%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV249^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n116.star.cs.uiowa.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory   : 32286.75MB
% OS       : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:33:57 EDT 2014

% Result   : Timeout 300.09s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV249^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n116.star.cs.uiowa.edu
% % Model    : x86_64 x86_64
% % CPU      : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory   : 32286.75MB
% % OS       : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 08:37:21 CDT 2014
% % CPUTime  : 300.09 
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x1222c68>, <kernel.DependentProduct object at 0x1222518>) of role type named cX
% Using role type
% Declaring cX:(fofType->Prop)
% FOF formula ((forall (Xw:((fofType->Prop)->Prop)), (((and (Xw (fun (Xx:fofType)=> False))) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw Xr)->(Xw (fun (Xt:fofType)=> ((or (Xr Xt)) (((eq fofType) Xt) Xx)))))))->(Xw cX)))->(forall (Xw:(((fofType->Prop)->Prop)->Prop)), (((and (Xw (fun (Xx:(fofType->Prop))=> False))) (forall (Xr:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx)))))))) of role conjecture named cTHM626_pme
% Conjecture to prove = ((forall (Xw:((fofType->Prop)->Prop)), (((and (Xw (fun (Xx:fofType)=> False))) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw Xr)->(Xw (fun (Xt:fofType)=> ((or (Xr Xt)) (((eq fofType) Xt) Xx)))))))->(Xw cX)))->(forall (Xw:(((fofType->Prop)->Prop)->Prop)), (((and (Xw (fun (Xx:(fofType->Prop))=> False))) (forall (Xr:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx)))))))):Prop
% Parameter fofType_DUMMY:fofType.
% We need to prove ['((forall (Xw:((fofType->Prop)->Prop)), (((and (Xw (fun (Xx:fofType)=> False))) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw Xr)->(Xw (fun (Xt:fofType)=> ((or (Xr Xt)) (((eq fofType) Xt) Xx)))))))->(Xw cX)))->(forall (Xw:(((fofType->Prop)->Prop)->Prop)), (((and (Xw (fun (Xx:(fofType->Prop))=> False))) (forall (Xr:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx))))))))']
% Parameter fofType:Type.
% Parameter cX:(fofType->Prop).
% Trying to prove ((forall (Xw:((fofType->Prop)->Prop)), (((and (Xw (fun (Xx:fofType)=> False))) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw Xr)->(Xw (fun (Xt:fofType)=> ((or (Xr Xt)) (((eq fofType) Xt) Xx)))))))->(Xw cX)))->(forall (Xw:(((fofType->Prop)->Prop)->Prop)), (((and (Xw (fun (Xx:(fofType->Prop))=> False))) (forall (Xr:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx))))))))
% Found eta_expansion0000:=(eta_expansion000 Xw):((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (x:(fofType->Prop))=> (forall (Xx:fofType), ((x Xx)->(Xr Xx))))))
% Found (eta_expansion000 Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found ((eta_expansion00 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found (((eta_expansion0 Prop) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found ((((eta_expansion (fofType->Prop)) Prop) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found ((((eta_expansion (fofType->Prop)) Prop) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx))))):(((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx))))) (fun (x:(fofType->Prop))=> (forall (Xx:fofType), ((x Xx)->(cX Xx)))))
% Found (eta_expansion_dep00 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx))))) b)
% Found ((eta_expansion_dep0 (fun (x2:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx))))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x2:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx))))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x2:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx))))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x2:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx))))) b)
% Found eta_expansion0000:=(eta_expansion000 Xw):((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (x:(fofType->Prop))=> (forall (Xx:fofType), ((x Xx)->(Xr Xx))))))
% Found (eta_expansion000 Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found ((eta_expansion00 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found (((eta_expansion0 Prop) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found ((((eta_expansion (fofType->Prop)) Prop) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found ((((eta_expansion (fofType->Prop)) Prop) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found eta_expansion0000:=(eta_expansion000 Xw):((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (x:(fofType->Prop))=> (forall (Xx:fofType), ((x Xx)->(Xr Xx))))))
% Found (eta_expansion000 Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found ((eta_expansion00 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found (((eta_expansion0 Prop) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found ((((eta_expansion (fofType->Prop)) Prop) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found ((((eta_expansion (fofType->Prop)) Prop) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found eq_ref00:=(eq_ref0 (forall (Xr:(fofType->Prop)) (Xx:fofType), ((forall (Xw0:(((fofType->Prop)->Prop)->Prop)), (((and (Xw0 (fun (Xx:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw0 Xr0)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))))->(forall (Xw0:(((fofType->Prop)->Prop)->Prop)), (((and (Xw0 (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr0)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))))):(((eq Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((forall (Xw0:(((fofType->Prop)->Prop)->Prop)), (((and (Xw0 (fun (Xx:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw0 Xr0)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))))->(forall (Xw0:(((fofType->Prop)->Prop)->Prop)), (((and (Xw0 (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr0)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))))) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((forall (Xw0:(((fofType->Prop)->Prop)->Prop)), (((and (Xw0 (fun (Xx:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw0 Xr0)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))))->(forall (Xw0:(((fofType->Prop)->Prop)->Prop)), (((and (Xw0 (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr0)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))))
% Found (eq_ref0 (forall (Xr:(fofType->Prop)) (Xx:fofType), ((forall (Xw0:(((fofType->Prop)->Prop)->Prop)), (((and (Xw0 (fun (Xx:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw0 Xr0)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))))->(forall (Xw0:(((fofType->Prop)->Prop)->Prop)), (((and (Xw0 (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr0)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))))) as proof of (((eq Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((forall (Xw0:(((fofType->Prop)->Prop)->Prop)), (((and (Xw0 (fun (Xx:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw0 Xr0)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))))->(forall (Xw0:(((fofType->Prop)->Prop)->Prop)), (((and (Xw0 (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr0)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((forall (Xw0:(((fofType->Prop)->Prop)->Prop)), (((and (Xw0 (fun (Xx:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw0 Xr0)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))))->(forall (Xw0:(((fofType->Prop)->Prop)->Prop)), (((and (Xw0 (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr0)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))))) as proof of (((eq Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((forall (Xw0:(((fofType->Prop)->Prop)->Prop)), (((and (Xw0 (fun (Xx:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw0 Xr0)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))))->(forall (Xw0:(((fofType->Prop)->Prop)->Prop)), (((and (Xw0 (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr0)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((forall (Xw0:(((fofType->Prop)->Prop)->Prop)), (((and (Xw0 (fun (Xx:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw0 Xr0)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))))->(forall (Xw0:(((fofType->Prop)->Prop)->Prop)), (((and (Xw0 (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr0)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))))) as proof of (((eq Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((forall (Xw0:(((fofType->Prop)->Prop)->Prop)), (((and (Xw0 (fun (Xx:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw0 Xr0)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))))->(forall (Xw0:(((fofType->Prop)->Prop)->Prop)), (((and (Xw0 (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr0)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((forall (Xw0:(((fofType->Prop)->Prop)->Prop)), (((and (Xw0 (fun (Xx:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw0 Xr0)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))))->(forall (Xw0:(((fofType->Prop)->Prop)->Prop)), (((and (Xw0 (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr0)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))))) as proof of (((eq Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((forall (Xw0:(((fofType->Prop)->Prop)->Prop)), (((and (Xw0 (fun (Xx:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw0 Xr0)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))))->(forall (Xw0:(((fofType->Prop)->Prop)->Prop)), (((and (Xw0 (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr0)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xr:(fofType->Prop)) (Xx:fofType), ((((and (Xw (fun (Xx:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))))->(((and (Xw (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))):(((eq Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((((and (Xw (fun (Xx:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))))->(((and (Xw (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((((and (Xw (fun (Xx:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))))->(((and (Xw (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))))
