%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV309^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n099.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 13:34:01 EDT 2014

% Result   : Timeout 300.08s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV309^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n099.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 08:48:01 CDT 2014
% % CPUTime  : 300.08 
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula ((forall (P:(fofType->Prop)) (Q:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((iff (P X)) (Q X))->(((eq (fofType->Prop)) P) Q)))))->(forall (K:((fofType->Prop)->(fofType->Prop))), ((or ((ex (fofType->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))) ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))))) of role conjecture named cTHM1_pme
% Conjecture to prove = ((forall (P:(fofType->Prop)) (Q:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((iff (P X)) (Q X))->(((eq (fofType->Prop)) P) Q)))))->(forall (K:((fofType->Prop)->(fofType->Prop))), ((or ((ex (fofType->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))) ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))))):Prop
% Parameter fofType_DUMMY:fofType.
% We need to prove ['((forall (P:(fofType->Prop)) (Q:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((iff (P X)) (Q X))->(((eq (fofType->Prop)) P) Q)))))->(forall (K:((fofType->Prop)->(fofType->Prop))), ((or ((ex (fofType->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))) ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))))))']
% Parameter fofType:Type.
% Trying to prove ((forall (P:(fofType->Prop)) (Q:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((iff (P X)) (Q X))->(((eq (fofType->Prop)) P) Q)))))->(forall (K:((fofType->Prop)->(fofType->Prop))), ((or ((ex (fofType->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))) ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))))))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))):(((eq Prop) ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))) ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))) b)
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))):(((eq Prop) ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))) ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))) b)
% Found eq_ref00:=(eq_ref0 a):(((eq ((fofType->Prop)->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found eta_expansion000:=(eta_expansion00 a):(((eq ((fofType->Prop)->Prop)) a) (fun (x:(fofType->Prop))=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found ((eta_expansion0 Prop) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found (((eta_expansion (fofType->Prop)) Prop) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found (((eta_expansion (fofType->Prop)) Prop) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found (((eta_expansion (fofType->Prop)) Prop) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found eq_ref00:=(eq_ref0 a):(((eq ((fofType->Prop)->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found eta_expansion000:=(eta_expansion00 a):(((eq ((fofType->Prop)->Prop)) a) (fun (x:(fofType->Prop))=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found ((eta_expansion0 Prop) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found (((eta_expansion (fofType->Prop)) Prop) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found (((eta_expansion (fofType->Prop)) Prop) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found (((eta_expansion (fofType->Prop)) Prop) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))):(((eq Prop) ((ex (fofType->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))) ((ex (fofType->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False)))))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))):(((eq ((fofType->Prop)->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) (fun (x:(fofType->Prop))=> (((eq (fofType->Prop)) (K x)) x)))
% Found (eta_expansion_dep00 (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->Prop))=> Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x1:(fofType->Prop))=> Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x1:(fofType->Prop))=> Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x1:(fofType->Prop))=> Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) b)
% Found eq_ref00:=(eq_ref0 (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False)))))):(((eq ((fofType->Prop)->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False)))))) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))
% Found (eq_ref0 (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False)))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False)))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False)))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False)))))) b)
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))))))))):(((eq Prop) ((ex (fofType->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))))))))) ((ex (fofType->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx))))))))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))))))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))))))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))))))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))))))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))))))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))))))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))))))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))))))))) b)
% Found eq_ref00:=(eq_ref0 (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))):(((eq ((fofType->Prop)->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found (eq_ref0 (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx))))))))):(((eq ((fofType->Prop)->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx))))))))) (fun (x:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K x) Xx)->((K Y) Xx)))))))))
% Found (eta_expansion_dep00 (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx))))))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx))))))))) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->Prop))=> Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx))))))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx))))))))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x1:(fofType->Prop))=> Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx))))))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx))))))))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x1:(fofType->Prop))=> Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx))))))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx))))))))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x1:(fofType->Prop))=> Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx))))))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx))))))))) b)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K x0)) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K x0)) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K x0)) x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K x0)) x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->Prop)), (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K x0)) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K x0)) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K x0)) x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K x0)) x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->Prop)), (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x)))
