48 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 74.20/74.48 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 74.20/74.48 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 74.20/74.48 Found (((eq_trans00000 ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f0 x)))=> x02)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 74.20/74.48 Found (((eq_trans00000 ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f0 x)))=> x02)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 74.20/74.48 Found ((((fun (x0:(((eq Prop) (f0 x)) b)) (x00:(((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((eq_trans0000 x0) x00) (fun (x1:Prop)=> ((P0 (f0 x))->(P0 x1))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f0 x)))=> x02)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 74.20/74.48 Found ((((fun (x0:(((eq Prop) (f0 x)) b)) (x00:(((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_trans000 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) (fun (x1:Prop)=> ((P0 (f0 x))->(P0 x1))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f0 x)))=> x02)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 74.20/74.49 Found ((((fun (x0:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (x00:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((((eq_trans00 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) (fun (x1:Prop)=> ((P0 (f0 x))->(P0 x1))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (fun (x02:(P0 (f0 x)))=> x02)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 74.20/74.49 Found ((((fun (x0:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (x00:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((((eq_trans0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) (fun (x1:Prop)=> ((P0 (f0 x))->(P0 x1))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (fun (x02:(P0 (f0 x)))=> x02)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 74.20/74.49 Found ((((fun (x0:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (x00:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((((((eq_trans Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) (fun (x1:Prop)=> ((P0 (f0 x))->(P0 x1))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (fun (x02:(P0 (f0 x)))=> x02)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 74.20/74.49 Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (x00:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((((((eq_trans Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) (fun (x1:Prop)=> ((P0 (f0 x))->(P0 x1))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (fun (x02:(P0 (f0 x)))=> x02))) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 74.20/74.49 Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (x00:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((((((eq_trans Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) (fun (x1:Prop)=> ((P0 (f0 x))->(P0 x1))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (fun (x02:(P0 (f0 x)))=> x02))) as proof of (forall (P:(Prop->Prop)), ((P (f0 x))->(P ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))))
% 74.33/74.65 Found x02:(P0 (f0 x))
% 74.33/74.65 Found (fun (x02:(P0 (f0 x)))=> x02) as proof of (P0 (f0 x))
% 74.33/74.65 Found (fun (x02:(P0 (f0 x)))=> x02) as proof of (P1 (f0 x))
% 74.33/74.65 Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% 74.33/74.65 Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 74.33/74.65 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 74.33/74.65 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 74.33/74.65 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 74.33/74.65 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 74.33/74.65 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 74.33/74.65 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 74.33/74.65 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 74.33/74.65 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 74.33/74.65 Found (((eq_trans00000 ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f0 x)))=> x02)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 74.33/74.65 Found (((eq_trans00000 ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f0 x)))=> x02)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 74.33/74.65 Found ((((fun (x0:(((eq Prop) (f0 x)) b)) (x00:(((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((eq_trans0000 x0) x00) (fun (x1:Prop)=> ((P0 (f0 x))->(P0 x1))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f0 x)))=> x02)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 74.33/74.65 Found ((((fun (x0:(((eq Prop) (f0 x)) b)) (x00:(((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_trans000 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) (fun (x1:Prop)=> ((P0 (f0 x))->(P0 x1))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f0 x)))=> x02)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 74.33/74.66 Found ((((fun (x0:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (x00:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((((eq_trans00 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) (fun (x1:Prop)=> ((P0 (f0 x))->(P0 x1))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (fun (x02:(P0 (f0 x)))=> x02)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 74.33/74.66 Found ((((fun (x0:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (x00:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((((eq_trans0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) (fun (x1:Prop)=> ((P0 (f0 x))->(P0 x1))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (fun (x02:(P0 (f0 x)))=> x02)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 74.33/74.66 Found ((((fun (x0:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (x00:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((((((eq_trans Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) (fun (x1:Prop)=> ((P0 (f0 x))->(P0 x1))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (fun (x02:(P0 (f0 x)))=> x02)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 74.33/74.67 Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (x00:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((((((eq_trans Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) (fun (x1:Prop)=> ((P0 (f0 x))->(P0 x1))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (fun (x02:(P0 (f0 x)))=> x02))) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 74.33/74.67 Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (x00:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((((((eq_trans Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x0) x00) (fun (x1:Prop)=> ((P0 (f0 x))->(P0 x1))))) ((eq_ref Prop) (f0 x))) ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))) (fun (x02:(P0 (f0 x)))=> x02))) as proof of (forall (P:(Prop->Prop)), ((P (f0 x))->(P ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))))
% 80.67/80.92 Found x1:(cP a)
% 80.67/80.92 Instantiate: b:=a:fofType
% 80.67/80.92 Found (fun (x2:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))=> x1) as proof of (P b)
% 80.67/80.92 Found (fun (x1:(cP a)) (x2:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))=> x1) as proof of ((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy)))->(P b))
% 80.67/80.92 Found (fun (x1:(cP a)) (x2:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))=> x1) as proof of ((cP a)->((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy)))->(P b)))
% 80.67/80.92 Found (and_rect00 (fun (x1:(cP a)) (x2:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))=> x1)) as proof of (P b)
% 80.67/80.92 Found ((and_rect0 (P b)) (fun (x1:(cP a)) (x2:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))=> x1)) as proof of (P b)
% 80.67/80.92 Found (((fun (P0:Type) (x1:((cP a)->((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy)))->P0)))=> (((((and_rect (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy)))) P0) x1) x0)) (P b)) (fun (x1:(cP a)) (x2:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))=> x1)) as proof of (P b)
% 80.67/80.92 Found (((fun (P0:Type) (x1:((cP a)->((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy)))->P0)))=> (((((and_rect (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy)))) P0) x1) x0)) (P b)) (fun (x1:(cP a)) (x2:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))=> x1)) as proof of (P b)
% 80.67/80.92 Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% 80.67/80.92 Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% 80.67/80.92 Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% 80.67/80.92 Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% 80.67/80.92 Found (fun (x:(fofType->Prop))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% 80.67/80.92 Found (fun (x:(fofType->Prop))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(fofType->Prop)), (((eq Prop) (f1 x)) (f0 x)))
% 80.67/80.92 Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% 80.67/80.92 Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% 80.67/80.92 Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% 80.67/80.92 Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% 80.67/80.92 Found (fun (x:(fofType->Prop))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% 80.67/80.92 Found (fun (x:(fofType->Prop))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(fofType->Prop)), (((eq Prop) (f1 x)) (f0 x)))
% 80.67/80.92 Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% 80.67/80.92 Found (eq_ref0 a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 80.67/80.92 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 81.23/81.47 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 81.23/81.47 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 81.23/81.47 Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% 81.23/81.47 Found (eq_ref0 a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 81.23/81.47 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 81.23/81.47 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 81.23/81.47 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 81.23/81.47 Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% 81.23/81.47 Found (eq_ref0 a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 81.23/81.47 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 81.23/81.47 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 81.23/81.47 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 81.23/81.47 Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% 81.23/81.47 Found (eq_ref0 a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 81.23/81.47 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 81.23/81.47 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 81.23/81.47 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 81.23/81.47 Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% 81.23/81.47 Found (eq_ref0 a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 81.23/81.47 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 81.23/81.47 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 81.23/81.47 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 81.23/81.47 Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% 81.23/81.47 Found (eq_ref0 a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 93.27/93.55 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 93.27/93.55 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 93.27/93.55 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))
% 93.27/93.55 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 93.27/93.55 Found (eq_ref0 b) as proof of (((eq Prop) b) (f0 x))
% 93.27/93.55 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 93.27/93.55 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 93.27/93.55 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 93.27/93.55 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))):(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 93.27/93.56 Found (eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 93.27/93.56 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 93.27/93.56 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 93.27/93.56 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 93.27/93.56 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 93.27/93.56 Found (eq_ref0 b) as proof of (((eq Prop) b) (f0 x))
% 93.27/93.56 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 93.27/93.56 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 93.27/93.56 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 93.27/93.56 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))):(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 93.59/93.87 Found (eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 93.59/93.87 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 93.59/93.87 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 93.59/93.87 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 93.59/93.87 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 93.59/93.87 Found (eq_ref0 b) as proof of (((eq Prop) b) (f0 x))
% 93.59/93.87 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 93.59/93.87 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 93.59/93.87 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 93.59/93.87 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))):(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 93.59/93.87 Found (eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 93.59/93.87 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 93.59/93.87 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 94.08/94.36 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 94.08/94.36 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 94.08/94.36 Found (eq_ref0 b) as proof of (((eq Prop) b) (f0 x))