% Found (eq_ref0 (forall (Xr:(fofType->Prop)) (Xx:fofType), ((((and (Xw (fun (Xx:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))))->(((and (Xw (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))) as proof of (((eq Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((((and (Xw (fun (Xx:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))))->(((and (Xw (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((((and (Xw (fun (Xx:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))))->(((and (Xw (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))) as proof of (((eq Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((((and (Xw (fun (Xx:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))))->(((and (Xw (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((((and (Xw (fun (Xx:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))))->(((and (Xw (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))) as proof of (((eq Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((((and (Xw (fun (Xx:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))))->(((and (Xw (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((((and (Xw (fun (Xx:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))))->(((and (Xw (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))) as proof of (((eq Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((((and (Xw (fun (Xx:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))))->(((and (Xw (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))):(((eq Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))
% Found (eq_ref0 (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))) as proof of (((eq Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))) as proof of (((eq Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))) as proof of (((eq Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))) as proof of (((eq Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))) b)
% Found eq_ref000:=(eq_ref00 Xw):((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))))
% Found (eq_ref00 Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found ((eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found (((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found (((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found ex_ind:(forall (A:Type) (F:(A->Prop)) (P:Prop), ((forall (x:A), ((F x)->P))->(((ex A) F)->P)))
% Instantiate: b:=(forall (A:Type) (F:(A->Prop)) (P:Prop), ((forall (x:A), ((F x)->P))->(((ex A) F)->P))):Prop
% Found ex_ind as proof of b
% Found eta_expansion000:=(eta_expansion00 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx))))):(((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx))))) (fun (x:(fofType->Prop))=> (forall (Xx:fofType), ((x Xx)->(cX Xx)))))
% Found (eta_expansion00 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx))))) b)
% Found ((eta_expansion0 Prop) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx))))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx))))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx))))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx))))) b)
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b
% Found eq_ref000:=(eq_ref00 Xw):((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))))
% Found (eq_ref00 Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found ((eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found (((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found (((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found eq_ref00:=(eq_ref0 (forall (Xr:(fofType->Prop)) (Xx:fofType), (((Xw (fun (Xx:(fofType->Prop))=> False))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))))->((Xw (fun (Xx0:(fofType->Prop))=> False))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))))):(((eq Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), (((Xw (fun (Xx:(fofType->Prop))=> False))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))))->((Xw (fun (Xx0:(fofType->Prop))=> False))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))))) (forall (Xr:(fofType->Prop)) (Xx:fofType), (((Xw (fun (Xx:(fofType->Prop))=> False))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))))->((Xw (fun (Xx0:(fofType->Prop))=> False))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))))
% Found (eq_ref0 (forall (Xr:(fofType->Prop)) (Xx:fofType), (((Xw (fun (Xx:(fofType->Prop))=> False))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))))->((Xw (fun (Xx0:(fofType->Prop))=> False))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))))) as proof of (((eq Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), (((Xw (fun (Xx:(fofType->Prop))=> False))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))))->((Xw (fun (Xx0:(fofType->Prop))=> False))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), (((Xw (fun (Xx:(fofType->Prop))=> False))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))))->((Xw (fun (Xx0:(fofType->Prop))=> False))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))))) as proof of (((eq Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), (((Xw (fun (Xx:(fofType->Prop))=> False))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))))->((Xw (fun (Xx0:(fofType->Prop))=> False))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), (((Xw (fun (Xx:(fofType->Prop))=> False))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))))->((Xw (fun (Xx0:(fofType->Prop))=> False))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))))) as proof of (((eq Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), (((Xw (fun (Xx:(fofType->Prop))=> False))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))))->((Xw (fun (Xx0:(fofType->Prop))=> False))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), (((Xw (fun (Xx:(fofType->Prop))=> False))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))))->((Xw (fun (Xx0:(fofType->Prop))=> False))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))))) as proof of (((eq Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), (((Xw (fun (Xx:(fofType->Prop))=> False))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))))->((Xw (fun (Xx0:(fofType->Prop))=> False))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))))) b)
% Found eq_sym0:=(eq_sym Prop):(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a)))
% Instantiate: b:=(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))):Prop
% Found eq_sym0 as proof of b
% Found eq_ref00:=(eq_ref0 (forall (Xr:(fofType->Prop)) (Xx:fofType), (((forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))):(((eq Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), (((forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))) (forall (Xr:(fofType->Prop)) (Xx:fofType), (((forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))))
% Found (eq_ref0 (forall (Xr:(fofType->Prop)) (Xx:fofType), (((forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))) as proof of (((eq Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), (((forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), (((forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))) as proof of (((eq Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), (((forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), (((forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))) as proof of (((eq Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), (((forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), (((forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))) as proof of (((eq Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), (((forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))) b)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 Xw):((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (x:(fofType->Prop))=> (forall (Xx:fofType), ((x Xx)->(Xr Xx))))))
% Found (eta_expansion_dep000 Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found ((eta_expansion_dep00 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found (((eta_expansion_dep0 (fun (x5:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found ((((eta_expansion_dep (fofType->Prop)) (fun (x5:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found ((((eta_expansion_dep (fofType->Prop)) (fun (x5:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 Xw):((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (x:(fofType->Prop))=> (forall (Xx:fofType), ((x Xx)->(Xr Xx))))))
% Found (eta_expansion_dep000 Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found ((eta_expansion_dep00 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found (((eta_expansion_dep0 (fun (x5:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found ((((eta_expansion_dep (fofType->Prop)) (fun (x5:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found ((((eta_expansion_dep (fofType->Prop)) (fun (x5:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found eq_ref00:=(eq_ref0 (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))):(((eq Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))
% Found (eq_ref0 (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))) as proof of (((eq Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))) as proof of (((eq Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))) as proof of (((eq Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))) as proof of (((eq Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))) b)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 Xw):((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (x:(fofType->Prop))=> (forall (Xx:fofType), ((x Xx)->(Xr Xx))))))
% Found (eta_expansion_dep000 Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found ((eta_expansion_dep00 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found (((eta_expansion_dep0 (fun (x5:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found ((((eta_expansion_dep (fofType->Prop)) (fun (x5:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found ((((eta_expansion_dep (fofType->Prop)) (fun (x5:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 Xw):((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (x:(fofType->Prop))=> (forall (Xx:fofType), ((x Xx)->(Xr Xx))))))
% Found (eta_expansion_dep000 Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found ((eta_expansion_dep00 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found (((eta_expansion_dep0 (fun (x5:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found ((((eta_expansion_dep (fofType->Prop)) (fun (x5:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found ((((eta_expansion_dep (fofType->Prop)) (fun (x5:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found eq_ref00:=(eq_ref0 (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))):(((eq Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))