% Found eq_ref00:=(eq_ref0 a):(((eq ((fofType->Prop)->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))
% Found eq_ref00:=(eq_ref0 a):(((eq ((fofType->Prop)->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K x0)) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K x0)) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K x0)) x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K x0)) x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->Prop)), (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K x0)) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K x0)) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K x0)) x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K x0)) x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->Prop)), (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x)))
% Found eq_ref00:=(eq_ref0 a):(((eq ((fofType->Prop)->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) 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) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False)))))):(((eq ((fofType->Prop)->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False)))))) (fun (x:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K x) Xx)->((K Y) Xx)))->False))))))
% Found (eta_expansion_dep00 (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False)))))) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->Prop))=> Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False)))))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x1:(fofType->Prop))=> Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False)))))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x1:(fofType->Prop))=> Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False)))))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x1:(fofType->Prop))=> Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False)))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False)))))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))):(((eq ((fofType->Prop)->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) (fun (x:(fofType->Prop))=> (((eq (fofType->Prop)) (K x)) x)))
% Found (eta_expansion00 (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) b)
% Found ((eta_expansion0 Prop) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) b)
% Found eq_ref00:=(eq_ref0 (K x0)):(((eq (fofType->Prop)) (K x0)) (K x0))
% Found (eq_ref0 (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found ((eq_ref (fofType->Prop)) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found ((eq_ref (fofType->Prop)) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found ((eq_ref (fofType->Prop)) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found eta_expansion000:=(eta_expansion00 (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))):(((eq ((fofType->Prop)->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) (fun (x:(fofType->Prop))=> (((eq (fofType->Prop)) (K x)) x)))
% Found (eta_expansion00 (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) b)
% Found ((eta_expansion0 Prop) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))) b)
% Found eq_ref00:=(eq_ref0 (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx))))))))):(((eq ((fofType->Prop)->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx))))))))) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))))))))
% Found (eq_ref0 (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx))))))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx))))))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx))))))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx))))))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx))))))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx))))))))) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx))))))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx))))))))) b)
% Found eta_expansion000:=(eta_expansion00 (K x0)):(((eq (fofType->Prop)) (K x0)) (fun (x:fofType)=> ((K x0) x)))
% Found (eta_expansion00 (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found ((eta_expansion0 Prop) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found (((eta_expansion fofType) Prop) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found (((eta_expansion fofType) Prop) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found (((eta_expansion fofType) Prop) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K x0)) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K x0)) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K x0)) x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K x0)) x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->Prop)), (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K x0)) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K x0)) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K x0)) x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K x0)) x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->Prop)), (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K x0)) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K x0)) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K x0)) x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K x0)) x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->Prop)), (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K x0)) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K x0)) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K x0)) x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K x0)) x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->Prop)), (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x)))
% Found eq_ref000:=(eq_ref00 x0):((x0 Xx)->(x0 Xx))
% Found (eq_ref00 x0) as proof of ((x0 Xx)->(x1 Xx))
% Found ((eq_ref0 Xx) x0) as proof of ((x0 Xx)->(x1 Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(x1 Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(x1 Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of ((x0 Xx)->(x1 Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of (forall (Xx:fofType), ((x0 Xx)->(x1 Xx)))