% 94.08/94.36 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 94.08/94.36 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 94.08/94.36 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 94.08/94.36 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))):(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 94.08/94.36 Found (eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 94.08/94.36 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 94.08/94.36 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 94.08/94.36 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 94.08/94.36 Found eq_ref00:=(eq_ref0 (fun (A:(fofType->Prop))=> ((and (forall (Xx:fofType), ((A (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (A Xz))))))):(((eq ((fofType->Prop)->Prop)) (fun (A:(fofType->Prop))=> ((and (forall (Xx:fofType), ((A (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (A Xz))))))) (fun (A:(fofType->Prop))=> ((and (forall (Xx:fofType), ((A (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (A Xz)))))))
% 94.54/94.79 Found (eq_ref0 (fun (A:(fofType->Prop))=> ((and (forall (Xx:fofType), ((A (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (A Xz))))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (A:(fofType->Prop))=> ((and (forall (Xx:fofType), ((A (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (A Xz))))))) b0)
% 94.54/94.79 Found ((eq_ref ((fofType->Prop)->Prop)) (fun (A:(fofType->Prop))=> ((and (forall (Xx:fofType), ((A (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (A Xz))))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (A:(fofType->Prop))=> ((and (forall (Xx:fofType), ((A (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (A Xz))))))) b0)
% 94.54/94.79 Found ((eq_ref ((fofType->Prop)->Prop)) (fun (A:(fofType->Prop))=> ((and (forall (Xx:fofType), ((A (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (A Xz))))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (A:(fofType->Prop))=> ((and (forall (Xx:fofType), ((A (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (A Xz))))))) b0)
% 94.54/94.79 Found ((eq_ref ((fofType->Prop)->Prop)) (fun (A:(fofType->Prop))=> ((and (forall (Xx:fofType), ((A (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (A Xz))))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (A:(fofType->Prop))=> ((and (forall (Xx:fofType), ((A (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (A Xz))))))) b0)
% 94.54/94.79 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 94.54/94.79 Found (eq_ref0 b) as proof of (((eq Prop) b) (f0 x))
% 94.54/94.79 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 94.54/94.79 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 94.54/94.79 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 94.54/94.79 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))):(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 94.54/94.79 Found (eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 94.54/94.79 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 95.05/95.34 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 95.05/95.34 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 95.05/95.34 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 95.05/95.34 Found (eq_ref0 b) as proof of (((eq Prop) b) (f0 x))
% 95.05/95.34 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 95.05/95.34 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 95.05/95.34 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 95.05/95.34 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))):(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 95.05/95.34 Found (eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 95.05/95.34 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 95.05/95.34 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 95.05/95.34 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 95.05/95.34 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 95.05/95.34 Found (eq_ref0 b) as proof of (((eq Prop) b) (f0 x))
% 95.63/95.88 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 95.63/95.88 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 95.63/95.88 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 95.63/95.88 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))):(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 95.63/95.88 Found (eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 95.63/95.88 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 95.63/95.88 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 95.63/95.88 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 95.63/95.88 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 95.63/95.88 Found (eq_ref0 b) as proof of (((eq Prop) b) (f0 x))
% 95.63/95.88 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 95.63/95.88 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 95.63/95.88 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 95.63/95.88 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))):(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 95.63/95.88 Found (eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 162.66/162.94 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 162.66/162.94 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 162.66/162.94 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 162.66/162.94 Found iff_refl:=(fun (A:Prop)=> ((((conj (A->A)) (A->A)) (fun (H:A)=> H)) (fun (H:A)=> H))):(forall (P:Prop), ((iff P) P))
% 162.66/162.94 Instantiate: b:=(forall (P:Prop), ((iff P) P)):Prop
% 162.66/162.94 Found iff_refl as proof of b
% 162.66/162.94 Found iff_refl as proof of a0
% 162.66/162.94 Found ((conj00 (fun (Xx:fofType) (x02:(x (f Xx)))=> ((x02 Xx) True))) iff_refl) as proof of ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) a0)
% 162.66/162.94 Found (((conj0 a0) (fun (Xx:fofType) (x02:(x (f Xx)))=> ((x02 Xx) True))) iff_refl) as proof of ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) a0)
% 162.66/162.94 Found ((((conj (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) a0) (fun (Xx:fofType) (x02:(x (f Xx)))=> ((x02 Xx) True))) iff_refl) as proof of ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) a0)
% 162.66/162.94 Found ((((conj (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) a0) (fun (Xx:fofType) (x02:(x (f Xx)))=> ((x02 Xx) True))) iff_refl) as proof of ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) a0)
% 162.66/162.94 Found ((((conj (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) a0) (fun (Xx:fofType) (x02:(x (f Xx)))=> ((x02 Xx) True))) iff_refl) as proof of (P a0)
% 162.66/162.94 Found x00:=(x0 Xx):(cP Xx)
% 162.66/162.94 Found (x0 Xx) as proof of (cP Xx)
% 162.66/162.94 Found (x0 Xx) as proof of (cP Xx)
% 162.66/162.94 Found (fun (x0:(x (f Xx)))=> (x0 Xx)) as proof of (cP Xx)
% 162.66/162.94 Found (fun (Xx:fofType) (x0:(x (f Xx)))=> (x0 Xx)) as proof of ((x (f Xx))->(cP Xx))
% 162.66/162.94 Found (fun (Xx:fofType) (x0:(x (f Xx)))=> (x0 Xx)) as proof of (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))
% 162.66/162.94 Found eta_expansion_dep000:=(eta_expansion_dep00 f0):(((eq (fofType->Prop)) f0) (fun (x:fofType)=> (f0 x)))
% 162.66/162.94 Found (eta_expansion_dep00 f0) as proof of (((eq (fofType->Prop)) f0) b)
% 162.66/162.94 Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 162.66/162.94 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 162.66/162.94 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 162.66/162.94 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 162.66/162.94 Found eta_expansion000:=(eta_expansion00 f0):(((eq (fofType->Prop)) f0) (fun (x:fofType)=> (f0 x)))
% 162.66/162.94 Found (eta_expansion00 f0) as proof of (((eq (fofType->Prop)) f0) b)
% 162.66/162.94 Found ((eta_expansion0 Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 162.66/162.94 Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 162.66/162.94 Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 162.66/162.94 Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 162.66/162.94 Found eta_expansion_dep000:=(eta_expansion_dep00 f0):(((eq (fofType->Prop)) f0) (fun (x:fofType)=> (f0 x)))
% 162.66/162.94 Found (eta_expansion_dep00 f0) as proof of (((eq (fofType->Prop)) f0) b)
% 171.26/171.56 Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 171.26/171.56 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 171.26/171.56 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 171.26/171.56 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 171.26/171.56 Found eq_ref00:=(eq_ref0 f0):(((eq (fofType->Prop)) f0) f0)
% 171.26/171.56 Found (eq_ref0 f0) as proof of (((eq (fofType->Prop)) f0) b)
% 171.26/171.56 Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 171.26/171.56 Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 171.26/171.56 Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 171.26/171.56 Found eq_ref00:=(eq_ref0 f0):(((eq (fofType->Prop)) f0) f0)
% 171.26/171.56 Found (eq_ref0 f0) as proof of (((eq (fofType->Prop)) f0) b)
% 171.26/171.56 Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 171.26/171.56 Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 171.26/171.56 Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 171.26/171.56 Found eta_expansion_dep000:=(eta_expansion_dep00 f0):(((eq (fofType->Prop)) f0) (fun (x:fofType)=> (f0 x)))
% 171.26/171.56 Found (eta_expansion_dep00 f0) as proof of (((eq (fofType->Prop)) f0) b)
% 171.26/171.56 Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 171.26/171.56 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 171.26/171.56 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 171.26/171.56 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 171.26/171.56 Found eta_expansion_dep000:=(eta_expansion_dep00 f0):(((eq (fofType->Prop)) f0) (fun (x:fofType)=> (f0 x)))
% 171.26/171.56 Found (eta_expansion_dep00 f0) as proof of (((eq (fofType->Prop)) f0) b)
% 171.26/171.56 Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 171.26/171.56 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 171.26/171.56 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 171.26/171.56 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 171.26/171.56 Found eta_expansion000:=(eta_expansion00 f0):(((eq (fofType->Prop)) f0) (fun (x:fofType)=> (f0 x)))
% 171.26/171.56 Found (eta_expansion00 f0) as proof of (((eq (fofType->Prop)) f0) b)
% 171.26/171.56 Found ((eta_expansion0 Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 171.26/171.56 Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 171.26/171.56 Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 171.26/171.56 Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 171.26/171.56 Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% 171.26/171.56 Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 171.26/171.56 Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 171.26/171.56 Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 171.26/171.56 Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (b x1))
% 171.26/171.56 Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% 171.26/171.56 Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% 171.26/171.56 Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 171.26/171.56 Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 171.26/171.56 Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 171.26/171.56 Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (b x1))
% 171.26/171.56 Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% 171.26/171.56 Found x0:(P0 (f0 x))
% 171.26/171.56 Instantiate: a0:=(f0 x):Prop
% 171.26/171.56 Found x0 as proof of (P1 a0)
% 171.26/171.56 Found x0:(P0 (f0 x))
% 171.26/171.56 Instantiate: a0:=(f0 x):Prop
% 171.26/171.56 Found x0 as proof of (P1 a0)
% 171.26/171.56 Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% 171.26/171.56 Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 174.22/174.51 Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 174.22/174.51 Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 174.22/174.51 Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (b x1))
% 174.22/174.51 Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% 174.22/174.51 Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% 174.22/174.51 Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 174.22/174.51 Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 174.22/174.51 Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 174.22/174.51 Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (b x1))
% 174.22/174.51 Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% 174.22/174.51 Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% 174.22/174.51 Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 174.22/174.51 Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 174.22/174.51 Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 174.22/174.51 Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (b x1))
% 174.22/174.51 Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% 174.22/174.51 Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% 174.22/174.51 Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 174.22/174.51 Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 174.22/174.51 Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 174.22/174.51 Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (b x1))