% Found (eq_ref0 (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))) as proof of (((eq Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))) as proof of (((eq Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))) as proof of (((eq Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))) b)
% Found ((eq_ref Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))) as proof of (((eq Prop) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))) b)
% 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)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b
% 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))
% Instantiate: b:=(forall (P:Prop), ((iff P) P)):Prop
% Found iff_refl as proof of b
% Found x1:(Xw (fun (Xx:(fofType->Prop))=> False))
% Instantiate: b:=(Xw (fun (Xx:(fofType->Prop))=> False)):Prop
% Found x1 as proof of b
% Found x1:=(x (fun (x2:(fofType->Prop))=> (Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(x2 Xx))))))):(((and (Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False))))) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx))))))
% Instantiate: b:=(((and (Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False))))) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx)))))):Prop
% Found x1 as proof of b
% Found x1:=(x (fun (x2:(fofType->Prop))=> (Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(x2 Xx))))))):(((and (Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False))))) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx))))))
% Instantiate: b:=(((and (Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False))))) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx)))))):Prop
% Found x1 as proof of b
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))):(((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) (fun (x:(fofType->Prop))=> (forall (Xx:fofType), ((x Xx)->False))))
% Found (eta_expansion_dep00 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found ((eta_expansion_dep0 (fun (x3:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x3:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x3:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x3:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found eq_ref00:=(eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))):(((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False))))
% Found (eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found eq_ref00:=(eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))):(((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False))))
% Found (eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found x2:(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Instantiate: b:=(fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))):((fofType->Prop)->Prop)
% Found x2 as proof of (P b)
% Found eq_ref00:=(eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))):(((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found x2:(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Instantiate: f:=(fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))):((fofType->Prop)->Prop)
% Found x2 as proof of (P f)
% Found x2:(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Instantiate: f:=(fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))):((fofType->Prop)->Prop)
% Found x2 as proof of (P f)
% Found eq_ref00:=(eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))):(((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False))))
% Found (eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found eta_expansion000:=(eta_expansion00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))):(((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) (fun (x:(fofType->Prop))=> (forall (Xx0:fofType), ((x Xx0)->(Xr Xx0)))))
% Found (eta_expansion00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) b)
% Found ((eta_expansion0 Prop) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) b)
% Found eq_ref00:=(eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))):(((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found x10:=(x1 x00):(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Instantiate: b:=(fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))):((fofType->Prop)->Prop)
% Found (x1 x00) as proof of (P b)
% Found (x1 x00) as proof of (P b)
% Found eq_ref000:=(eq_ref00 Xw):((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))))
% Found (eq_ref00 Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found ((eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found (((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found (((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found eta_expansion000:=(eta_expansion00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))):(((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) (fun (x:(fofType->Prop))=> (forall (Xx0:fofType), ((x Xx0)->(Xr Xx0)))))
% Found (eta_expansion00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) b)
% Found ((eta_expansion0 Prop) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) b)
% Found eq_ref00:=(eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))):(((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False))))
% Found (eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))):(((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) (fun (x:(fofType->Prop))=> (forall (Xx:fofType), ((x Xx)->False))))
% Found (eta_expansion00 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found ((eta_expansion0 Prop) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found x00:((and (Xw (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))
% Found x00 as proof of ((and (Xw (fun (Xx:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))):(((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) (fun (x:(fofType->Prop))=> (forall (Xx0:fofType), ((x Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (eta_expansion_dep00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found ((eta_expansion_dep0 (fun (x3:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x3:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x3:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x3:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found eq_ref00:=(eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))):(((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False))))
% Found (eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found x4:(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Instantiate: b:=(fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))):((fofType->Prop)->Prop)
% Found x4 as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))):(((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) (fun (x:(fofType->Prop))=> (forall (Xx0:fofType), ((x Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (eta_expansion_dep00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found ((eta_expansion_dep0 (fun (x6:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x6:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x6:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x6:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found x10:=(x1 x00):(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Instantiate: f:=(fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))):((fofType->Prop)->Prop)
% Found (x1 x00) as proof of (P f)
% Found (x1 x00) as proof of (P f)
% Found x10:=(x1 x00):(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Instantiate: f:=(fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))):((fofType->Prop)->Prop)
% Found (x1 x00) as proof of (P f)
% Found (x1 x00) as proof of (P f)
% Found x4:(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Instantiate: b:=(fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))):((fofType->Prop)->Prop)
% Found x4 as proof of (P b)
% Found eq_ref00:=(eq_ref0 (((fofType->Prop)->Prop)->((fofType->Prop)->((forall (Xw00:(((fofType->Prop)->Prop)->Prop)), (((and (Xw00 (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw00 Xr00)->(Xw00 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))->(forall (Xw00:(((fofType->Prop)->Prop)->Prop)), (((and (Xw00 (fun (Xx00:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw00 Xr00)->(Xw00 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00)))))))->(Xw00 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))))):(((eq Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((forall (Xw00:(((fofType->Prop)->Prop)->Prop)), (((and (Xw00 (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw00 Xr00)->(Xw00 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))->(forall (Xw00:(((fofType->Prop)->Prop)->Prop)), (((and (Xw00 (fun (Xx00:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw00 Xr00)->(Xw00 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00)))))))->(Xw00 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))))) (((fofType->Prop)->Prop)->((fofType->Prop)->((forall (Xw00:(((fofType->Prop)->Prop)->Prop)), (((and (Xw00 (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw00 Xr00)->(Xw00 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))->(forall (Xw00:(((fofType->Prop)->Prop)->Prop)), (((and (Xw00 (fun (Xx00:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw00 Xr00)->(Xw00 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00)))))))->(Xw00 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx))))))))))))
% Found (eq_ref0 (((fofType->Prop)->Prop)->((fofType->Prop)->((forall (Xw00:(((fofType->Prop)->Prop)->Prop)), (((and (Xw00 (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw00 Xr00)->(Xw00 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))->(forall (Xw00:(((fofType->Prop)->Prop)->Prop)), (((and (Xw00 (fun (Xx00:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw00 Xr00)->(Xw00 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00)))))))->(Xw00 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))))) as proof of (((eq Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((forall (Xw00:(((fofType->Prop)->Prop)->Prop)), (((and (Xw00 (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw00 Xr00)->(Xw00 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))->(forall (Xw00:(((fofType->Prop)->Prop)->Prop)), (((and (Xw00 (fun (Xx00:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw00 Xr00)->(Xw00 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00)))))))->(Xw00 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))))) b)