% Found eta_expansion000:=(eta_expansion00 (K x0)):(((eq (fofType->Prop)) (K x0)) (fun (x:fofType)=> ((K x0) x)))
% Found (eta_expansion00 (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found ((eta_expansion0 Prop) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found (((eta_expansion fofType) Prop) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found (((eta_expansion fofType) Prop) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found (((eta_expansion fofType) Prop) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found eq_ref000:=(eq_ref00 x0):((x0 Xx)->(x0 Xx))
% Found (eq_ref00 x0) as proof of ((x0 Xx)->(x1 Xx))
% Found ((eq_ref0 Xx) x0) as proof of ((x0 Xx)->(x1 Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(x1 Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(x1 Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of ((x0 Xx)->(x1 Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of (forall (Xx:fofType), ((x0 Xx)->(x1 Xx)))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found eq_ref00:=(eq_ref0 x0):(((eq (fofType->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found eta_expansion000:=(eta_expansion00 (K x0)):(((eq (fofType->Prop)) (K x0)) (fun (x:fofType)=> ((K x0) x)))
% Found (eta_expansion00 (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found ((eta_expansion0 Prop) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found (((eta_expansion fofType) Prop) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found (((eta_expansion fofType) Prop) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found (((eta_expansion fofType) Prop) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found eq_ref000:=(eq_ref00 P):((P ((K x0) x1))->(P ((K x0) x1)))
% Found (eq_ref00 P) as proof of (P0 ((K x0) x1))
% Found ((eq_ref0 ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found eq_ref000:=(eq_ref00 P):((P ((K x0) x1))->(P ((K x0) x1)))
% Found (eq_ref00 P) as proof of (P0 ((K x0) x1))
% Found ((eq_ref0 ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found eq_ref00:=(eq_ref0 ((K x0) x1)):(((eq Prop) ((K x0) x1)) ((K x0) x1))
% Found (eq_ref0 ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((K x0) x1)):(((eq Prop) ((K x0) x1)) ((K x0) x1))
% Found (eq_ref0 ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found eq_ref00:=(eq_ref0 x0):(((eq (fofType->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found eq_ref000:=(eq_ref00 P):((P ((K x0) x1))->(P ((K x0) x1)))
% Found (eq_ref00 P) as proof of (P0 ((K x0) x1))
% Found ((eq_ref0 ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found eq_ref000:=(eq_ref00 P):((P ((K x0) x1))->(P ((K x0) x1)))
% Found (eq_ref00 P) as proof of (P0 ((K x0) x1))
% Found ((eq_ref0 ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found eq_ref000:=(eq_ref00 x0):((x0 Xx)->(x0 Xx))
% Found (eq_ref00 x0) as proof of ((x0 Xx)->(x1 Xx))
% Found ((eq_ref0 Xx) x0) as proof of ((x0 Xx)->(x1 Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(x1 Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(x1 Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of ((x0 Xx)->(x1 Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of (forall (Xx:fofType), ((x0 Xx)->(x1 Xx)))
% Found eq_ref00:=(eq_ref0 ((K x0) x1)):(((eq Prop) ((K x0) x1)) ((K x0) x1))
% Found (eq_ref0 ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((K x0) x1)):(((eq Prop) ((K x0) x1)) ((K x0) x1))
% Found (eq_ref0 ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref000:=(eq_ref00 x0):((x0 Xx)->(x0 Xx))
% Found (eq_ref00 x0) as proof of ((x0 Xx)->(x1 Xx))
% Found ((eq_ref0 Xx) x0) as proof of ((x0 Xx)->(x1 Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(x1 Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(x1 Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of ((x0 Xx)->(x1 Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of (forall (Xx:fofType), ((x0 Xx)->(x1 Xx)))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found eq_ref00:=(eq_ref0 x0):(((eq (fofType->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found eq_ref000:=(eq_ref00 P):((P ((K x0) x1))->(P ((K x0) x1)))
% Found (eq_ref00 P) as proof of (P0 ((K x0) x1))
% Found ((eq_ref0 ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found eq_ref000:=(eq_ref00 P):((P ((K x0) x1))->(P ((K x0) x1)))
% Found (eq_ref00 P) as proof of (P0 ((K x0) x1))
% Found ((eq_ref0 ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found eq_ref00:=(eq_ref0 ((K x0) x1)):(((eq Prop) ((K x0) x1)) ((K x0) x1))
% Found (eq_ref0 ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((K x0) x1)):(((eq Prop) ((K x0) x1)) ((K x0) x1))
% Found (eq_ref0 ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found eq_ref00:=(eq_ref0 x0):(((eq (fofType->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found eq_ref000:=(eq_ref00 P):((P (x0 x1))->(P (x0 x1)))
% Found (eq_ref00 P) as proof of (P0 (x0 x1))
% Found ((eq_ref0 (x0 x1)) P) as proof of (P0 (x0 x1))
% Found (((eq_ref Prop) (x0 x1)) P) as proof of (P0 (x0 x1))
% Found (((eq_ref Prop) (x0 x1)) P) as proof of (P0 (x0 x1))
% Found eq_ref000:=(eq_ref00 P):((P (x0 x1))->(P (x0 x1)))
% Found (eq_ref00 P) as proof of (P0 (x0 x1))
% Found ((eq_ref0 (x0 x1)) P) as proof of (P0 (x0 x1))
% Found (((eq_ref Prop) (x0 x1)) P) as proof of (P0 (x0 x1))
% Found (((eq_ref Prop) (x0 x1)) P) as proof of (P0 (x0 x1))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found eq_ref000:=(eq_ref00 P):((P ((K x0) x1))->(P ((K x0) x1)))
% Found (eq_ref00 P) as proof of (P0 ((K x0) x1))
% Found ((eq_ref0 ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found eq_ref000:=(eq_ref00 P):((P ((K x0) x1))->(P ((K x0) x1)))
% Found (eq_ref00 P) as proof of (P0 ((K x0) x1))
% Found ((eq_ref0 ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found eq_ref00:=(eq_ref0 ((K x0) x1)):(((eq Prop) ((K x0) x1)) ((K x0) x1))
% Found (eq_ref0 ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((K x0) x1)):(((eq Prop) ((K x0) x1)) ((K x0) x1))
% Found (eq_ref0 ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref000:=(eq_ref00 P):((P (x0 x1))->(P (x0 x1)))
% Found (eq_ref00 P) as proof of (P0 (x0 x1))
% Found ((eq_ref0 (x0 x1)) P) as proof of (P0 (x0 x1))
% Found (((eq_ref Prop) (x0 x1)) P) as proof of (P0 (x0 x1))
% Found (((eq_ref Prop) (x0 x1)) P) as proof of (P0 (x0 x1))
% Found eq_ref000:=(eq_ref00 P):((P (x0 x1))->(P (x0 x1)))
% Found (eq_ref00 P) as proof of (P0 (x0 x1))
% Found ((eq_ref0 (x0 x1)) P) as proof of (P0 (x0 x1))
% Found (((eq_ref Prop) (x0 x1)) P) as proof of (P0 (x0 x1))
% Found (((eq_ref Prop) (x0 x1)) P) as proof of (P0 (x0 x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref000:=(eq_ref00 P):((P (x0 x1))->(P (x0 x1)))
% Found (eq_ref00 P) as proof of (P0 (x0 x1))
% Found ((eq_ref0 (x0 x1)) P) as proof of (P0 (x0 x1))
% Found (((eq_ref Prop) (x0 x1)) P) as proof of (P0 (x0 x1))