% 174.22/174.51 Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% 174.22/174.51 Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% 174.22/174.51 Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% 174.22/174.51 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% 174.22/174.51 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% 174.22/174.51 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% 174.22/174.51 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 174.22/174.51 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 174.22/174.51 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 174.22/174.51 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 174.22/174.51 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 174.22/174.51 Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% 174.22/174.51 Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% 174.22/174.51 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% 174.22/174.51 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% 174.22/174.51 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% 174.22/174.51 Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% 174.22/174.51 Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% 174.22/174.51 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% 174.22/174.51 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% 174.22/174.51 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% 174.22/174.51 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 174.22/174.51 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 174.22/174.51 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 186.14/186.48 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 186.14/186.48 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 186.14/186.48 Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% 186.14/186.48 Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% 186.14/186.48 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% 186.14/186.48 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% 186.14/186.48 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% 186.14/186.48 Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% 186.14/186.48 Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 186.14/186.48 Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 186.14/186.48 Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 186.14/186.48 Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (b x1))
% 186.14/186.48 Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% 186.14/186.48 Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% 186.14/186.48 Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 186.14/186.48 Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 186.14/186.48 Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 186.14/186.48 Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (b x1))
% 186.14/186.48 Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% 186.14/186.48 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 186.14/186.48 Found (eq_ref0 b) as proof of (((eq fofType) b) x00)
% 186.14/186.48 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x00)
% 186.14/186.48 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x00)
% 186.14/186.48 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x00)
% 186.14/186.48 Found eq_ref00:=(eq_ref0 a0):(((eq fofType) a0) a0)
% 186.14/186.48 Found (eq_ref0 a0) as proof of (((eq fofType) a0) b)
% 186.14/186.48 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 186.14/186.48 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 186.14/186.48 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 186.14/186.48 Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% 186.14/186.48 Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) b0)
% 186.14/186.48 Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% 186.14/186.48 Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% 186.14/186.48 Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% 186.14/186.48 Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% 186.14/186.48 Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) b0)
% 186.14/186.48 Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% 186.14/186.49 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% 186.14/186.49 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% 186.14/186.49 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% 186.14/186.49 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 186.14/186.49 Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% 186.14/186.49 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% 186.14/186.49 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% 186.14/186.49 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% 186.14/186.49 Found eq_ref00:=(eq_ref0 a0):(((eq fofType) a0) a0)
% 186.14/186.49 Found (eq_ref0 a0) as proof of (((eq fofType) a0) b)
% 186.14/186.49 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 186.14/186.49 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 186.14/186.49 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 186.14/186.49 Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% 186.14/186.49 Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) b0)
% 195.73/196.06 Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% 195.73/196.06 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% 195.73/196.06 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% 195.73/196.06 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% 195.73/196.06 Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% 195.73/196.06 Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) b0)
% 195.73/196.06 Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% 195.73/196.06 Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% 195.73/196.06 Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% 195.73/196.06 Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% 195.73/196.06 Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) b0)
% 195.73/196.06 Found ((eta_expansion_dep0 (fun (x4:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% 195.73/196.06 Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% 195.73/196.06 Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% 195.73/196.06 Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% 195.73/196.06 Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:fofType)=> (x Xz))):(((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) (fun (x0:fofType)=> (x x0)))
% 195.73/196.06 Found (eta_expansion_dep00 (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b0)
% 195.73/196.06 Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b0)
% 195.73/196.06 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b0)
% 195.73/196.06 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b0)
% 195.73/196.06 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b0)
% 195.73/196.06 Found eta_expansion_dep000:=(eta_expansion_dep00 f0):(((eq (fofType->Prop)) f0) (fun (x:fofType)=> (f0 x)))
% 195.73/196.06 Found (eta_expansion_dep00 f0) as proof of (((eq (fofType->Prop)) f0) b)
% 195.73/196.06 Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 195.73/196.06 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 195.73/196.06 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 195.73/196.06 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 195.73/196.06 Found eq_ref00:=(eq_ref0 f0):(((eq (fofType->Prop)) f0) f0)
% 195.73/196.06 Found (eq_ref0 f0) as proof of (((eq (fofType->Prop)) f0) b)
% 195.73/196.06 Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 195.73/196.06 Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 195.73/196.06 Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 195.73/196.06 Found eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% 195.73/196.06 Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) b0)
% 195.73/196.06 Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% 195.73/196.06 Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% 195.73/196.06 Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% 195.73/196.06 Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% 195.73/196.06 Found eta_expansion000:=(eta_expansion00 (fun (Xz:fofType)=> (x Xz))):(((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) (fun (x0:fofType)=> (x x0)))
% 195.73/196.06 Found (eta_expansion00 (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b0)
% 195.73/196.06 Found ((eta_expansion0 Prop) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b0)
% 201.35/201.66 Found (((eta_expansion fofType) Prop) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b0)
% 201.35/201.66 Found (((eta_expansion fofType) Prop) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b0)
% 201.35/201.66 Found (((eta_expansion fofType) Prop) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b0)
% 201.35/201.66 Found eq_ref00:=(eq_ref0 (f1 x1)):(((eq Prop) (f1 x1)) (f1 x1))
% 201.35/201.66 Found (eq_ref0 (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 201.35/201.66 Found ((eq_ref Prop) (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 201.35/201.66 Found ((eq_ref Prop) (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 201.35/201.66 Found (fun (x1:fofType)=> ((eq_ref Prop) (f1 x1))) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 201.35/201.66 Found (fun (x1:fofType)=> ((eq_ref Prop) (f1 x1))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% 201.35/201.66 Found eq_ref00:=(eq_ref0 (f1 x1)):(((eq Prop) (f1 x1)) (f1 x1))
% 201.35/201.66 Found (eq_ref0 (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 201.35/201.66 Found ((eq_ref Prop) (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 201.35/201.66 Found ((eq_ref Prop) (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 201.35/201.66 Found (fun (x1:fofType)=> ((eq_ref Prop) (f1 x1))) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 201.35/201.66 Found (fun (x1:fofType)=> ((eq_ref Prop) (f1 x1))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% 201.35/201.66 Found eta_expansion000:=(eta_expansion00 f0):(((eq (fofType->Prop)) f0) (fun (x:fofType)=> (f0 x)))
% 201.35/201.66 Found (eta_expansion00 f0) as proof of (((eq (fofType->Prop)) f0) b)
% 201.35/201.66 Found ((eta_expansion0 Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 201.35/201.66 Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 201.35/201.66 Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 201.35/201.66 Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 201.35/201.66 Found eta_expansion_dep000:=(eta_expansion_dep00 f0):(((eq (fofType->Prop)) f0) (fun (x:fofType)=> (f0 x)))
% 201.35/201.66 Found (eta_expansion_dep00 f0) as proof of (((eq (fofType->Prop)) f0) b)
% 201.35/201.66 Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 201.35/201.66 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 201.35/201.66 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 201.35/201.66 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 201.35/201.66 Found eta_expansion000:=(eta_expansion00 f0):(((eq (fofType->Prop)) f0) (fun (x:fofType)=> (f0 x)))
% 201.35/201.66 Found (eta_expansion00 f0) as proof of (((eq (fofType->Prop)) f0) b)
% 201.35/201.66 Found ((eta_expansion0 Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 201.35/201.66 Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 201.35/201.66 Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 201.35/201.66 Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 201.35/201.66 Found eta_expansion_dep000:=(eta_expansion_dep00 f0):(((eq (fofType->Prop)) f0) (fun (x:fofType)=> (f0 x)))
% 201.35/201.66 Found (eta_expansion_dep00 f0) as proof of (((eq (fofType->Prop)) f0) b)
% 201.35/201.66 Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 201.35/201.66 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 201.35/201.66 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 201.35/201.66 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 201.35/201.66 Found eq_ref00:=(eq_ref0 (f1 x1)):(((eq Prop) (f1 x1)) (f1 x1))
% 201.35/201.66 Found (eq_ref0 (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 201.35/201.66 Found ((eq_ref Prop) (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 201.35/201.66 Found ((eq_ref Prop) (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 201.35/201.66 Found (fun (x1:fofType)=> ((eq_ref Prop) (f1 x1))) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 201.35/201.66 Found (fun (x1:fofType)=> ((eq_ref Prop) (f1 x1))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% 206.08/206.41 Found eq_ref00:=(eq_ref0 (f1 x1)):(((eq Prop) (f1 x1)) (f1 x1))
% 206.08/206.41 Found (eq_ref0 (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 206.08/206.41 Found ((eq_ref Prop) (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 206.08/206.41 Found ((eq_ref Prop) (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 206.08/206.41 Found (fun (x1:fofType)=> ((eq_ref Prop) (f1 x1))) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 206.08/206.41 Found (fun (x1:fofType)=> ((eq_ref Prop) (f1 x1))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% 206.08/206.41 Found eq_ref00:=(eq_ref0 (f1 x1)):(((eq Prop) (f1 x1)) (f1 x1))
% 206.08/206.41 Found (eq_ref0 (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 206.08/206.41 Found ((eq_ref Prop) (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 206.08/206.41 Found ((eq_ref Prop) (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 206.08/206.41 Found (fun (x1:fofType)=> ((eq_ref Prop) (f1 x1))) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 206.08/206.41 Found (fun (x1:fofType)=> ((eq_ref Prop) (f1 x1))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% 206.08/206.41 Found eq_ref00:=(eq_ref0 (f1 x1)):(((eq Prop) (f1 x1)) (f1 x1))