% Found ((eq_ref Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((forall (Xw00:(((fofType->Prop)->Prop)->Prop)), (((and (Xw00 (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw00 Xr00)->(Xw00 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))->(forall (Xw00:(((fofType->Prop)->Prop)->Prop)), (((and (Xw00 (fun (Xx00:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw00 Xr00)->(Xw00 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00)))))))->(Xw00 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))))) as proof of (((eq Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((forall (Xw00:(((fofType->Prop)->Prop)->Prop)), (((and (Xw00 (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw00 Xr00)->(Xw00 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))->(forall (Xw00:(((fofType->Prop)->Prop)->Prop)), (((and (Xw00 (fun (Xx00:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw00 Xr00)->(Xw00 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00)))))))->(Xw00 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))))) b)
% Found ((eq_ref Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((forall (Xw00:(((fofType->Prop)->Prop)->Prop)), (((and (Xw00 (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw00 Xr00)->(Xw00 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))->(forall (Xw00:(((fofType->Prop)->Prop)->Prop)), (((and (Xw00 (fun (Xx00:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw00 Xr00)->(Xw00 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00)))))))->(Xw00 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))))) as proof of (((eq Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((forall (Xw00:(((fofType->Prop)->Prop)->Prop)), (((and (Xw00 (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw00 Xr00)->(Xw00 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))->(forall (Xw00:(((fofType->Prop)->Prop)->Prop)), (((and (Xw00 (fun (Xx00:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw00 Xr00)->(Xw00 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00)))))))->(Xw00 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))))) b)
% Found ((eq_ref Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((forall (Xw00:(((fofType->Prop)->Prop)->Prop)), (((and (Xw00 (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw00 Xr00)->(Xw00 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))->(forall (Xw00:(((fofType->Prop)->Prop)->Prop)), (((and (Xw00 (fun (Xx00:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw00 Xr00)->(Xw00 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00)))))))->(Xw00 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))))) as proof of (((eq Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((forall (Xw00:(((fofType->Prop)->Prop)->Prop)), (((and (Xw00 (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw00 Xr00)->(Xw00 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))->(forall (Xw00:(((fofType->Prop)->Prop)->Prop)), (((and (Xw00 (fun (Xx00:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw00 Xr00)->(Xw00 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00)))))))->(Xw00 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))))) b)
% Found eq_ref00:=(eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))):(((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found x4:(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Instantiate: f:=(fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))):((fofType->Prop)->Prop)
% Found x4 as proof of (P f)
% Found x4:(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Instantiate: f:=(fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))):((fofType->Prop)->Prop)
% Found x4 as proof of (P f)
% Found eq_ref00:=(eq_ref0 (((fofType->Prop)->Prop)->((fofType->Prop)->((((and (Xw0 (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))->(((and (Xw0 (fun (Xx00:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx))))))))))):(((eq Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((((and (Xw0 (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))->(((and (Xw0 (fun (Xx00:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx))))))))))) (((fofType->Prop)->Prop)->((fofType->Prop)->((((and (Xw0 (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))->(((and (Xw0 (fun (Xx00:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))))
% Found (eq_ref0 (((fofType->Prop)->Prop)->((fofType->Prop)->((((and (Xw0 (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))->(((and (Xw0 (fun (Xx00:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx))))))))))) as proof of (((eq Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((((and (Xw0 (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))->(((and (Xw0 (fun (Xx00:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx))))))))))) b)
% Found ((eq_ref Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((((and (Xw0 (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))->(((and (Xw0 (fun (Xx00:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx))))))))))) as proof of (((eq Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((((and (Xw0 (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))->(((and (Xw0 (fun (Xx00:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx))))))))))) b)
% Found ((eq_ref Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((((and (Xw0 (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))->(((and (Xw0 (fun (Xx00:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx))))))))))) as proof of (((eq Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((((and (Xw0 (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))->(((and (Xw0 (fun (Xx00:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx))))))))))) b)
% Found ((eq_ref Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((((and (Xw0 (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))->(((and (Xw0 (fun (Xx00:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx))))))))))) as proof of (((eq Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((((and (Xw0 (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))->(((and (Xw0 (fun (Xx00:(fofType->Prop))=> False))) (forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx))))))))))) b)
% Found x10:(Xw (fun (Xx0:(fofType->Prop))=> False))
% Found x10 as proof of (Xw (fun (Xx:(fofType->Prop))=> False))
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))):(((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) (fun (x:(fofType->Prop))=> (forall (Xx0:fofType), ((x Xx0)->(Xr Xx0)))))
% Found (eta_expansion_dep00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) b)
% Found ((eta_expansion_dep0 (fun (x5:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x5:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x5:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x5:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) b)
% Found x20:(forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))
% Found x20 as proof of (forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx))))))
% Found eta_expansion000:=(eta_expansion00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))):(((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) (fun (x:(fofType->Prop))=> (forall (Xx0:fofType), ((x Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (eta_expansion00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found ((eta_expansion0 Prop) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found x30:=(x3 x20):(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Instantiate: b:=(fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))):((fofType->Prop)->Prop)
% Found (x3 x20) as proof of (P b)
% Found (x3 x20) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))):(((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False))))
% Found (eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))):(((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) (fun (x:(fofType->Prop))=> (forall (Xx0:fofType), ((x Xx0)->(Xr Xx0)))))
% Found (eta_expansion_dep00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) b)
% Found ((eta_expansion_dep0 (fun (x5:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x5:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x5:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x5:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) b)
% Found eq_ref000:=(eq_ref00 Xw):((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))))
% Found (eq_ref00 Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found ((eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found (((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found (((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))):(((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) (fun (x:(fofType->Prop))=> (forall (Xx0:fofType), ((x Xx0)->(Xr Xx0)))))
% Found (eta_expansion_dep00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) b)
% Found ((eta_expansion_dep0 (fun (x5:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x5:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x5:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x5:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) b)
% Found eq_ref00:=(eq_ref0 (((fofType->Prop)->Prop)->((fofType->Prop)->((Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))):(((eq Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))) (((fofType->Prop)->Prop)->((fofType->Prop)->((Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx))))))))))
% Found (eq_ref0 (((fofType->Prop)->Prop)->((fofType->Prop)->((Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))) as proof of (((eq Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))) b)
% Found ((eq_ref Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))) as proof of (((eq Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))) b)
% Found ((eq_ref Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))) as proof of (((eq Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))) b)
% Found ((eq_ref Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))) as proof of (((eq Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))) b)
% Found eq_ref00:=(eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))):(((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found x210:=(x21 x20):(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Instantiate: b:=(fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))):((fofType->Prop)->Prop)
% Found (x21 x20) as proof of (P b)