% Found (((eq_ref Prop) (x0 x1)) P) as proof of (P0 (x0 x1))
% Found eq_ref000:=(eq_ref00 P):((P (x0 x1))->(P (x0 x1)))
% Found (eq_ref00 P) as proof of (P0 (x0 x1))
% Found ((eq_ref0 (x0 x1)) P) as proof of (P0 (x0 x1))
% Found (((eq_ref Prop) (x0 x1)) P) as proof of (P0 (x0 x1))
% Found (((eq_ref Prop) (x0 x1)) P) as proof of (P0 (x0 x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found eq_ref00:=(eq_ref0 ((K x0) x1)):(((eq Prop) ((K x0) x1)) ((K x0) x1))
% Found (eq_ref0 ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((K x0) x1)):(((eq Prop) ((K x0) x1)) ((K x0) x1))
% Found (eq_ref0 ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found eq_ref000:=(eq_ref00 P):((P (x0 x1))->(P (x0 x1)))
% Found (eq_ref00 P) as proof of (P0 (x0 x1))
% Found ((eq_ref0 (x0 x1)) P) as proof of (P0 (x0 x1))
% Found (((eq_ref Prop) (x0 x1)) P) as proof of (P0 (x0 x1))
% Found (((eq_ref Prop) (x0 x1)) P) as proof of (P0 (x0 x1))
% Found eq_ref000:=(eq_ref00 P):((P (x0 x1))->(P (x0 x1)))
% Found (eq_ref00 P) as proof of (P0 (x0 x1))
% Found ((eq_ref0 (x0 x1)) P) as proof of (P0 (x0 x1))
% Found (((eq_ref Prop) (x0 x1)) P) as proof of (P0 (x0 x1))
% Found (((eq_ref Prop) (x0 x1)) P) as proof of (P0 (x0 x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x0) x1))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((K x0) x1)):(((eq Prop) ((K x0) x1)) ((K x0) x1))
% Found (eq_ref0 ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found eq_ref00:=(eq_ref0 ((K x0) x1)):(((eq Prop) ((K x0) x1)) ((K x0) x1))
% Found (eq_ref0 ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% 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) ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))))
% 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) ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U))))
% Found eta_expansion000:=(eta_expansion00 (K x0)):(((eq (fofType->Prop)) (K x0)) (fun (x:fofType)=> ((K x0) x)))
% Found (eta_expansion00 (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found ((eta_expansion0 Prop) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found (((eta_expansion fofType) Prop) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found (((eta_expansion fofType) Prop) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found (((eta_expansion fofType) Prop) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% 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) ((ex (fofType->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->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 (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq ((fofType->Prop)->Prop)) a) (fun (x:(fofType->Prop))=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->Prop))=> Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x1:(fofType->Prop))=> Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x1:(fofType->Prop))=> Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x1:(fofType->Prop))=> Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% 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 eta_expansion000:=(eta_expansion00 a):(((eq ((fofType->Prop)->Prop)) a) (fun (x:(fofType->Prop))=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% Found ((eta_expansion0 Prop) a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% Found (((eta_expansion (fofType->Prop)) Prop) a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% Found (((eta_expansion (fofType->Prop)) Prop) a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% Found (((eta_expansion (fofType->Prop)) Prop) a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq ((fofType->Prop)->Prop)) b) (fun (x:(fofType->Prop))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found ((eta_expansion_dep0 (fun (x1:(fofType->Prop))=> Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x1:(fofType->Prop))=> Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x1:(fofType->Prop))=> Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x1:(fofType->Prop))=> Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% 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 (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((K x0) x1)):(((eq Prop) ((K x0) x1)) ((K x0) x1))
% Found (eq_ref0 ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((K x0) x1)):(((eq Prop) ((K x0) x1)) ((K x0) x1))
% Found (eq_ref0 ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (K x0)):(((eq (fofType->Prop)) (K x0)) (fun (x:fofType)=> ((K x0) x)))
% Found (eta_expansion_dep00 (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% 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) ((ex (fofType->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) 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 (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) 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_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_ref00:=(eq_ref0 b):(((eq ((fofType->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found x1:(P (K x0))
% Instantiate: b:=(K x0):(fofType->Prop)
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 x0):(((eq (fofType->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx)))->False)))):(((eq ((fofType->Prop)->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx)))->False)))) (fun (x:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(x Xx)))) ((forall (Xx:fofType), (((K x0) Xx)->((K x) Xx)))->False))))
% Found (eta_expansion_dep00 (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx)))->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx)))->False)))) b)
% Found ((eta_expansion_dep0 (fun (x2:(fofType->Prop))=> Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx)))->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx)))->False)))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x2:(fofType->Prop))=> Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx)))->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx)))->False)))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x2:(fofType->Prop))=> Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx)))->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx)))->False)))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x2:(fofType->Prop))=> Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx)))->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx)))->False)))) b)
% Found x1:(P (K x0))
% Instantiate: f:=(K x0):(fofType->Prop)
% Found x1 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (x0 x)))
% Found x1:(P (K x0))
% Instantiate: f:=(K x0):(fofType->Prop)
% Found x1 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (x0 x)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((K x0) x1)):(((eq Prop) ((K x0) x1)) ((K x0) x1))
% Found (eq_ref0 ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((K x0) x1)):(((eq Prop) ((K x0) x1)) ((K x0) x1))