% 206.08/206.41 Found (eq_ref0 (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 206.08/206.41 Found ((eq_ref Prop) (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 206.08/206.41 Found ((eq_ref Prop) (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 206.08/206.41 Found (fun (x1:fofType)=> ((eq_ref Prop) (f1 x1))) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 206.08/206.41 Found (fun (x1:fofType)=> ((eq_ref Prop) (f1 x1))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% 206.08/206.41 Found x00:(cP a)
% 206.08/206.41 Instantiate: a0:=a:fofType
% 206.08/206.41 Found x00 as proof of (P a0)
% 206.08/206.41 Found eq_ref00:=(eq_ref0 f0):(((eq (fofType->Prop)) f0) f0)
% 206.08/206.41 Found (eq_ref0 f0) as proof of (((eq (fofType->Prop)) f0) b)
% 206.08/206.41 Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 206.08/206.41 Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 206.08/206.41 Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 206.08/206.41 Found eq_ref00:=(eq_ref0 f0):(((eq (fofType->Prop)) f0) f0)
% 206.08/206.41 Found (eq_ref0 f0) as proof of (((eq (fofType->Prop)) f0) b)
% 206.08/206.41 Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 206.08/206.41 Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 206.08/206.41 Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 206.08/206.41 Found eq_ref00:=(eq_ref0 (f1 x1)):(((eq Prop) (f1 x1)) (f1 x1))
% 206.08/206.41 Found (eq_ref0 (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 206.08/206.41 Found ((eq_ref Prop) (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 206.08/206.41 Found ((eq_ref Prop) (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 206.08/206.41 Found (fun (x1:fofType)=> ((eq_ref Prop) (f1 x1))) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 206.08/206.41 Found (fun (x1:fofType)=> ((eq_ref Prop) (f1 x1))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% 206.08/206.41 Found eq_ref00:=(eq_ref0 (f1 x1)):(((eq Prop) (f1 x1)) (f1 x1))
% 206.08/206.41 Found (eq_ref0 (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 206.08/206.41 Found ((eq_ref Prop) (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 206.08/206.41 Found ((eq_ref Prop) (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 206.08/206.41 Found (fun (x1:fofType)=> ((eq_ref Prop) (f1 x1))) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 206.08/206.41 Found (fun (x1:fofType)=> ((eq_ref Prop) (f1 x1))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% 206.08/206.41 Found eq_ref00:=(eq_ref0 (f1 x1)):(((eq Prop) (f1 x1)) (f1 x1))
% 206.08/206.41 Found (eq_ref0 (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 206.08/206.41 Found ((eq_ref Prop) (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 206.08/206.41 Found ((eq_ref Prop) (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 206.08/206.41 Found (fun (x1:fofType)=> ((eq_ref Prop) (f1 x1))) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 206.08/206.41 Found (fun (x1:fofType)=> ((eq_ref Prop) (f1 x1))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% 206.08/206.41 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 206.08/206.41 Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% 206.08/206.41 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% 206.08/206.41 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% 206.08/206.41 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% 206.08/206.41 Found eq_ref00:=(eq_ref0 a0):(((eq fofType) a0) a0)
% 206.08/206.41 Found (eq_ref0 a0) as proof of (((eq fofType) a0) b)
% 208.34/208.70 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 208.34/208.70 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 208.34/208.70 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 208.34/208.70 Found eq_ref00:=(eq_ref0 a0):(((eq fofType) a0) a0)
% 208.34/208.70 Found (eq_ref0 a0) as proof of (((eq fofType) a0) b)
% 208.34/208.70 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 208.34/208.70 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 208.34/208.70 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 208.34/208.70 Found eq_ref00:=(eq_ref0 (f1 x1)):(((eq Prop) (f1 x1)) (f1 x1))
% 208.34/208.70 Found (eq_ref0 (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 208.34/208.70 Found ((eq_ref Prop) (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 208.34/208.70 Found ((eq_ref Prop) (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 208.34/208.70 Found (fun (x1:fofType)=> ((eq_ref Prop) (f1 x1))) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 208.34/208.70 Found (fun (x1:fofType)=> ((eq_ref Prop) (f1 x1))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% 208.34/208.70 Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% 208.34/208.70 Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 208.34/208.70 Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 208.34/208.70 Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 208.34/208.70 Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (b x1))
% 208.34/208.70 Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% 208.34/208.70 Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% 208.34/208.70 Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 208.34/208.70 Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 208.34/208.70 Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 208.34/208.70 Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (b x1))
% 208.34/208.70 Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% 208.34/208.70 Found x1:(cP a)
% 208.34/208.70 Instantiate: a0:=a:fofType
% 208.34/208.70 Found (fun (x2:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))=> x1) as proof of (P a0)
% 208.34/208.70 Found (fun (x1:(cP a)) (x2:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))=> x1) as proof of ((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy)))->(P a0))
% 208.34/208.70 Found (fun (x1:(cP a)) (x2:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))=> x1) as proof of ((cP a)->((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy)))->(P a0)))
% 208.34/208.70 Found (and_rect00 (fun (x1:(cP a)) (x2:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))=> x1)) as proof of (P a0)
% 208.34/208.70 Found ((and_rect0 (P a0)) (fun (x1:(cP a)) (x2:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))=> x1)) as proof of (P a0)
% 208.34/208.70 Found (((fun (P0:Type) (x1:((cP a)->((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy)))->P0)))=> (((((and_rect (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy)))) P0) x1) x0)) (P a0)) (fun (x1:(cP a)) (x2:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))=> x1)) as proof of (P a0)
% 208.34/208.70 Found (((fun (P0:Type) (x1:((cP a)->((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy)))->P0)))=> (((((and_rect (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy)))) P0) x1) x0)) (P a0)) (fun (x1:(cP a)) (x2:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))=> x1)) as proof of (P a0)
% 208.34/208.70 Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:fofType)=> (x Xz))):(((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) (fun (x0:fofType)=> (x x0)))
% 208.34/208.70 Found (eta_expansion_dep00 (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b0)
% 208.34/208.70 Found ((eta_expansion_dep0 (fun (x4:fofType)=> Prop)) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b0)
% 211.15/211.47 Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b0)
% 211.15/211.47 Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b0)
% 211.15/211.47 Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b0)
% 211.15/211.47 Found eq_ref00:=(eq_ref0 (f1 x1)):(((eq Prop) (f1 x1)) (f1 x1))
% 211.15/211.47 Found (eq_ref0 (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 211.15/211.47 Found ((eq_ref Prop) (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 211.15/211.47 Found ((eq_ref Prop) (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 211.15/211.47 Found (fun (x1:fofType)=> ((eq_ref Prop) (f1 x1))) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 211.15/211.47 Found (fun (x1:fofType)=> ((eq_ref Prop) (f1 x1))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% 211.15/211.47 Found eq_ref00:=(eq_ref0 (f1 x1)):(((eq Prop) (f1 x1)) (f1 x1))
% 211.15/211.47 Found (eq_ref0 (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 211.15/211.47 Found ((eq_ref Prop) (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 211.15/211.47 Found ((eq_ref Prop) (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 211.15/211.47 Found (fun (x1:fofType)=> ((eq_ref Prop) (f1 x1))) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 211.15/211.47 Found (fun (x1:fofType)=> ((eq_ref Prop) (f1 x1))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% 211.15/211.47 Found eq_ref00:=(eq_ref0 (f1 x1)):(((eq Prop) (f1 x1)) (f1 x1))
% 211.15/211.47 Found (eq_ref0 (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 211.15/211.47 Found ((eq_ref Prop) (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 211.15/211.47 Found ((eq_ref Prop) (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 211.15/211.47 Found (fun (x1:fofType)=> ((eq_ref Prop) (f1 x1))) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 211.15/211.47 Found (fun (x1:fofType)=> ((eq_ref Prop) (f1 x1))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% 211.15/211.47 Found eq_ref00:=(eq_ref0 (f1 x1)):(((eq Prop) (f1 x1)) (f1 x1))
% 211.15/211.47 Found (eq_ref0 (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 211.15/211.47 Found ((eq_ref Prop) (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 211.15/211.47 Found ((eq_ref Prop) (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 211.15/211.47 Found (fun (x1:fofType)=> ((eq_ref Prop) (f1 x1))) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 211.15/211.47 Found (fun (x1:fofType)=> ((eq_ref Prop) (f1 x1))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% 211.15/211.47 Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% 211.15/211.47 Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 211.15/211.47 Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 211.15/211.47 Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 211.15/211.47 Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (b x1))
% 211.15/211.47 Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% 211.15/211.47 Found eta_expansion000:=(eta_expansion00 f0):(((eq (fofType->Prop)) f0) (fun (x:fofType)=> (f0 x)))
% 211.15/211.47 Found (eta_expansion00 f0) as proof of (((eq (fofType->Prop)) f0) b)
% 211.15/211.47 Found ((eta_expansion0 Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 211.15/211.47 Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 211.15/211.47 Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 211.15/211.47 Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 211.15/211.47 Found eta_expansion000:=(eta_expansion00 f0):(((eq (fofType->Prop)) f0) (fun (x:fofType)=> (f0 x)))
% 211.15/211.47 Found (eta_expansion00 f0) as proof of (((eq (fofType->Prop)) f0) b)
% 211.15/211.47 Found ((eta_expansion0 Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 211.15/211.47 Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 211.15/211.47 Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 211.15/211.47 Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 211.15/211.47 Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% 211.15/211.47 Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 211.15/211.47 Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 211.74/212.07 Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 211.74/212.07 Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (b x1))
% 211.74/212.07 Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% 211.74/212.07 Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% 211.74/212.07 Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 211.74/212.07 Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 211.74/212.07 Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 211.74/212.07 Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (b x1))
% 211.74/212.07 Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% 211.74/212.07 Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% 211.74/212.07 Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 211.74/212.07 Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 211.74/212.07 Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 211.74/212.07 Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (b x1))
% 211.74/212.07 Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% 211.74/212.07 Found x1:(cP a)
% 211.74/212.07 Instantiate: a0:=a:fofType
% 211.74/212.07 Found (fun (x2:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))=> x1) as proof of (P a0)
% 211.74/212.07 Found (fun (x1:(cP a)) (x2:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))=> x1) as proof of ((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy)))->(P a0))
% 211.74/212.07 Found (fun (x1:(cP a)) (x2:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))=> x1) as proof of ((cP a)->((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy)))->(P a0)))
% 211.74/212.07 Found (and_rect00 (fun (x1:(cP a)) (x2:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))=> x1)) as proof of (P a0)
% 211.74/212.07 Found ((and_rect0 (P a0)) (fun (x1:(cP a)) (x2:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))=> x1)) as proof of (P a0)
% 211.74/212.07 Found (((fun (P0:Type) (x1:((cP a)->((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy)))->P0)))=> (((((and_rect (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy)))) P0) x1) x0)) (P a0)) (fun (x1:(cP a)) (x2:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))=> x1)) as proof of (P a0)