% Found ((x2 x10) x20) as proof of (P b)
% Found ((x2 x10) x20) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))):(((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False))))
% Found (eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found eq_ref000:=(eq_ref00 Xw):((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))))
% Found (eq_ref00 Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found ((eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found (((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found (((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))) Xw) as proof of (P (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))):(((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) (fun (x:(fofType->Prop))=> (forall (Xx0:fofType), ((x Xx0)->(Xr Xx0)))))
% Found (eta_expansion_dep00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) b)
% Found ((eta_expansion_dep0 (fun (x5:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x5:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x5:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x5:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) b)
% Found eq_ref00:=(eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))):(((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False))))
% Found (eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False)))) b)
% Found x30:=(x3 x20):(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Instantiate: f:=(fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))):((fofType->Prop)->Prop)
% Found (x3 x20) as proof of (P f)
% Found (x3 x20) as proof of (P f)
% Found x30:=(x3 x20):(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Instantiate: f:=(fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))):((fofType->Prop)->Prop)
% Found (x3 x20) as proof of (P f)
% Found (x3 x20) as proof of (P f)
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(P0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(P0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found x2:(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Instantiate: b:=(fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))):((fofType->Prop)->Prop)
% Found x2 as proof of (P b)
% Found eq_ref00:=(eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))):(((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found eq_ref00:=(eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))):(((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found x10:=(x1 x00):(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Found (x1 x00) as proof of (P b)
% Found (x1 x00) as proof of (P b)
% Found (x1 x00) as proof of (P b)
% Found x00:((and (Xw (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))
% Found x00 as proof of ((and (Xw (fun (Xx:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Found eta_expansion000:=(eta_expansion00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))):(((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) (fun (x:(fofType->Prop))=> (forall (Xx0:fofType), ((x Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (eta_expansion00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found ((eta_expansion0 Prop) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Found eq_ref00:=(eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))):(((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))):(((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) (fun (x:(fofType->Prop))=> (forall (Xx0:fofType), ((x Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (eta_expansion00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found ((eta_expansion0 Prop) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found x210:=(x21 x20):(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Instantiate: f:=(fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))):((fofType->Prop)->Prop)
% Found (x21 x20) as proof of (P f)
% Found ((x2 x10) x20) as proof of (P f)
% Found ((x2 x10) x20) as proof of (P f)
% Found x210:=(x21 x20):(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Instantiate: f:=(fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))):((fofType->Prop)->Prop)
% Found (x21 x20) as proof of (P f)
% Found ((x2 x10) x20) as proof of (P f)
% Found ((x2 x10) x20) as proof of (P f)
% Found x4:(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Instantiate: f:=(fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))):((fofType->Prop)->Prop)
% Found x4 as proof of (P f)
% Found x4 as proof of (P f)
% Found x4 as proof of (P f)
% Found x4:(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Instantiate: f:=(fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))):((fofType->Prop)->Prop)
% Found x4 as proof of (P f)
% Found x4 as proof of (P f)
% Found x4 as proof of (P f)
% Found x00:((and (Xw (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))
% Found x00 as proof of ((and (Xw (fun (Xx:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))
% Found eq_ref00:=(eq_ref0 (((fofType->Prop)->Prop)->((fofType->Prop)->(((Xw0 (fun (Xx0:(fofType->Prop))=> False))->((forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))->((Xw0 (fun (Xx00:(fofType->Prop))=> False))->((forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))))):(((eq Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->(((Xw0 (fun (Xx0:(fofType->Prop))=> False))->((forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))->((Xw0 (fun (Xx00:(fofType->Prop))=> False))->((forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))))) (((fofType->Prop)->Prop)->((fofType->Prop)->(((Xw0 (fun (Xx0:(fofType->Prop))=> False))->((forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))->((Xw0 (fun (Xx00:(fofType->Prop))=> False))->((forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx))))))))))))
% Found (eq_ref0 (((fofType->Prop)->Prop)->((fofType->Prop)->(((Xw0 (fun (Xx0:(fofType->Prop))=> False))->((forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))->((Xw0 (fun (Xx00:(fofType->Prop))=> False))->((forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))))) as proof of (((eq Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->(((Xw0 (fun (Xx0:(fofType->Prop))=> False))->((forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))->((Xw0 (fun (Xx00:(fofType->Prop))=> False))->((forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))))) b)
% Found ((eq_ref Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->(((Xw0 (fun (Xx0:(fofType->Prop))=> False))->((forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))->((Xw0 (fun (Xx00:(fofType->Prop))=> False))->((forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))))) as proof of (((eq Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->(((Xw0 (fun (Xx0:(fofType->Prop))=> False))->((forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))->((Xw0 (fun (Xx00:(fofType->Prop))=> False))->((forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))))) b)
% Found ((eq_ref Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->(((Xw0 (fun (Xx0:(fofType->Prop))=> False))->((forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))->((Xw0 (fun (Xx00:(fofType->Prop))=> False))->((forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))))) as proof of (((eq Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->(((Xw0 (fun (Xx0:(fofType->Prop))=> False))->((forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))->((Xw0 (fun (Xx00:(fofType->Prop))=> False))->((forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))))) b)
% Found ((eq_ref Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->(((Xw0 (fun (Xx0:(fofType->Prop))=> False))->((forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))->((Xw0 (fun (Xx00:(fofType->Prop))=> False))->((forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))))) as proof of (((eq Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->(((Xw0 (fun (Xx0:(fofType->Prop))=> False))->((forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))->((Xw0 (fun (Xx00:(fofType->Prop))=> False))->((forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))))) b)
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(P0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(P0 (fun (x:(fofType->Prop))=> (forall (Xx0:fofType), ((x Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eta_expansion00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (((eta_expansion0 Prop) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((((eta_expansion (fofType->Prop)) Prop) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((((eta_expansion (fofType->Prop)) Prop) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(P0 (fun (x:(fofType->Prop))=> (forall (Xx0:fofType), ((x Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))
% Found (eta_expansion_dep000 P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eta_expansion_dep00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (((eta_expansion_dep0 (fun (x3:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((((eta_expansion_dep (fofType->Prop)) (fun (x3:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((((eta_expansion_dep (fofType->Prop)) (fun (x3:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(P0 (fun (x:(fofType->Prop))=> (forall (Xx0:fofType), ((x Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))
% Found (eta_expansion_dep000 P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eta_expansion_dep00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (((eta_expansion_dep0 (fun (x3:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((((eta_expansion_dep (fofType->Prop)) (fun (x3:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((((eta_expansion_dep (fofType->Prop)) (fun (x3:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found eq_ref00:=(eq_ref0 (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))):(((eq Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0))))
% Found (eq_ref0 (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) b)
% Found ((eq_ref Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) b)
% Found ((eq_ref Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) b)
% Found ((eq_ref Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x2 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x2 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x2 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x2 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found eq_ref00:=(eq_ref0 (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))):(((eq Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0))))