% Found (eq_ref0 ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found x1:(P (K x0))
% Instantiate: b:=(K x0):(fofType->Prop)
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 x0):(((eq (fofType->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (K x0)):(((eq (fofType->Prop)) (K x0)) (fun (x:fofType)=> ((K x0) x)))
% Found (eta_expansion_dep00 (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx))))))):(((eq ((fofType->Prop)->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx))))))) (fun (x:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(x Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K x) Xx)))))))
% Found (eta_expansion_dep00 (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx))))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx))))))) b)
% Found ((eta_expansion_dep0 (fun (x2:(fofType->Prop))=> Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx))))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx))))))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x2:(fofType->Prop))=> Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx))))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx))))))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x2:(fofType->Prop))=> Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx))))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx))))))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x2:(fofType->Prop))=> Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx))))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx))))))) b)
% Found x1:(P (K x0))
% Instantiate: f:=(K x0):(fofType->Prop)
% Found x1 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (x0 x)))
% Found x1:(P (K x0))
% Instantiate: f:=(K x0):(fofType->Prop)
% Found x1 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (x0 x)))
% Found classic0:=(classic ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))):((or ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))) (not ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))))
% Found (classic ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))) as proof of ((or ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))) b)
% Found (classic ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))) as proof of ((or ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))) b)
% Found (classic ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))) as proof of ((or ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))) b)
% Found (classic ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))) as proof of (P 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 (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq ((fofType->Prop)->Prop)) a) (fun (x:(fofType->Prop))=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->Prop))=> Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x1:(fofType->Prop))=> Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x1:(fofType->Prop))=> Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x1:(fofType->Prop))=> Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% 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 eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq ((fofType->Prop)->Prop)) a) (fun (x:(fofType->Prop))=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->Prop))=> Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x1:(fofType->Prop))=> Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x1:(fofType->Prop))=> Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x1:(fofType->Prop))=> Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) 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 (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% 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 (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found eta_expansion000:=(eta_expansion00 (K x0)):(((eq (fofType->Prop)) (K x0)) (fun (x:fofType)=> ((K x0) x)))
% Found (eta_expansion00 (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found ((eta_expansion0 Prop) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found (((eta_expansion fofType) Prop) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found (((eta_expansion fofType) Prop) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found (((eta_expansion fofType) Prop) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found eq_ref00:=(eq_ref0 (K x0)):(((eq (fofType->Prop)) (K x0)) (K x0))
% Found (eq_ref0 (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found ((eq_ref (fofType->Prop)) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found ((eq_ref (fofType->Prop)) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found ((eq_ref (fofType->Prop)) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found eq_ref00:=(eq_ref0 x0):(((eq (fofType->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found eq_ref00:=(eq_ref0 x0):(((eq (fofType->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found x1:(P x0)
% Instantiate: x0:=(K b):(fofType->Prop)
% Found x1 as proof of (P0 b)
% Found eta_expansion000:=(eta_expansion00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion0 Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and (forall (Xx:fofType), ((x0 Xx)->(x1 Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K x1) Xx))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (forall (Xx:fofType), ((x0 Xx)->(x1 Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K x1) Xx))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (forall (Xx:fofType), ((x0 Xx)->(x1 Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K x1) Xx))))))
% Found (fun (x1:(fofType->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (forall (Xx:fofType), ((x0 Xx)->(x1 Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K x1) Xx))))))
% Found (fun (x1:(fofType->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(fofType->Prop)), (((eq Prop) (f x)) ((and (forall (Xx:fofType), ((x0 Xx)->(x Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K x) Xx)))))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and (forall (Xx:fofType), ((x0 Xx)->(x1 Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K x1) Xx))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (forall (Xx:fofType), ((x0 Xx)->(x1 Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K x1) Xx))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (forall (Xx:fofType), ((x0 Xx)->(x1 Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K x1) Xx))))))
% Found (fun (x1:(fofType->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (forall (Xx:fofType), ((x0 Xx)->(x1 Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K x1) Xx))))))
% Found (fun (x1:(fofType->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(fofType->Prop)), (((eq Prop) (f x)) ((and (forall (Xx:fofType), ((x0 Xx)->(x Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K x) Xx)))))))