% 211.74/212.07 Found (((fun (P0:Type) (x1:((cP a)->((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy)))->P0)))=> (((((and_rect (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy)))) P0) x1) x0)) (P a0)) (fun (x1:(cP a)) (x2:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))=> x1)) as proof of (P a0)
% 211.74/212.07 Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:fofType)=> (x Xz))):(((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) (fun (x0:fofType)=> (x x0)))
% 211.74/212.07 Found (eta_expansion_dep00 (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b0)
% 211.74/212.07 Found ((eta_expansion_dep0 (fun (x4:fofType)=> Prop)) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b0)
% 211.74/212.07 Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b0)
% 211.74/212.07 Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b0)
% 211.74/212.07 Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b0)
% 211.74/212.07 Found eq_ref00:=(eq_ref0 (f1 x1)):(((eq Prop) (f1 x1)) (f1 x1))
% 211.74/212.07 Found (eq_ref0 (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 215.85/216.24 Found ((eq_ref Prop) (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 215.85/216.24 Found ((eq_ref Prop) (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 215.85/216.24 Found (fun (x1:fofType)=> ((eq_ref Prop) (f1 x1))) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 215.85/216.24 Found (fun (x1:fofType)=> ((eq_ref Prop) (f1 x1))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% 215.85/216.24 Found eq_ref00:=(eq_ref0 (f1 x1)):(((eq Prop) (f1 x1)) (f1 x1))
% 215.85/216.24 Found (eq_ref0 (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 215.85/216.24 Found ((eq_ref Prop) (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 215.85/216.24 Found ((eq_ref Prop) (f1 x1)) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 215.85/216.24 Found (fun (x1:fofType)=> ((eq_ref Prop) (f1 x1))) as proof of (((eq Prop) (f1 x1)) (f0 x1))
% 215.85/216.24 Found (fun (x1:fofType)=> ((eq_ref Prop) (f1 x1))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% 215.85/216.24 Found x01:(cP a)
% 215.85/216.24 Instantiate: a0:=a:fofType
% 215.85/216.24 Found x01 as proof of (P a0)
% 215.85/216.24 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 215.85/216.24 Found (eq_ref0 b) as proof of (((eq fofType) b) x02)
% 215.85/216.24 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x02)
% 215.85/216.24 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x02)
% 215.85/216.24 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x02)
% 215.85/216.24 Found eq_ref00:=(eq_ref0 a0):(((eq fofType) a0) a0)
% 215.85/216.24 Found (eq_ref0 a0) as proof of (((eq fofType) a0) b)
% 215.85/216.24 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 215.85/216.24 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 215.85/216.24 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 215.85/216.24 Found eq_ref00:=(eq_ref0 (f1 x3)):(((eq Prop) (f1 x3)) (f1 x3))
% 215.85/216.24 Found (eq_ref0 (f1 x3)) as proof of (((eq Prop) (f1 x3)) (f0 x3))
% 215.85/216.24 Found ((eq_ref Prop) (f1 x3)) as proof of (((eq Prop) (f1 x3)) (f0 x3))
% 215.85/216.24 Found ((eq_ref Prop) (f1 x3)) as proof of (((eq Prop) (f1 x3)) (f0 x3))
% 215.85/216.24 Found (fun (x3:fofType)=> ((eq_ref Prop) (f1 x3))) as proof of (((eq Prop) (f1 x3)) (f0 x3))
% 215.85/216.24 Found (fun (x3:fofType)=> ((eq_ref Prop) (f1 x3))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% 215.85/216.24 Found eq_ref00:=(eq_ref0 (f1 x3)):(((eq Prop) (f1 x3)) (f1 x3))
% 215.85/216.24 Found (eq_ref0 (f1 x3)) as proof of (((eq Prop) (f1 x3)) (f0 x3))
% 215.85/216.24 Found ((eq_ref Prop) (f1 x3)) as proof of (((eq Prop) (f1 x3)) (f0 x3))
% 215.85/216.24 Found ((eq_ref Prop) (f1 x3)) as proof of (((eq Prop) (f1 x3)) (f0 x3))
% 215.85/216.24 Found (fun (x3:fofType)=> ((eq_ref Prop) (f1 x3))) as proof of (((eq Prop) (f1 x3)) (f0 x3))
% 215.85/216.24 Found (fun (x3:fofType)=> ((eq_ref Prop) (f1 x3))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% 215.85/216.24 Found x1:(cP a)
% 215.85/216.24 Instantiate: a0:=a:fofType
% 215.85/216.24 Found x1 as proof of (P a0)
% 215.85/216.24 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 215.85/216.24 Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% 215.85/216.24 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% 215.85/216.24 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% 215.85/216.24 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% 215.85/216.24 Found eq_ref00:=(eq_ref0 a0):(((eq fofType) a0) a0)
% 215.85/216.24 Found (eq_ref0 a0) as proof of (((eq fofType) a0) b)
% 215.85/216.24 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 215.85/216.24 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 215.85/216.24 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 215.85/216.24 Found eq_ref00:=(eq_ref0 a0):(((eq fofType) a0) a0)
% 215.85/216.24 Found (eq_ref0 a0) as proof of (((eq fofType) a0) b)
% 215.85/216.24 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 215.85/216.24 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 215.85/216.24 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 215.85/216.24 Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% 215.85/216.24 Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 215.85/216.24 Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 215.85/216.24 Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 215.85/216.24 Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (b x1))
% 215.85/216.24 Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% 215.85/216.24 Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% 215.85/216.24 Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 215.85/216.24 Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 218.57/218.95 Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (b x1))
% 218.57/218.95 Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (b x1))
% 218.57/218.95 Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% 218.57/218.95 Found eta_expansion_dep000:=(eta_expansion_dep00 f0):(((eq (fofType->Prop)) f0) (fun (x:fofType)=> (f0 x)))
% 218.57/218.95 Found (eta_expansion_dep00 f0) as proof of (((eq (fofType->Prop)) f0) b)
% 218.57/218.95 Found ((eta_expansion_dep0 (fun (x4:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 218.57/218.95 Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 218.57/218.95 Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 218.57/218.95 Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 218.57/218.95 Found eta_expansion000:=(eta_expansion00 f0):(((eq (fofType->Prop)) f0) (fun (x:fofType)=> (f0 x)))
% 218.57/218.95 Found (eta_expansion00 f0) as proof of (((eq (fofType->Prop)) f0) b)
% 218.57/218.95 Found ((eta_expansion0 Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 218.57/218.95 Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 218.57/218.95 Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 218.57/218.95 Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 218.57/218.95 Found eta_expansion_dep000:=(eta_expansion_dep00 f0):(((eq (fofType->Prop)) f0) (fun (x:fofType)=> (f0 x)))
% 218.57/218.95 Found (eta_expansion_dep00 f0) as proof of (((eq (fofType->Prop)) f0) b)
% 218.57/218.95 Found ((eta_expansion_dep0 (fun (x4:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 218.57/218.95 Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 218.57/218.95 Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 218.57/218.95 Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 218.57/218.95 Found eq_ref00:=(eq_ref0 f0):(((eq (fofType->Prop)) f0) f0)
% 218.57/218.95 Found (eq_ref0 f0) as proof of (((eq (fofType->Prop)) f0) b)
% 218.57/218.95 Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 218.57/218.95 Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 218.57/218.95 Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 218.57/218.95 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 218.57/218.95 Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% 218.57/218.95 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% 218.57/218.95 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% 218.57/218.95 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% 218.57/218.95 Found eq_ref00:=(eq_ref0 a0):(((eq fofType) a0) a0)
% 218.57/218.95 Found (eq_ref0 a0) as proof of (((eq fofType) a0) b)
% 218.57/218.95 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 218.57/218.95 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 218.57/218.95 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 218.57/218.95 Found eq_ref00:=(eq_ref0 a0):(((eq fofType) a0) a0)
% 218.57/218.95 Found (eq_ref0 a0) as proof of (((eq fofType) a0) b)
% 218.57/218.95 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 218.57/218.95 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 218.57/218.95 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 218.57/218.95 Found eq_ref00:=(eq_ref0 (f1 x3)):(((eq Prop) (f1 x3)) (f1 x3))
% 218.57/218.95 Found (eq_ref0 (f1 x3)) as proof of (((eq Prop) (f1 x3)) (f0 x3))
% 218.57/218.95 Found ((eq_ref Prop) (f1 x3)) as proof of (((eq Prop) (f1 x3)) (f0 x3))
% 218.57/218.95 Found ((eq_ref Prop) (f1 x3)) as proof of (((eq Prop) (f1 x3)) (f0 x3))
% 218.57/218.95 Found (fun (x3:fofType)=> ((eq_ref Prop) (f1 x3))) as proof of (((eq Prop) (f1 x3)) (f0 x3))
% 218.57/218.95 Found (fun (x3:fofType)=> ((eq_ref Prop) (f1 x3))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% 218.57/218.95 Found eq_ref00:=(eq_ref0 (f1 x3)):(((eq Prop) (f1 x3)) (f1 x3))
% 218.57/218.95 Found (eq_ref0 (f1 x3)) as proof of (((eq Prop) (f1 x3)) (f0 x3))
% 218.57/218.95 Found ((eq_ref Prop) (f1 x3)) as proof of (((eq Prop) (f1 x3)) (f0 x3))
% 218.57/218.95 Found ((eq_ref Prop) (f1 x3)) as proof of (((eq Prop) (f1 x3)) (f0 x3))
% 218.57/218.95 Found (fun (x3:fofType)=> ((eq_ref Prop) (f1 x3))) as proof of (((eq Prop) (f1 x3)) (f0 x3))
% 218.57/218.95 Found (fun (x3:fofType)=> ((eq_ref Prop) (f1 x3))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% 220.34/220.71 Found eq_ref00:=(eq_ref0 (f1 x3)):(((eq Prop) (f1 x3)) (f1 x3))
% 220.34/220.71 Found (eq_ref0 (f1 x3)) as proof of (((eq Prop) (f1 x3)) (f0 x3))
% 220.34/220.71 Found ((eq_ref Prop) (f1 x3)) as proof of (((eq Prop) (f1 x3)) (f0 x3))
% 220.34/220.71 Found ((eq_ref Prop) (f1 x3)) as proof of (((eq Prop) (f1 x3)) (f0 x3))
% 220.34/220.71 Found (fun (x3:fofType)=> ((eq_ref Prop) (f1 x3))) as proof of (((eq Prop) (f1 x3)) (f0 x3))
% 220.34/220.71 Found (fun (x3:fofType)=> ((eq_ref Prop) (f1 x3))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% 220.34/220.71 Found eq_ref00:=(eq_ref0 (f1 x3)):(((eq Prop) (f1 x3)) (f1 x3))
% 220.34/220.71 Found (eq_ref0 (f1 x3)) as proof of (((eq Prop) (f1 x3)) (f0 x3))
% 220.34/220.71 Found ((eq_ref Prop) (f1 x3)) as proof of (((eq Prop) (f1 x3)) (f0 x3))
% 220.34/220.71 Found ((eq_ref Prop) (f1 x3)) as proof of (((eq Prop) (f1 x3)) (f0 x3))
% 220.34/220.71 Found (fun (x3:fofType)=> ((eq_ref Prop) (f1 x3))) as proof of (((eq Prop) (f1 x3)) (f0 x3))
% 220.34/220.71 Found (fun (x3:fofType)=> ((eq_ref Prop) (f1 x3))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% 220.34/220.71 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))):(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 220.34/220.71 Found (eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 220.34/220.71 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 220.34/220.71 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 220.34/220.71 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 220.34/220.71 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))):(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 220.61/220.96 Found (eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 220.61/220.96 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 220.61/220.96 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 220.61/220.96 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 220.61/220.96 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))):(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 220.61/220.96 Found (eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 220.61/220.96 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 220.61/220.96 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 220.61/220.96 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 222.82/223.16 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))):(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 222.82/223.16 Found (eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 222.82/223.16 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 222.82/223.16 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 222.82/223.16 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 222.82/223.16 Found eta_expansion000:=(eta_expansion00 f0):(((eq (fofType->Prop)) f0) (fun (x:fofType)=> (f0 x)))
% 222.82/223.16 Found (eta_expansion00 f0) as proof of (((eq (fofType->Prop)) f0) b)
% 222.82/223.16 Found ((eta_expansion0 Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 222.82/223.16 Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 222.82/223.16 Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 222.82/223.16 Found (((eta_expansion fofType) Prop) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 222.82/223.16 Found eq_ref00:=(eq_ref0 f0):(((eq (fofType->Prop)) f0) f0)
% 222.82/223.16 Found (eq_ref0 f0) as proof of (((eq (fofType->Prop)) f0) b)
% 222.82/223.16 Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 222.82/223.16 Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 222.82/223.16 Found ((eq_ref (fofType->Prop)) f0) as proof of (((eq (fofType->Prop)) f0) b)
% 222.82/223.16 Found eq_ref00:=(eq_ref0 (f1 x3)):(((eq Prop) (f1 x3)) (f1 x3))