% Found (eq_ref0 (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) b)
% Found ((eq_ref Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) b)
% Found ((eq_ref Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) b)
% Found ((eq_ref Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x2 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x2 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x2 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x2 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found eq_ref00:=(eq_ref0 (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))):(((eq Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0))))
% Found (eq_ref0 (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) b)
% Found ((eq_ref Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) b)
% Found ((eq_ref Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) b)
% Found ((eq_ref Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x2 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x2 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x2 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x2 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found eq_ref00:=(eq_ref0 (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))):(((eq Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0))))
% Found (eq_ref0 (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) b)
% Found ((eq_ref Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) b)
% Found ((eq_ref Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) b)
% Found ((eq_ref Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x2 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x2 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x2 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x2 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx)))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx)))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx)))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx)))))
% Found eq_ref00:=(eq_ref0 a):(((eq ((fofType->Prop)->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% Found eq_ref000:=(eq_ref00 P0):((P0 (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0))))->(P0 (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))))
% Found (eq_ref00 P0) as proof of (P1 (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0))))
% Found ((eq_ref0 (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) P0) as proof of (P1 (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0))))
% Found (((eq_ref Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) P0) as proof of (P1 (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0))))
% Found (((eq_ref Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) P0) as proof of (P1 (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0))))->(P0 (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))))
% Found (eq_ref00 P0) as proof of (P1 (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0))))
% Found ((eq_ref0 (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) P0) as proof of (P1 (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0))))
% Found (((eq_ref Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) P0) as proof of (P1 (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0))))
% Found (((eq_ref Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) P0) as proof of (P1 (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0))))->(P0 (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))))
% Found (eq_ref00 P0) as proof of (P1 (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0))))
% Found ((eq_ref0 (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) P0) as proof of (P1 (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0))))
% Found (((eq_ref Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) P0) as proof of (P1 (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0))))
% Found (((eq_ref Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) P0) as proof of (P1 (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0))))->(P0 (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))))
% Found (eq_ref00 P0) as proof of (P1 (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0))))
% Found ((eq_ref0 (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) P0) as proof of (P1 (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0))))
% Found (((eq_ref Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) P0) as proof of (P1 (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0))))
% Found (((eq_ref Prop) (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0)))) P0) as proof of (P1 (forall (Xx0:fofType), ((x2 Xx0)->(Xr Xx0))))
% Found eq_ref00:=(eq_ref0 (((fofType->Prop)->Prop)->((fofType->Prop)->(((forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))->((forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx))))))))))):(((eq Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->(((forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))->((forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx))))))))))) (((fofType->Prop)->Prop)->((fofType->Prop)->(((forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))->((forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))))
% Found (eq_ref0 (((fofType->Prop)->Prop)->((fofType->Prop)->(((forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))->((forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx))))))))))) as proof of (((eq Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->(((forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))->((forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx))))))))))) b)
% Found ((eq_ref Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->(((forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))->((forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx))))))))))) as proof of (((eq Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->(((forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))->((forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx))))))))))) b)
% Found ((eq_ref Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->(((forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))->((forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx))))))))))) as proof of (((eq Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->(((forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))->((forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx))))))))))) b)
% Found ((eq_ref Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->(((forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))->((forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx))))))))))) as proof of (((eq Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->(((forall (Xr00:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))->((forall (Xr00:((fofType->Prop)->Prop)) (Xx00:(fofType->Prop)), ((Xw0 Xr00)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr00 Xt)) (((eq (fofType->Prop)) Xt) Xx00))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx))))))))))) b)
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((forall (Xw0:(((fofType->Prop)->Prop)->Prop)), (((and (Xw0 (fun (Xx:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw0 Xr0)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))))->(forall (Xw0:(((fofType->Prop)->Prop)->Prop)), (((and (Xw0 (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr0)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((forall (Xw0:(((fofType->Prop)->Prop)->Prop)), (((and (Xw0 (fun (Xx:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw0 Xr0)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))))->(forall (Xw0:(((fofType->Prop)->Prop)->Prop)), (((and (Xw0 (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr0)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((forall (Xw0:(((fofType->Prop)->Prop)->Prop)), (((and (Xw0 (fun (Xx:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw0 Xr0)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))))->(forall (Xw0:(((fofType->Prop)->Prop)->Prop)), (((and (Xw0 (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr0)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((forall (Xw0:(((fofType->Prop)->Prop)->Prop)), (((and (Xw0 (fun (Xx:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw0 Xr0)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))))->(forall (Xw0:(((fofType->Prop)->Prop)->Prop)), (((and (Xw0 (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw0 Xr0)->(Xw0 (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))))
% Found eq_ref000:=(eq_ref00 Xw):((Xw Xr0)->(Xw Xr0))
% Found (eq_ref00 Xw) as proof of (P Xr0)
% Found ((eq_ref0 Xr0) Xw) as proof of (P Xr0)
% Found (((eq_ref ((fofType->Prop)->Prop)) Xr0) Xw) as proof of (P Xr0)
% Found (((eq_ref ((fofType->Prop)->Prop)) Xr0) Xw) as proof of (P Xr0)
% Found x10:=(x1 x00):(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Found (x1 x00) as proof of (P f)
% Found (x1 x00) as proof of (P f)
% Found (x1 x00) as proof of (P f)
% Found x10:=(x1 x00):(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Found (x1 x00) as proof of (P f)
% Found (x1 x00) as proof of (P f)
% Found (x1 x00) as proof of (P f)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(P0 (fun (x:(fofType->Prop))=> (forall (Xx0:fofType), ((x Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eta_expansion00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (((eta_expansion0 Prop) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((((eta_expansion (fofType->Prop)) Prop) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((((eta_expansion (fofType->Prop)) Prop) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(P0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found relational_choice:(forall (A:Type) (B:Type) (R:(A->(B->Prop))), ((forall (x:A), ((ex B) (fun (y:B)=> ((R x) y))))->((ex (A->(B->Prop))) (fun (R':(A->(B->Prop)))=> ((and ((((subrelation A) B) R') R)) (forall (x:A), ((ex B) ((unique B) (fun (y:B)=> ((R' x) y))))))))))
% Instantiate: a:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), ((forall (x:A), ((ex B) (fun (y:B)=> ((R x) y))))->((ex (A->(B->Prop))) (fun (R':(A->(B->Prop)))=> ((and ((((subrelation A) B) R') R)) (forall (x:A), ((ex B) ((unique B) (fun (y:B)=> ((R' x) y)))))))))):Prop
% Found relational_choice as proof of a