% Found classic0:=(classic ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))):((or ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))) (not ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))))
% Found (classic ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))) as proof of ((or ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))) b)
% Found (classic ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))) as proof of ((or ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))) b)
% Found (classic ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))) as proof of ((or ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))) b)
% Found (classic ((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))) as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) 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 (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) 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 eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq ((fofType->Prop)->Prop)) a) (fun (x:(fofType->Prop))=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->Prop))=> Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x1:(fofType->Prop))=> Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x1:(fofType->Prop))=> Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x1:(fofType->Prop))=> Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) 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 (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% 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 (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (K x0)):(((eq (fofType->Prop)) (K x0)) (fun (x:fofType)=> ((K x0) x)))
% Found (eta_expansion_dep00 (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found x1:(P (K x0))
% Instantiate: b:=(K x0):(fofType->Prop)
% Found x1 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found x1:(P x0)
% Instantiate: b:=x0:(fofType->Prop)
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (K x0)):(((eq (fofType->Prop)) (K x0)) (K x0))
% Found (eq_ref0 (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found ((eq_ref (fofType->Prop)) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found ((eq_ref (fofType->Prop)) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found ((eq_ref (fofType->Prop)) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx)))->False)))):(((eq ((fofType->Prop)->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx)))->False)))) (fun (x:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(x Xx)))) ((forall (Xx:fofType), (((K x0) Xx)->((K x) Xx)))->False))))
% Found (eta_expansion00 (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx)))->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx)))->False)))) b)
% Found ((eta_expansion0 Prop) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx)))->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx)))->False)))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx)))->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx)))->False)))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx)))->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx)))->False)))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx)))->False)))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx)))->False)))) b)
% Found eq_ref000:=(eq_ref00 P):((P ((K x0) x1))->(P ((K x0) x1)))
% Found (eq_ref00 P) as proof of (P0 ((K x0) x1))
% Found ((eq_ref0 ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found eq_ref000:=(eq_ref00 P):((P ((K x0) x1))->(P ((K x0) x1)))
% Found (eq_ref00 P) as proof of (P0 ((K x0) x1))
% Found ((eq_ref0 ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found eq_ref000:=(eq_ref00 P):((P ((K x0) x1))->(P ((K x0) x1)))
% Found (eq_ref00 P) as proof of (P0 ((K x0) x1))
% Found ((eq_ref0 ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found eq_ref000:=(eq_ref00 P):((P ((K x0) x1))->(P ((K x0) x1)))
% Found (eq_ref00 P) as proof of (P0 ((K x0) x1))
% Found ((eq_ref0 ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found eq_ref00:=(eq_ref0 ((K x0) x1)):(((eq Prop) ((K x0) x1)) ((K x0) x1))
% Found (eq_ref0 ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((K x0) x1)):(((eq Prop) ((K x0) x1)) ((K x0) x1))
% Found (eq_ref0 ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((K x0) x1)):(((eq Prop) ((K x0) x1)) ((K x0) x1))
% Found (eq_ref0 ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((K x0) x1)):(((eq Prop) ((K x0) x1)) ((K x0) x1))
% Found (eq_ref0 ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found eq_ref00:=(eq_ref0 x0):(((eq (fofType->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found eq_ref00:=(eq_ref0 x0):(((eq (fofType->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found x1:(P x0)
% Instantiate: x0:=(K b):(fofType->Prop)
% Found x1 as proof of (P0 b)
% Found eta_expansion000:=(eta_expansion00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion0 Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found x1:(P (K x0))
% Instantiate: f:=(K x0):(fofType->Prop)
% Found x1 as proof of (P0 f)
% Found x1:(P (K x0))
% Instantiate: f:=(K x0):(fofType->Prop)
% Found x1 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (x0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (x0 x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found x1:(P (K x0))
% Instantiate: b:=(K x0):(fofType->Prop)
% Found x1 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref (fofType->Prop)) b) P) as proof of (P0 b)
% Found (((eq_ref (fofType->Prop)) b) P) as proof of (P0 b)
% Found x1:(P x0)
% Instantiate: b:=x0:(fofType->Prop)
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (K x0)):(((eq (fofType->Prop)) (K x0)) (K x0))
% Found (eq_ref0 (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found ((eq_ref (fofType->Prop)) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found ((eq_ref (fofType->Prop)) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found ((eq_ref (fofType->Prop)) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (K x0)):(((eq (fofType->Prop)) (K x0)) (fun (x:fofType)=> ((K x0) x)))
% Found (eta_expansion_dep00 (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b0)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (fofType->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (fofType->Prop)) b0) x0)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) x0)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) x0)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) x0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx))))))):(((eq ((fofType->Prop)->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx))))))) (fun (x:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(x Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K x) Xx)))))))