% 222.82/223.16 Found (eq_ref0 (f1 x3)) as proof of (((eq Prop) (f1 x3)) (f0 x3))
% 222.82/223.16 Found ((eq_ref Prop) (f1 x3)) as proof of (((eq Prop) (f1 x3)) (f0 x3))
% 222.82/223.16 Found ((eq_ref Prop) (f1 x3)) as proof of (((eq Prop) (f1 x3)) (f0 x3))
% 222.82/223.16 Found (fun (x3:fofType)=> ((eq_ref Prop) (f1 x3))) as proof of (((eq Prop) (f1 x3)) (f0 x3))
% 227.32/227.67 Found (fun (x3:fofType)=> ((eq_ref Prop) (f1 x3))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% 227.32/227.67 Found eq_ref00:=(eq_ref0 (f1 x3)):(((eq Prop) (f1 x3)) (f1 x3))
% 227.32/227.67 Found (eq_ref0 (f1 x3)) as proof of (((eq Prop) (f1 x3)) (f0 x3))
% 227.32/227.67 Found ((eq_ref Prop) (f1 x3)) as proof of (((eq Prop) (f1 x3)) (f0 x3))
% 227.32/227.67 Found ((eq_ref Prop) (f1 x3)) as proof of (((eq Prop) (f1 x3)) (f0 x3))
% 227.32/227.67 Found (fun (x3:fofType)=> ((eq_ref Prop) (f1 x3))) as proof of (((eq Prop) (f1 x3)) (f0 x3))
% 227.32/227.67 Found (fun (x3:fofType)=> ((eq_ref Prop) (f1 x3))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% 227.32/227.67 Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% 227.32/227.67 Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) (b x3))
% 227.32/227.67 Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (b x3))
% 227.32/227.67 Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (b x3))
% 227.32/227.67 Found (fun (x3:fofType)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) (b x3))
% 227.32/227.67 Found (fun (x3:fofType)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% 227.32/227.67 Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% 227.32/227.67 Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) (b x3))
% 227.32/227.67 Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (b x3))
% 227.32/227.67 Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (b x3))
% 227.32/227.67 Found (fun (x3:fofType)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) (b x3))
% 227.32/227.67 Found (fun (x3:fofType)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% 227.32/227.67 Found eq_ref000:=(eq_ref00 P0):((P0 (f Xx0))->(P0 (f Xx0)))
% 227.32/227.67 Found (eq_ref00 P0) as proof of (P1 (f Xx0))
% 227.32/227.67 Found ((eq_ref0 (f Xx0)) P0) as proof of (P1 (f Xx0))
% 227.32/227.67 Found (((eq_ref fofType) (f Xx0)) P0) as proof of (P1 (f Xx0))
% 227.32/227.67 Found (((eq_ref fofType) (f Xx0)) P0) as proof of (P1 (f Xx0))
% 227.32/227.67 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 227.32/227.67 Found (eq_ref0 b) as proof of (((eq fofType) b) x00)
% 227.32/227.67 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x00)
% 227.32/227.67 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x00)
% 227.32/227.67 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x00)
% 227.32/227.67 Found eq_ref00:=(eq_ref0 a0):(((eq fofType) a0) a0)
% 227.32/227.67 Found (eq_ref0 a0) as proof of (((eq fofType) a0) b)
% 227.32/227.67 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 227.32/227.67 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 227.32/227.67 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 227.32/227.67 Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% 227.32/227.67 Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) (b x3))
% 227.32/227.67 Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (b x3))
% 227.32/227.67 Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (b x3))
% 227.32/227.67 Found (fun (x3:fofType)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) (b x3))
% 227.32/227.67 Found (fun (x3:fofType)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% 227.32/227.67 Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% 227.32/227.67 Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) (b x3))
% 227.32/227.67 Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (b x3))
% 227.32/227.67 Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (b x3))
% 227.32/227.67 Found (fun (x3:fofType)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) (b x3))
% 227.32/227.67 Found (fun (x3:fofType)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% 227.32/227.67 Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% 227.32/227.67 Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) (b x3))
% 227.32/227.67 Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (b x3))
% 227.32/227.67 Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (b x3))
% 227.32/227.67 Found (fun (x3:fofType)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) (b x3))
% 227.32/227.67 Found (fun (x3:fofType)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% 227.32/227.67 Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% 227.32/227.67 Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) (b x3))
% 227.32/227.67 Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (b x3))
% 227.32/227.67 Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (b x3))
% 237.61/237.98 Found (fun (x3:fofType)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) (b x3))
% 237.61/237.98 Found (fun (x3:fofType)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% 237.61/237.98 Found eq_ref000:=(eq_ref00 P0):((P0 Xx)->(P0 Xx))
% 237.61/237.98 Found (eq_ref00 P0) as proof of (P1 Xx)
% 237.61/237.98 Found ((eq_ref0 Xx) P0) as proof of (P1 Xx)
% 237.61/237.98 Found (((eq_ref fofType) Xx) P0) as proof of (P1 Xx)
% 237.61/237.98 Found (((eq_ref fofType) Xx) P0) as proof of (P1 Xx)
% 237.61/237.98 Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% 237.61/237.98 Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) (b x3))
% 237.61/237.98 Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (b x3))
% 237.61/237.98 Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (b x3))
% 237.61/237.98 Found (fun (x3:fofType)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) (b x3))
% 237.61/237.98 Found (fun (x3:fofType)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% 237.61/237.98 Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% 237.61/237.98 Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) (b x3))
% 237.61/237.98 Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (b x3))
% 237.61/237.98 Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (b x3))
% 237.61/237.98 Found (fun (x3:fofType)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) (b x3))
% 237.61/237.98 Found (fun (x3:fofType)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (b x)))
% 237.61/237.98 Found eq_ref000:=(eq_ref00 P0):((P0 (f Xx0))->(P0 (f Xx0)))
% 237.61/237.98 Found (eq_ref00 P0) as proof of (P1 (f Xx0))
% 237.61/237.98 Found ((eq_ref0 (f Xx0)) P0) as proof of (P1 (f Xx0))
% 237.61/237.98 Found (((eq_ref fofType) (f Xx0)) P0) as proof of (P1 (f Xx0))
% 237.61/237.98 Found (((eq_ref fofType) (f Xx0)) P0) as proof of (P1 (f Xx0))
% 237.61/237.98 Found eq_ref000:=(eq_ref00 P0):((P0 (f Xx0))->(P0 (f Xx0)))
% 237.61/237.98 Found (eq_ref00 P0) as proof of (P1 (f Xx0))
% 237.61/237.98 Found ((eq_ref0 (f Xx0)) P0) as proof of (P1 (f Xx0))
% 237.61/237.98 Found (((eq_ref fofType) (f Xx0)) P0) as proof of (P1 (f Xx0))
% 237.61/237.98 Found (((eq_ref fofType) (f Xx0)) P0) as proof of (P1 (f Xx0))
% 237.61/237.98 Found x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 237.61/237.98 Instantiate: b:=((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))):Prop
% 237.61/237.98 Found x0 as proof of (P3 b)
% 237.61/237.98 Found x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 237.61/237.98 Instantiate: b:=((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))):Prop
% 237.61/237.98 Found x0 as proof of (P3 b)
% 237.61/237.98 Found x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 237.61/237.98 Instantiate: b:=((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))):Prop
% 237.61/237.98 Found x0 as proof of (P3 b)
% 237.61/237.98 Found x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 237.61/237.98 Instantiate: b:=((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))):Prop
% 237.61/237.98 Found x0 as proof of (P3 b)
% 237.61/237.98 Found eq_ref000:=(eq_ref00 P0):((P0 Xx)->(P0 Xx))
% 237.61/237.98 Found (eq_ref00 P0) as proof of (P1 Xx)
% 237.61/237.98 Found ((eq_ref0 Xx) P0) as proof of (P1 Xx)
% 237.61/237.98 Found (((eq_ref fofType) Xx) P0) as proof of (P1 Xx)
% 237.61/237.98 Found (((eq_ref fofType) Xx) P0) as proof of (P1 Xx)
% 238.82/239.22 Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% 238.82/239.22 Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 238.82/239.22 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 238.82/239.22 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 238.82/239.22 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 238.82/239.22 Found ((eq_sym0100 ((eq_ref Prop) (f0 x))) x0) as proof of (P2 (f0 x))
% 238.82/239.22 Found ((eq_sym0100 ((eq_ref Prop) (f0 x))) x0) as proof of (P2 (f0 x))
% 238.82/239.22 Found (((fun (x00:(((eq Prop) (f0 x)) b))=> ((eq_sym010 x00) P2)) ((eq_ref Prop) (f0 x))) x0) as proof of (P2 (f0 x))
% 238.82/239.22 Found (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((eq_sym01 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0) as proof of (P2 (f0 x))
% 238.82/239.22 Found (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0) as proof of (P2 (f0 x))
% 238.82/239.22 Found (fun (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0)) as proof of (P2 (f0 x))
% 238.82/239.22 Found (fun (P2:(Prop->Prop)) (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0)) as proof of ((P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))->(P2 (f0 x)))
% 238.82/239.22 Found (fun (P2:(Prop->Prop)) (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0)) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x))
% 238.82/239.24 Found ((eq_sym0000 (fun (P2:(Prop->Prop)) (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0))) (fun (x01:(P0 (f0 x)))=> x01)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 238.82/239.24 Found ((eq_sym0000 (fun (P2:(Prop->Prop)) (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0))) (fun (x01:(P0 (f0 x)))=> x01)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 238.82/239.24 Found (((fun (x0:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)))=> ((eq_sym000 x0) (fun (x1:Prop)=> ((P0 (f0 x))->(P0 x1))))) (fun (P2:(Prop->Prop)) (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0))) (fun (x01:(P0 (f0 x)))=> x01)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 238.82/239.24 Found (((fun (x0:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)))=> (((eq_sym00 (f0 x)) x0) (fun (x1:Prop)=> ((P0 (f0 x))->(P0 x1))))) (fun (P2:(Prop->Prop)) (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0))) (fun (x01:(P0 (f0 x)))=> x01)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 238.82/239.25 Found (((fun (x0:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)))=> ((((eq_sym0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) x0) (fun (x1:Prop)=> ((P0 (f0 x))->(P0 x1))))) (fun (P2:(Prop->Prop)) (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0))) (fun (x01:(P0 (f0 x)))=> x01)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 238.82/239.25 Found (((fun (x0:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)))=> (((((eq_sym Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) x0) (fun (x1:Prop)=> ((P0 (f0 x))->(P0 x1))))) (fun (P2:(Prop->Prop)) (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((((eq_sym Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0))) (fun (x01:(P0 (f0 x)))=> x01)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 238.82/239.27 Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)))=> (((((eq_sym Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) x0) (fun (x1:Prop)=> ((P0 (f0 x))->(P0 x1))))) (fun (P2:(Prop->Prop)) (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((((eq_sym Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0))) (fun (x01:(P0 (f0 x)))=> x01))) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 238.82/239.27 Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% 238.82/239.27 Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 238.82/239.27 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 238.82/239.27 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 238.82/239.27 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 238.82/239.27 Found ((eq_sym0100 ((eq_ref Prop) (f0 x))) x0) as proof of (P2 (f0 x))
% 238.82/239.27 Found ((eq_sym0100 ((eq_ref Prop) (f0 x))) x0) as proof of (P2 (f0 x))
% 238.82/239.27 Found (((fun (x00:(((eq Prop) (f0 x)) b))=> ((eq_sym010 x00) P2)) ((eq_ref Prop) (f0 x))) x0) as proof of (P2 (f0 x))
% 238.82/239.27 Found (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((eq_sym01 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0) as proof of (P2 (f0 x))
% 238.82/239.27 Found (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0) as proof of (P2 (f0 x))
% 238.82/239.27 Found (fun (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0)) as proof of (P2 (f0 x))
% 238.82/239.27 Found (fun (P2:(Prop->Prop)) (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0)) as proof of ((P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))->(P2 (f0 x)))
% 238.92/239.29 Found (fun (P2:(Prop->Prop)) (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0)) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x))