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((((and (Xw (fun (Xx:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))))->(((and (Xw (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((((and (Xw (fun (Xx:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))))->(((and (Xw (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((((and (Xw (fun (Xx:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))))->(((and (Xw (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((((and (Xw (fun (Xx:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))))->(((and (Xw (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))))
% Found x00:((and (Xw (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))
% Found x00 as proof of ((and (Xw (fun (Xx:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))
% 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)))
% Instantiate: a:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of a
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(P0 (fun (x:(fofType->Prop))=> (forall (Xx0:fofType), ((x Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eta_expansion00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (((eta_expansion0 Prop) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((((eta_expansion (fofType->Prop)) Prop) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((((eta_expansion (fofType->Prop)) Prop) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(P0 (fun (x:(fofType->Prop))=> (forall (Xx0:fofType), ((x Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))
% Found (eta_expansion_dep000 P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eta_expansion_dep00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (((eta_expansion_dep0 (fun (x5:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((((eta_expansion_dep (fofType->Prop)) (fun (x5:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((((eta_expansion_dep (fofType->Prop)) (fun (x5:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Found eq_ref00:=(eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))):(((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Found eq_ref00:=(eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))):(((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% 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)))
% Instantiate: a:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of a
% Found eq_ref00:=(eq_ref0 (((fofType->Prop)->Prop)->((fofType->Prop)->((Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))):(((eq Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))) (((fofType->Prop)->Prop)->((fofType->Prop)->((Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx))))))))))
% Found (eq_ref0 (((fofType->Prop)->Prop)->((fofType->Prop)->((Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))) as proof of (((eq Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))) b)
% Found ((eq_ref Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))) as proof of (((eq Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))) b)
% Found ((eq_ref Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))) as proof of (((eq Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))) b)
% Found ((eq_ref Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))) as proof of (((eq Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Found eq_ref00:=(eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))):(((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Found eta_expansion000:=(eta_expansion00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))):(((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) (fun (x:(fofType->Prop))=> (forall (Xx0:fofType), ((x Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (eta_expansion00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found ((eta_expansion0 Prop) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) b)
% Found eq_ref00:=(eq_ref0 (((fofType->Prop)->Prop)->((fofType->Prop)->((Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))):(((eq Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))) (((fofType->Prop)->Prop)->((fofType->Prop)->((Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx))))))))))
% Found (eq_ref0 (((fofType->Prop)->Prop)->((fofType->Prop)->((Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))) as proof of (((eq Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))) b)
% Found ((eq_ref Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))) as proof of (((eq Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))) b)
% Found ((eq_ref Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))) as proof of (((eq Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))) b)
% Found ((eq_ref Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))) as proof of (((eq Prop) (((fofType->Prop)->Prop)->((fofType->Prop)->((Xw0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(Xw0 (fun (R:(fofType->Prop))=> (forall (Xx00:fofType), ((R Xx00)->((or (Xr Xx00)) (((eq fofType) Xx00) Xx)))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx)))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx)))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx)))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx)))))
% Found eq_ref00:=(eq_ref0 a):(((eq ((fofType->Prop)->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% Found eq_ref000:=(eq_ref00 Xw):((Xw Xr0)->(Xw Xr0))
% Found (eq_ref00 Xw) as proof of (P Xr0)
% Found ((eq_ref0 Xr0) Xw) as proof of (P Xr0)
% Found (((eq_ref ((fofType->Prop)->Prop)) Xr0) Xw) as proof of (P Xr0)
% Found (((eq_ref ((fofType->Prop)->Prop)) Xr0) Xw) as proof of (P Xr0)
% Found eq_ref000:=(eq_ref00 Xw):((Xw Xr0)->(Xw Xr0))
% Found (eq_ref00 Xw) as proof of (P Xr0)
% Found ((eq_ref0 Xr0) Xw) as proof of (P Xr0)
% Found (((eq_ref ((fofType->Prop)->Prop)) Xr0) Xw) as proof of (P Xr0)
% Found (((eq_ref ((fofType->Prop)->Prop)) Xr0) Xw) as proof of (P Xr0)
% Found x00:((and (Xw (fun (Xx0:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0)))))))
% Found x00 as proof of ((and (Xw (fun (Xx:(fofType->Prop))=> False))) (forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx)))))))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xr:(fofType->Prop)) (Xx:fofType), (((Xw (fun (Xx:(fofType->Prop))=> False))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))))->((Xw (fun (Xx0:(fofType->Prop))=> False))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xr:(fofType->Prop)) (Xx:fofType), (((Xw (fun (Xx:(fofType->Prop))=> False))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))))->((Xw (fun (Xx0:(fofType->Prop))=> False))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xr:(fofType->Prop)) (Xx:fofType), (((Xw (fun (Xx:(fofType->Prop))=> False))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))))->((Xw (fun (Xx0:(fofType->Prop))=> False))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xr:(fofType->Prop)) (Xx:fofType), (((Xw (fun (Xx:(fofType->Prop))=> False))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))))->((Xw (fun (Xx0:(fofType->Prop))=> False))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xr:(fofType->Prop)) (Xx:fofType), (((forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xr:(fofType->Prop)) (Xx:fofType), (((forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xr:(fofType->Prop)) (Xx:fofType), (((forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xr:(fofType->Prop)) (Xx:fofType), (((forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx))))))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(P0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(P0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(P0 (fun (x:(fofType->Prop))=> (forall (Xx0:fofType), ((x Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))
% Found (eta_expansion_dep000 P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eta_expansion_dep00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (((eta_expansion_dep0 (fun (x5:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((((eta_expansion_dep (fofType->Prop)) (fun (x5:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((((eta_expansion_dep (fofType->Prop)) (fun (x5:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(P0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found x1:=(x (fun (x2:(fofType->Prop))=> ((Xw (fun (Xx:(fofType->Prop))=> False))->((forall (Xr:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr Xt)) (((eq (fofType->Prop)) Xt) Xx))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(x2 Xx))))))))):(((and ((Xw (fun (Xx:(fofType->Prop))=> False))->((forall (Xr:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr Xt)) (((eq (fofType->Prop)) Xt) Xx))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False))))))) (forall (Xr:(fofType->Prop)) (Xx:fofType), (((Xw (fun (Xx:(fofType->Prop))=> False))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))))->((Xw (fun (Xx0:(fofType->Prop))=> False))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))))->((Xw (fun (Xx:(fofType->Prop))=> False))->((forall (Xr:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr Xt)) (((eq (fofType->Prop)) Xt) Xx))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx))))))))
% Instantiate: a:=(((and ((Xw (fun (Xx:(fofType->Prop))=> False))->((forall (Xr:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr Xt)) (((eq (fofType->Prop)) Xt) Xx))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->False))))))) (forall (Xr:(fofType->Prop)) (Xx:fofType), (((Xw (fun (Xx:(fofType->Prop))=> False))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))))->((Xw (fun (Xx0:(fofType->Prop))=> False))->((forall (Xr0:((fofType->Prop)->Prop)) (Xx0:(fofType->Prop)), ((Xw Xr0)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr0 Xt)) (((eq (fofType->Prop)) Xt) Xx0))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))))->((Xw (fun (Xx:(fofType->Prop))=> False))->((forall (Xr:((fofType->Prop)->Prop)) (Xx:(fofType->Prop)), ((Xw Xr)->(Xw (fun (Xt:(fofType->Prop))=> ((or (Xr Xt)) (((eq (fofType->Prop)) Xt) Xx))))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(cX Xx)))))))):Prop
% Found x1 as proof of a
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found eq_ref00:=(eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))):(((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found (eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found eq_ref00:=(eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))):(((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0)))))