% Found (eta_expansion_dep00 (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx))))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx))))))) b)
% Found ((eta_expansion_dep0 (fun (x2:(fofType->Prop))=> Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx))))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx))))))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x2:(fofType->Prop))=> Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx))))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx))))))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x2:(fofType->Prop))=> Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx))))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx))))))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x2:(fofType->Prop))=> Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx))))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((x0 Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K Y) Xx))))))) b)
% Found eq_ref00:=(eq_ref0 (K x0)):(((eq (fofType->Prop)) (K x0)) (K x0))
% Found (eq_ref0 (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found ((eq_ref (fofType->Prop)) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found ((eq_ref (fofType->Prop)) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found ((eq_ref (fofType->Prop)) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found x1:(P x0)
% Instantiate: f:=x0:(fofType->Prop)
% Found x1 as proof of (P0 f)
% Found x1:(P x0)
% Instantiate: f:=x0:(fofType->Prop)
% Found x1 as proof of (P0 f)
% Found eq_ref000:=(eq_ref00 P):((P ((K x0) x1))->(P ((K x0) x1)))
% Found (eq_ref00 P) as proof of (P0 ((K x0) x1))
% Found ((eq_ref0 ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found eq_ref000:=(eq_ref00 P):((P ((K x0) x1))->(P ((K x0) x1)))
% Found (eq_ref00 P) as proof of (P0 ((K x0) x1))
% Found ((eq_ref0 ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found eq_ref000:=(eq_ref00 P):((P ((K x0) x1))->(P ((K x0) x1)))
% Found (eq_ref00 P) as proof of (P0 ((K x0) x1))
% Found ((eq_ref0 ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found eq_ref000:=(eq_ref00 P):((P ((K x0) x1))->(P ((K x0) x1)))
% Found (eq_ref00 P) as proof of (P0 ((K x0) x1))
% Found ((eq_ref0 ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((K x0) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((K x0) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((K x0) x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((K x0) x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((K x0) x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((K x0) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((K x0) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((K x0) x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((K x0) x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((K x0) x)))
% Found eq_ref00:=(eq_ref0 ((K x0) x1)):(((eq Prop) ((K x0) x1)) ((K x0) x1))
% Found (eq_ref0 ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((K x0) x1)):(((eq Prop) ((K x0) x1)) ((K x0) x1))
% Found (eq_ref0 ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((K x0) x1)):(((eq Prop) ((K x0) x1)) ((K x0) x1))
% Found (eq_ref0 ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((K x0) x1)):(((eq Prop) ((K x0) x1)) ((K x0) x1))
% Found (eq_ref0 ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found x2:(P ((K x0) x1))
% Instantiate: b:=((K x0) x1):Prop
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found x2:(P ((K x0) x1))
% Instantiate: b:=((K x0) x1):Prop
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found x1:(P (K x0))
% Instantiate: f:=(K x0):(fofType->Prop)
% Found x1 as proof of (P0 f)
% Found x1:(P (K x0))
% Instantiate: f:=(K x0):(fofType->Prop)
% Found x1 as proof of (P0 f)
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref (fofType->Prop)) b) P) as proof of (P0 b)
% Found (((eq_ref (fofType->Prop)) b) P) as proof of (P0 b)
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref (fofType->Prop)) b) P) as proof of (P0 b)
% Found (((eq_ref (fofType->Prop)) b) P) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (x0 x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (x0 x)))
% Found eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (fofType->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (fofType->Prop)) b0) x0)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) x0)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) x0)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) x0)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found eq_ref00:=(eq_ref0 x0):(((eq (fofType->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found eq_ref00:=(eq_ref0 x0):(((eq (fofType->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found eq_ref00:=(eq_ref0 x0):(((eq (fofType->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref00:=(eq_ref0 a):(((eq ((fofType->Prop)->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))
% Found eq_ref00:=(eq_ref0 a):(((eq ((fofType->Prop)->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) ((forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))->False))))))
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref (fofType->Prop)) b) P) as proof of (P0 b)
% Found (((eq_ref (fofType->Prop)) b) P) as proof of (P0 b)
% Found x1:(P x0)
% Instantiate: x0:=(K a):(fofType->Prop)
% Found x1 as proof of (P0 a)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) x0)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) x0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) x0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) x0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) x0)
% Found x1:(P x0)
% Instantiate: x0:=(K a):(fofType->Prop)
% Found x1 as proof of (P0 (K a))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) x0)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) x0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) x0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) x0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) x0)
% Found eq_ref00:=(eq_ref0 (K x0)):(((eq (fofType->Prop)) (K x0)) (K x0))
% Found (eq_ref0 (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found ((eq_ref (fofType->Prop)) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found ((eq_ref (fofType->Prop)) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found ((eq_ref (fofType->Prop)) (K x0)) as proof of (((eq (fofType->Prop)) (K x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq ((fofType->Prop)->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) 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) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (X:(fofType->Prop))=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (forall (Xx:fofType), ((X Xx)->(Y Xx)))) (not (forall (Xx:fofType), (((K X) Xx)->((K Y) Xx)))))))))
% Found x1:(P x0)