% 238.92/239.29 Found (eq_sym0000 (fun (P2:(Prop->Prop)) (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0))) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 238.92/239.29 Found (eq_sym0000 (fun (P2:(Prop->Prop)) (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0))) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 238.92/239.29 Found ((fun (x0:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)))=> ((eq_sym000 x0) P0)) (fun (P2:(Prop->Prop)) (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0))) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 238.92/239.29 Found ((fun (x0:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)))=> (((eq_sym00 (f0 x)) x0) P0)) (fun (P2:(Prop->Prop)) (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0))) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 238.92/239.29 Found ((fun (x0:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)))=> ((((eq_sym0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) x0) P0)) (fun (P2:(Prop->Prop)) (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0))) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 238.92/239.29 Found ((fun (x0:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)))=> (((((eq_sym Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) x0) P0)) (fun (P2:(Prop->Prop)) (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((((eq_sym Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0))) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 238.97/239.33 Found (fun (P0:(Prop->Prop))=> ((fun (x0:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)))=> (((((eq_sym Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) x0) P0)) (fun (P2:(Prop->Prop)) (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((((eq_sym Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0)))) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 238.97/239.33 Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% 238.97/239.33 Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 238.97/239.33 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 238.97/239.33 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 238.97/239.33 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 238.97/239.33 Found ((eq_sym0100 ((eq_ref Prop) (f0 x))) x0) as proof of (P2 (f0 x))
% 238.97/239.33 Found ((eq_sym0100 ((eq_ref Prop) (f0 x))) x0) as proof of (P2 (f0 x))
% 238.97/239.33 Found (((fun (x00:(((eq Prop) (f0 x)) b))=> ((eq_sym010 x00) P2)) ((eq_ref Prop) (f0 x))) x0) as proof of (P2 (f0 x))
% 238.97/239.33 Found (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((eq_sym01 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0) as proof of (P2 (f0 x))
% 238.97/239.33 Found (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0) as proof of (P2 (f0 x))
% 238.97/239.35 Found (fun (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0)) as proof of (P2 (f0 x))
% 238.97/239.35 Found (fun (P2:(Prop->Prop)) (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0)) as proof of ((P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))->(P2 (f0 x)))
% 238.97/239.35 Found (fun (P2:(Prop->Prop)) (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0)) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x))
% 238.97/239.35 Found ((eq_sym0000 (fun (P2:(Prop->Prop)) (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0))) (fun (x01:(P0 (f0 x)))=> x01)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 238.97/239.35 Found ((eq_sym0000 (fun (P2:(Prop->Prop)) (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0))) (fun (x01:(P0 (f0 x)))=> x01)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 238.97/239.35 Found (((fun (x0:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)))=> ((eq_sym000 x0) (fun (x1:Prop)=> ((P0 (f0 x))->(P0 x1))))) (fun (P2:(Prop->Prop)) (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0))) (fun (x01:(P0 (f0 x)))=> x01)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 238.97/239.35 Found (((fun (x0:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)))=> (((eq_sym00 (f0 x)) x0) (fun (x1:Prop)=> ((P0 (f0 x))->(P0 x1))))) (fun (P2:(Prop->Prop)) (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0))) (fun (x01:(P0 (f0 x)))=> x01)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 238.97/239.35 Found (((fun (x0:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)))=> ((((eq_sym0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) x0) (fun (x1:Prop)=> ((P0 (f0 x))->(P0 x1))))) (fun (P2:(Prop->Prop)) (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0))) (fun (x01:(P0 (f0 x)))=> x01)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 238.97/239.36 Found (((fun (x0:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)))=> (((((eq_sym Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) x0) (fun (x1:Prop)=> ((P0 (f0 x))->(P0 x1))))) (fun (P2:(Prop->Prop)) (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((((eq_sym Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0))) (fun (x01:(P0 (f0 x)))=> x01)) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 238.97/239.36 Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)))=> (((((eq_sym Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) x0) (fun (x1:Prop)=> ((P0 (f0 x))->(P0 x1))))) (fun (P2:(Prop->Prop)) (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((((eq_sym Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0))) (fun (x01:(P0 (f0 x)))=> x01))) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 239.06/239.45 Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% 239.06/239.45 Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 239.06/239.45 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 239.06/239.45 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 239.06/239.45 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) b)
% 239.06/239.45 Found ((eq_sym0100 ((eq_ref Prop) (f0 x))) x0) as proof of (P2 (f0 x))
% 239.06/239.45 Found ((eq_sym0100 ((eq_ref Prop) (f0 x))) x0) as proof of (P2 (f0 x))
% 239.06/239.45 Found (((fun (x00:(((eq Prop) (f0 x)) b))=> ((eq_sym010 x00) P2)) ((eq_ref Prop) (f0 x))) x0) as proof of (P2 (f0 x))
% 239.06/239.45 Found (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((eq_sym01 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0) as proof of (P2 (f0 x))
% 239.06/239.45 Found (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0) as proof of (P2 (f0 x))
% 239.06/239.45 Found (fun (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0)) as proof of (P2 (f0 x))
% 239.06/239.45 Found (fun (P2:(Prop->Prop)) (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0)) as proof of ((P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))->(P2 (f0 x)))
% 239.06/239.45 Found (fun (P2:(Prop->Prop)) (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0)) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x))
% 239.06/239.47 Found (eq_sym0000 (fun (P2:(Prop->Prop)) (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0))) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 239.06/239.47 Found (eq_sym0000 (fun (P2:(Prop->Prop)) (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0))) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 239.06/239.47 Found ((fun (x0:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)))=> ((eq_sym000 x0) P0)) (fun (P2:(Prop->Prop)) (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0))) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 239.06/239.47 Found ((fun (x0:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)))=> (((eq_sym00 (f0 x)) x0) P0)) (fun (P2:(Prop->Prop)) (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0))) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 239.06/239.48 Found ((fun (x0:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)))=> ((((eq_sym0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) x0) P0)) (fun (P2:(Prop->Prop)) (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> ((((eq_sym0 (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0))) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 239.06/239.48 Found ((fun (x0:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)))=> (((((eq_sym Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) x0) P0)) (fun (P2:(Prop->Prop)) (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((((eq_sym Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0))) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 239.06/239.48 Found (fun (P0:(Prop->Prop))=> ((fun (x0:(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)))=> (((((eq_sym Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) (f0 x)) x0) P0)) (fun (P2:(Prop->Prop)) (x0:(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((fun (x00:(((eq Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))=> (((((eq_sym Prop) (f0 x)) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) x00) P2)) ((eq_ref Prop) (f0 x))) x0)))) as proof of ((P0 (f0 x))->(P0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 241.23/241.63 Found eq_ref000:=(eq_ref00 P0):((P0 Xx)->(P0 Xx))
% 241.23/241.63 Found (eq_ref00 P0) as proof of (P1 Xx)
% 241.23/241.63 Found ((eq_ref0 Xx) P0) as proof of (P1 Xx)
% 241.23/241.63 Found (((eq_ref fofType) Xx) P0) as proof of (P1 Xx)
% 241.23/241.63 Found (((eq_ref fofType) Xx) P0) as proof of (P1 Xx)
% 241.23/241.63 Found eq_ref000:=(eq_ref00 P0):((P0 Xx)->(P0 Xx))
% 241.23/241.63 Found (eq_ref00 P0) as proof of (P1 Xx)
% 241.23/241.63 Found ((eq_ref0 Xx) P0) as proof of (P1 Xx)
% 241.23/241.63 Found (((eq_ref fofType) Xx) P0) as proof of (P1 Xx)
% 241.23/241.63 Found (((eq_ref fofType) Xx) P0) as proof of (P1 Xx)
% 241.23/241.63 Found eq_ref000:=(eq_ref00 P2):((P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))->(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 241.23/241.63 Found (eq_ref00 P2) as proof of (P3 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 241.23/241.63 Found ((eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) P2) as proof of (P3 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 241.23/241.63 Found (((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) P2) as proof of (P3 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 241.23/241.63 Found (((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) P2) as proof of (P3 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 241.23/241.63 Found eq_ref000:=(eq_ref00 P2):((P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))->(P2 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))))
% 241.71/242.08 Found (eq_ref00 P2) as proof of (P3 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 241.71/242.08 Found ((eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) P2) as proof of (P3 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 241.71/242.08 Found (((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) P2) as proof of (P3 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 241.71/242.08 Found (((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) P2) as proof of (P3 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 241.71/242.08 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 241.71/242.08 Found (eq_ref0 b) as proof of (((eq Prop) b) (f0 x))
% 241.71/242.08 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 241.71/242.08 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 241.71/242.08 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 241.71/242.08 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))):(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 241.71/242.08 Found (eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 241.71/242.08 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 241.71/242.08 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 242.26/242.68 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 242.26/242.68 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 242.26/242.68 Found (eq_ref0 b) as proof of (((eq Prop) b) (f0 x))
% 242.26/242.68 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 242.26/242.68 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 242.26/242.68 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 242.26/242.68 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))):(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 242.26/242.68 Found (eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 242.26/242.68 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 242.26/242.68 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 242.26/242.68 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 242.26/242.68 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 242.26/242.68 Found (eq_ref0 b) as proof of (((eq Prop) b) (f0 x))
% 242.26/242.68 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 242.26/242.68 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 242.26/242.68 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 242.26/242.68 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))):(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 242.56/243.00 Found (eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 242.56/243.00 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 242.56/243.00 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 242.56/243.00 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 242.56/243.00 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 242.56/243.00 Found (eq_ref0 b) as proof of (((eq Prop) b) (f0 x))
% 242.56/243.00 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 242.56/243.00 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 242.56/243.00 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f0 x))
% 242.56/243.00 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))):(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 242.56/243.00 Found (eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 242.56/243.00 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 242.56/243.00 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 252.18/252.62 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b)
% 252.18/252.62 Found eq_ref000:=(eq_ref00 P0):((P0 (f Xx0))->(P0 (f Xx0)))