% Found (eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->(Xr Xx0))))) b)
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))
% Found eq_ref00:=(eq_ref0 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))):(((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0))))
% Found (eq_ref0 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) b)
% Found ((eq_ref Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) b)
% Found ((eq_ref Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) b)
% Found ((eq_ref Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x4 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x4 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x4 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x4 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found eq_ref00:=(eq_ref0 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))):(((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0))))
% Found (eq_ref0 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) b)
% Found ((eq_ref Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) b)
% Found ((eq_ref Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) b)
% Found ((eq_ref Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x4 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x4 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x4 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x4 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found eq_ref00:=(eq_ref0 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))):(((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0))))
% Found (eq_ref0 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) b)
% Found ((eq_ref Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) b)
% Found ((eq_ref Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) b)
% Found ((eq_ref Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x4 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x4 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x4 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x4 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found eq_ref00:=(eq_ref0 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))):(((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0))))
% Found (eq_ref0 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) b)
% Found ((eq_ref Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) b)
% Found ((eq_ref Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) b)
% Found ((eq_ref Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x4 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x4 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x4 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x4 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(P0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(P0 (fun (x:(fofType->Prop))=> (forall (Xx0:fofType), ((x Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eta_expansion00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (((eta_expansion0 Prop) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((((eta_expansion (fofType->Prop)) Prop) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((((eta_expansion (fofType->Prop)) Prop) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(P0 (fun (x:(fofType->Prop))=> (forall (Xx0:fofType), ((x Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))
% Found (eta_expansion_dep000 P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eta_expansion_dep00 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (((eta_expansion_dep0 (fun (x5:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((((eta_expansion_dep (fofType->Prop)) (fun (x5:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((((eta_expansion_dep (fofType->Prop)) (fun (x5:(fofType->Prop))=> Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))->(P0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found ((eq_ref0 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found (((eq_ref ((fofType->Prop)->Prop)) (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))) P0) as proof of (P1 (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx))))))
% Found eq_sym:=(fun (T:Type) (a:T) (b:T) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq T) x) a))) ((eq_ref T) a))):(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a)))
% Instantiate: a:=(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a))):Prop
% Found eq_sym as proof of a
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xr:(fofType->Prop)) (Xx:fofType), ((Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))->(Xw (fun (R:(fofType->Prop))=> (forall (Xx0:fofType), ((R Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))))))
% Found eq_ref00:=(eq_ref0 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))):(((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0))))
% Found (eq_ref0 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) b)
% Found ((eq_ref Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) b)
% Found ((eq_ref Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) b)
% Found ((eq_ref Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x4 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x4 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x4 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x4 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found eq_ref00:=(eq_ref0 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))):(((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0))))
% Found (eq_ref0 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) b)
% Found ((eq_ref Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) b)
% Found ((eq_ref Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) b)
% Found ((eq_ref Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x4 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x4 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x4 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x4 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found eq_ref00:=(eq_ref0 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))):(((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0))))
% Found (eq_ref0 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) b)
% Found ((eq_ref Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) b)
% Found ((eq_ref Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) b)
% Found ((eq_ref Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x4 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x4 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x4 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x4 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found eq_ref00:=(eq_ref0 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))):(((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0))))
% Found (eq_ref0 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) b)
% Found ((eq_ref Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) b)
% Found ((eq_ref Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) b)
% Found ((eq_ref Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) as proof of (((eq Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x4 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x4 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x4 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:fofType), ((x4 Xx0)->((or (Xr Xx0)) (((eq fofType) Xx0) Xx)))))
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: a:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of a
% Found eq_ref000:=(eq_ref00 P0):((P0 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0))))->(P0 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))))
% Found (eq_ref00 P0) as proof of (P1 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0))))
% Found ((eq_ref0 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) P0) as proof of (P1 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0))))
% Found (((eq_ref Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) P0) as proof of (P1 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0))))
% Found (((eq_ref Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) P0) as proof of (P1 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0))))->(P0 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))))
% Found (eq_ref00 P0) as proof of (P1 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0))))
% Found ((eq_ref0 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) P0) as proof of (P1 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0))))
% Found (((eq_ref Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) P0) as proof of (P1 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0))))
% Found (((eq_ref Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) P0) as proof of (P1 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0))))->(P0 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))))
% Found (eq_ref00 P0) as proof of (P1 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0))))
% Found ((eq_ref0 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) P0) as proof of (P1 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0))))
% Found (((eq_ref Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) P0) as proof of (P1 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0))))
% Found (((eq_ref Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) P0) as proof of (P1 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0))))->(P0 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))))
% Found (eq_ref00 P0) as proof of (P1 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0))))
% Found ((eq_ref0 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) P0) as proof of (P1 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0))))
% Found (((eq_ref Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) P0) as proof of (P1 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0))))
% Found (((eq_ref Prop) (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0)))) P0) as proof of (P1 (forall (Xx0:fofType), ((x4 Xx0)->(Xr Xx0))))
% Found x10:(Xw (fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))))
% Instantiate: b:=(fun (R:(fofType->Prop))=> (forall (Xx:fofType), ((R Xx)->(Xr Xx)))):((fofType->Prop)->Prop)
% Found x10 as proof of (P b)
% Found eq_sym0:=(eq_sym Prop):(forall (a:Prop) (b:
% EOF
%------------------------------------------------------------------------------