% Instantiate: f:=x0:(fofType->Prop)
% Found x1 as proof of (P0 f)
% Found x1:(P x0)
% Instantiate: f:=x0:(fofType->Prop)
% Found x1 as proof of (P0 f)
% Found eq_ref000:=(eq_ref00 P1):((P1 x0)->(P1 x0))
% Found (eq_ref00 P1) as proof of (P2 x0)
% Found ((eq_ref0 x0) P1) as proof of (P2 x0)
% Found (((eq_ref (fofType->Prop)) x0) P1) as proof of (P2 x0)
% Found (((eq_ref (fofType->Prop)) x0) P1) as proof of (P2 x0)
% Found eq_ref000:=(eq_ref00 P1):((P1 x0)->(P1 x0))
% Found (eq_ref00 P1) as proof of (P2 x0)
% Found ((eq_ref0 x0) P1) as proof of (P2 x0)
% Found (((eq_ref (fofType->Prop)) x0) P1) as proof of (P2 x0)
% Found (((eq_ref (fofType->Prop)) x0) P1) as proof of (P2 x0)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((K x0) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((K x0) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((K x0) x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((K x0) x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((K x0) x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((K x0) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((K x0) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((K x0) x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((K x0) x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((K x0) x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and (forall (Xx:fofType), ((x0 Xx)->(x1 Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K x1) Xx))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (forall (Xx:fofType), ((x0 Xx)->(x1 Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K x1) Xx))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (forall (Xx:fofType), ((x0 Xx)->(x1 Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K x1) Xx))))))
% Found (fun (x1:(fofType->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (forall (Xx:fofType), ((x0 Xx)->(x1 Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K x1) Xx))))))
% Found (fun (x1:(fofType->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(fofType->Prop)), (((eq Prop) (f x)) ((and (forall (Xx:fofType), ((x0 Xx)->(x Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K x) Xx)))))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and (forall (Xx:fofType), ((x0 Xx)->(x1 Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K x1) Xx))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (forall (Xx:fofType), ((x0 Xx)->(x1 Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K x1) Xx))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (forall (Xx:fofType), ((x0 Xx)->(x1 Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K x1) Xx))))))
% Found (fun (x1:(fofType->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (forall (Xx:fofType), ((x0 Xx)->(x1 Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K x1) Xx))))))
% Found (fun (x1:(fofType->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(fofType->Prop)), (((eq Prop) (f x)) ((and (forall (Xx:fofType), ((x0 Xx)->(x Xx)))) (not (forall (Xx:fofType), (((K x0) Xx)->((K x) Xx)))))))
% Found x1:(P x0)
% Instantiate: x0:=(K b):(fofType->Prop)
% Found x1 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found eq_ref00:=(eq_ref0 x0):(((eq (fofType->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found eq_ref00:=(eq_ref0 x0):(((eq (fofType->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found x2:(P ((K x0) x1))
% Instantiate: b:=((K x0) x1):Prop
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found x2:(P ((K x0) x1))
% Instantiate: b:=((K x0) x1):Prop
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 x0):(((eq (fofType->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref000:=(eq_ref00 P):((P ((K x0) x1))->(P ((K x0) x1)))
% Found (eq_ref00 P) as proof of (P0 ((K x0) x1))
% Found ((eq_ref0 ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found eq_ref000:=(eq_ref00 P):((P ((K x0) x1))->(P ((K x0) x1)))
% Found (eq_ref00 P) as proof of (P0 ((K x0) x1))
% Found ((eq_ref0 ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P) as proof of (P0 ((K x0) x1))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((K x0) x1))->(P1 ((K x0) x1)))
% Found (eq_ref00 P1) as proof of (P2 ((K x0) x1))
% Found ((eq_ref0 ((K x0) x1)) P1) as proof of (P2 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P1) as proof of (P2 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P1) as proof of (P2 ((K x0) x1))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((K x0) x1))->(P1 ((K x0) x1)))
% Found (eq_ref00 P1) as proof of (P2 ((K x0) x1))
% Found ((eq_ref0 ((K x0) x1)) P1) as proof of (P2 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P1) as proof of (P2 ((K x0) x1))
% Found (((eq_ref Prop) ((K x0) x1)) P1) as proof of (P2 ((K x0) x1))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P b)->(P (fun (x:fofType)=> (b x))))
% Found (eta_expansion_dep000 P) as proof of (P0 b)
% Found ((eta_expansion_dep00 b) P) as proof of (P0 b)
% Found (((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) b) P) as proof of (P0 b)
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) P) as proof of (P0 b)
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) P) as proof of (P0 b)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P b)->(P (fun (x:fofType)=> (b x))))
% Found (eta_expansion_dep000 P) as proof of (P0 b)
% Found ((eta_expansion_dep00 b) P) as proof of (P0 b)
% Found (((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) b) P) as proof of (P0 b)
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) P) as proof of (P0 b)
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) P) as proof of (P0 b)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found eq_ref00:=(eq_ref0 ((K x0) x1)):(((eq Prop) ((K x0) x1)) ((K x0) x1))
% Found (eq_ref0 ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((K x0) x1)):(((eq Prop) ((K x0) x1)) ((K x0) x1))
% Found (eq_ref0 ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((K x0) x1)):(((eq Prop) ((K x0) x1)) ((K x0) x1))
% Found (eq_ref0 ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((K x0) x1)):(((eq Prop) ((K x0) x1)) ((K x0) x1))
% Found (eq_ref0 ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((K x0) x1)):(((eq Prop) ((K x0) x1)) ((K x0) x1))
% Found (eq_ref0 ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((K x0) x1)):(((eq Prop) ((K x0) x1)) ((K x0) x1))
% Found (eq_ref0 ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found ((eq_ref Prop) ((K x0) x1)) as proof of (((eq Prop) ((K x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found eq_ref00:=(eq_ref0 x0):(((eq (fofType->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found eq_ref00:=(eq_ref0 x0):(((eq (fofType->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x0))
% Fou
% EOF
%------------------------------------------------------------------------------