% 252.18/252.62 Found (eq_ref00 P0) as proof of (P1 (f Xx0))
% 252.18/252.62 Found ((eq_ref0 (f Xx0)) P0) as proof of (P1 (f Xx0))
% 252.18/252.62 Found (((eq_ref fofType) (f Xx0)) P0) as proof of (P1 (f Xx0))
% 252.18/252.62 Found (((eq_ref fofType) (f Xx0)) P0) as proof of (P1 (f Xx0))
% 252.18/252.62 Found eq_ref000:=(eq_ref00 P0):((P0 Xx)->(P0 Xx))
% 252.18/252.62 Found (eq_ref00 P0) as proof of (P1 Xx)
% 252.18/252.62 Found ((eq_ref0 Xx) P0) as proof of (P1 Xx)
% 252.18/252.62 Found (((eq_ref fofType) Xx) P0) as proof of (P1 Xx)
% 252.18/252.62 Found (((eq_ref fofType) Xx) P0) as proof of (P1 Xx)
% 252.18/252.62 Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% 252.18/252.62 Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% 252.18/252.62 Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% 252.18/252.62 Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% 252.18/252.62 Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% 252.18/252.62 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))):(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 252.18/252.62 Found (eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b0)
% 252.18/252.62 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b0)
% 252.18/252.62 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b0)
% 252.18/252.62 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b0)
% 252.18/252.62 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))):(((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz))))))
% 260.98/261.37 Found (eq_ref0 ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b0)
% 260.98/261.37 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b0)
% 260.98/261.37 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b0)
% 260.98/261.37 Found ((eq_ref Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) as proof of (((eq Prop) ((and (forall (Xx:fofType), ((x (f Xx))->(cP Xx)))) (((and (cP a)) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (f Xx)) (f Xy))->(((eq fofType) Xx) Xy))))->((ex fofType) (fun (Xz:fofType)=> (x Xz)))))) b0)
% 260.98/261.37 Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% 260.98/261.37 Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% 260.98/261.37 Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% 260.98/261.37 Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% 260.98/261.37 Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% 260.98/261.37 Found eq_ref000:=(eq_ref00 P0):((P0 Xx)->(P0 Xx))
% 260.98/261.37 Found (eq_ref00 P0) as proof of (P1 Xx)
% 260.98/261.37 Found ((eq_ref0 Xx) P0) as proof of (P1 Xx)
% 260.98/261.37 Found (((eq_ref fofType) Xx) P0) as proof of (P1 Xx)
% 260.98/261.37 Found (((eq_ref fofType) Xx) P0) as proof of (P1 Xx)
% 260.98/261.37 Found eq_ref000:=(eq_ref00 P0):((P0 (f Xx0))->(P0 (f Xx0)))
% 260.98/261.37 Found (eq_ref00 P0) as proof of (P1 (f Xx0))
% 260.98/261.37 Found ((eq_ref0 (f Xx0)) P0) as proof of (P1 (f Xx0))
% 260.98/261.37 Found (((eq_ref fofType) (f Xx0)) P0) as proof of (P1 (f Xx0))
% 260.98/261.37 Found (((eq_ref fofType) (f Xx0)) P0) as proof of (P1 (f Xx0))
% 260.98/261.37 Found eq_ref000:=(eq_ref00 P0):((P0 (f Xx0))->(P0 (f Xx0)))
% 260.98/261.37 Found (eq_ref00 P0) as proof of (P1 (f Xx0))
% 260.98/261.37 Found ((eq_ref0 (f Xx0)) P0) as proof of (P1 (f Xx0))
% 260.98/261.37 Found (((eq_ref fofType) (f Xx0)) P0) as proof of (P1 (f Xx0))
% 260.98/261.37 Found (((eq_ref fofType) (f Xx0)) P0) as proof of (P1 (f Xx0))
% 260.98/261.37 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 260.98/261.37 Found (eq_ref0 b) as proof of (((eq fofType) b) (f Xx))
% 260.98/261.37 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (f Xx))
% 260.98/261.37 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (f Xx))
% 260.98/261.37 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (f Xx))
% 260.98/261.37 Found eq_ref00:=(eq_ref0 (f Xx0)):(((eq fofType) (f Xx0)) (f Xx0))
% 260.98/261.37 Found (eq_ref0 (f Xx0)) as proof of (((eq fofType) (f Xx0)) b)
% 260.98/261.37 Found ((eq_ref fofType) (f Xx0)) as proof of (((eq fofType) (f Xx0)) b)
% 260.98/261.37 Found ((eq_ref fofType) (f Xx0)) as proof of (((eq fofType) (f Xx0)) b)
% 260.98/261.37 Found ((eq_ref fofType) (f Xx0)) as proof of (((eq fofType) (f Xx0)) b)
% 272.99/273.38 Found eta_expansion000:=(eta_expansion00 (fun (Xz:fofType)=> (x Xz))):(((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) (fun (x0:fofType)=> (x x0)))
% 272.99/273.38 Found (eta_expansion00 (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b0)
% 272.99/273.38 Found ((eta_expansion0 Prop) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b0)
% 272.99/273.38 Found (((eta_expansion fofType) Prop) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b0)
% 272.99/273.38 Found (((eta_expansion fofType) Prop) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b0)
% 272.99/273.38 Found (((eta_expansion fofType) Prop) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b0)
% 272.99/273.38 Found eq_ref00:=(eq_ref0 (f Xx0)):(((eq fofType) (f Xx0)) (f Xx0))
% 272.99/273.38 Found (eq_ref0 (f Xx0)) as proof of (P0 Xx0)
% 272.99/273.38 Found ((eq_ref fofType) (f Xx0)) as proof of (P0 Xx0)
% 272.99/273.38 Found ((eq_ref fofType) (f Xx0)) as proof of (P0 Xx0)
% 272.99/273.38 Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% 272.99/273.38 Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% 272.99/273.38 Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% 272.99/273.38 Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% 272.99/273.38 Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% 272.99/273.38 Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% 272.99/273.38 Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% 272.99/273.38 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 272.99/273.38 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 272.99/273.38 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 272.99/273.38 Found eq_ref000:=(eq_ref00 P0):((P0 (f Xx0))->(P0 (f Xx0)))
% 272.99/273.38 Found (eq_ref00 P0) as proof of (P1 (f Xx0))
% 272.99/273.38 Found ((eq_ref0 (f Xx0)) P0) as proof of (P1 (f Xx0))
% 272.99/273.38 Found (((eq_ref fofType) (f Xx0)) P0) as proof of (P1 (f Xx0))
% 272.99/273.38 Found (((eq_ref fofType) (f Xx0)) P0) as proof of (P1 (f Xx0))
% 272.99/273.38 Found eq_ref000:=(eq_ref00 P0):((P0 (f Xx0))->(P0 (f Xx0)))
% 272.99/273.38 Found (eq_ref00 P0) as proof of (P1 Xx0)
% 272.99/273.38 Found ((eq_ref0 (f Xx0)) P0) as proof of (P1 Xx0)
% 272.99/273.38 Found (((eq_ref fofType) (f Xx0)) P0) as proof of (P1 Xx0)
% 272.99/273.38 Found (((eq_ref fofType) (f Xx0)) P0) as proof of (P1 Xx0)
% 272.99/273.38 Found eq_ref000:=(eq_ref00 P0):((P0 (f Xx0))->(P0 (f Xx0)))
% 272.99/273.38 Found (eq_ref00 P0) as proof of (P1 (f Xx0))
% 272.99/273.38 Found ((eq_ref0 (f Xx0)) P0) as proof of (P1 (f Xx0))
% 272.99/273.38 Found (((eq_ref fofType) (f Xx0)) P0) as proof of (P1 (f Xx0))
% 272.99/273.38 Found (((eq_ref fofType) (f Xx0)) P0) as proof of (P1 (f Xx0))
% 272.99/273.38 Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:fofType)=> (x Xz))):(((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) (fun (x0:fofType)=> (x x0)))
% 272.99/273.38 Found (eta_expansion_dep00 (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b0)
% 272.99/273.38 Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b0)
% 272.99/273.38 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b0)
% 272.99/273.38 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b0)
% 272.99/273.38 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (Xz:fofType)=> (x Xz))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> (x Xz))) b0)
% 272.99/273.38 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 272.99/273.38 Found (eq_ref0 b) as proof of (((eq fofType) b) (f Xx))
% 272.99/273.38 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (f Xx))
% 272.99/273.38 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (f Xx))
% 272.99/273.38 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (f Xx))
% 272.99/273.38 Found eq_ref00:=(eq_ref0 (f Xx0)):(((eq fofType) (f Xx0)) (f Xx0))
% 272.99/273.38 Found (eq_ref0 (f Xx0)) as proof of (((eq fofType) (f Xx0)) b)
% 272.99/273.38 Found ((eq_ref fofType) (f Xx0)) as proof of (((eq fofType) (f Xx0)) b)
% 272.99/273.38 Found ((eq_ref fofType) (f Xx0)) as proof of (((eq fofType) (f Xx0)) b)
% 272.99/273.38 Found ((eq_ref fofType) (f Xx0)) as proof of (((eq fofType) (f Xx0)) b)
% 272.99/273.38 Found eq_ref00:=(eq_ref0 (f Xx0)):(((eq fofType) (f Xx0)) (f Xx0))
% 281.83/282.25 Found (eq_ref0 (f Xx0)) as proof of (((eq fofType) (f Xx0)) b)
% 281.83/282.25 Found ((eq_ref fofType) (f Xx0)) as proof of (((eq fofType) (f Xx0)) b)
% 281.83/282.25 Found ((eq_ref fofType) (f Xx0)) as proof of (((eq fofType) (f Xx0)) b)
% 281.83/282.25 Found ((eq_ref fofType) (f Xx0)) as proof of (((eq fofType) (f Xx0)) b)
% 281.83/282.25 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 281.83/282.25 Found (eq_ref0 b) as proof of (((eq fofType) b) (f Xx))
% 281.83/282.25 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (f Xx))
% 281.83/282.25 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (f Xx))
% 281.83/282.25 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (f Xx))
% 281.83/282.25 Found eq_ref00:=(eq_ref0 (f Xx0)):(((eq fofType) (f Xx0)) (f Xx0))
% 281.83/282.25 Found (eq_ref0 (f Xx0)) as proof of (P0 Xx0)
% 281.83/282.25 Found ((eq_ref fofType) (f Xx0)) as proof of (P0 Xx0)
% 281.83/282.25 Found ((eq_ref fofType) (f Xx0)) as proof of (P0 Xx0)
% 281.83/282.25 Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% 281.83/282.25 Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% 281.83/282.25 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 281.83/282.25 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 281.83/282.25 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 281.83/282.25 Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% 281.83/282.25 Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% 281.83/282.25 Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% 281.83/282.25 Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% 281.83/282.25 Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% 281.83/282.25 Found eq_ref000:=(eq_ref00 P0):((P0 (f Xx0))->(P0 (f Xx0)))
% 281.83/282.25 Found (eq_ref00 P0) as proof of (P1 (f Xx0))
% 281.83/282.25 Found ((eq_ref0 (f Xx0)) P0) as proof of (P1 (f Xx0))
% 281.83/282.25 Found (((eq_ref fofType) (f Xx0)) P0) as proof of (P1 (f Xx0))
% 281.83/282.25 Found (((eq_ref fofType) (f Xx0)) P0) as proof of (P1 (f Xx0))
% 281.83/282.25 Found eq_ref000:=(eq_ref00 P0):((P0 (f Xx0))->(P0 (f Xx0)))
% 281.83/282.25 Found (eq_ref00 P0) as proof of (P1 Xx0)
% 281.83/282.25 Found ((eq_ref0 (f Xx0)) P0) as proof of (P1 Xx0)
% 281.83/282.25 Found (((eq_ref fofType) (f Xx0)) P0) as proof of (P1 Xx0)
% 281.83/282.25 Found (((eq_ref fofType) (f Xx0)) P0) as proof of (P1 Xx0)
% 281.83/282.25 Found eq_ref000:=(eq_ref00 P0):((P0 (f Xx0))->(P0 (f Xx0)))
% 281.83/282.25 Found (eq_ref00 P0) as proof of (P1 (f Xx0))
% 281.83/282.25 Found ((eq_ref0 (f Xx0)) P0) as proof of (P1 (f Xx0))
% 281.83/282.25 Found (((eq_ref fofType) (f Xx0)) P0) as proof of (P1 (f Xx0))
% 281.83/282.25 Found (((eq_ref fofType) (f Xx0)) P0) as proof of (P1 (f Xx0))
% 281.83/282.25 Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% 281.83/282.25 Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% 281.83/282.25 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 281.83/282.25 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 281.83/282.25 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 281.83/282.25 Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% 281.83/282.25 Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% 281.83/282.25 Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% 281.83/282.25 Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% 281.83/282.25 Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% 281.83/282.25 Found eq_ref00:=(eq_ref0 (f Xx0)):(((eq fofType) (f Xx0)) (f Xx0))
% 281.83/282.25 Found (eq_ref0 (f Xx0)) as proof of (P0 Xx0)
% 281.83/282.25 Found ((eq_ref fofType) (f Xx0)) as proof of (P0 Xx0)
% 281.83/282.25 Found ((eq_ref fofType) (f Xx0)) as proof of (P0 Xx0)
% 281.83/282.25 Found eq_ref000:=(eq_ref00 P0):((P0 b)->(P0 b))
% 281.83/282.25 Found (eq_ref00 P0) as proof of (P1 b)
% 281.83/282.25 Found ((eq_ref0 b) P0) as proof of (P1 b)
% 281.83/282.25 Found (((eq_ref fofType) b) P0) as proof of (P1 b)
% 281.83/282.25 Found (((eq_ref fofType) b) P0) as proof of (P1 b)
% 281.83/282.25 Found eq_ref000:=(eq_ref00 P0):((P0 (f Xx))->(P0 (f Xx)))
% 281.83/282.25 Found (eq_ref00 P0) as proof of (P1 (f Xx))
% 281.83/282.25 Found ((eq_ref0 (f Xx)) P0) as proof of (P1 (f Xx))
% 281.83/282.25 Found (((eq_ref fofType) (f Xx)) P0) as proof of (P1 (f Xx))
% 281.83/282.25 Found (((eq_ref fofType) (f Xx)) P0) as proof of (P1 (f Xx))
% 281.83/282.25 Found eq_ref000:=(eq_ref00 P0):((P0 (f Xx0))->(P0 (f Xx0)))
% 281.83/282.25 Found (eq_ref00 P0) as proof of (P1 Xx0)
% 281.83/282.25 Found ((eq_ref0 (f Xx0)) P0) as proof of (P1 Xx0)
% 281.83/282.25 Found (((eq_ref fofType) (f Xx0)) P0) as proof of (P1 Xx0)
% 281.83/282.25 Found (((eq_ref fofType) (f Xx0)) P0) as proof of (P1 Xx0)
% 281.83/282.25 Found eq_ref000:=(eq_ref00 P0):((P0 (f Xx0))->(P0 (f Xx0)))
% 281.83/282.25 Found (eq_ref00 P0) as proof of (P1 (f Xx0))
% 281.83/282.25 Found ((eq_ref0 (f Xx0)) P0) as proof of (P1 (f Xx0))
% 281.83/282.25 Found (((eq_ref fofType) (f Xx0)) P0) as proof of (P1 (f Xx0))
% 281.83/282.25 Found (((eq_ref fofType) (f
%------------------------------------------------------------------------------