%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV308^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n093.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 300.02s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV308^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n093.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 08:47:51 CDT 2014
% % CPUTime  : 300.02 
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (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 cTHM1A_pme
% Conjecture to prove = (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 (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 (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 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 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 ((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 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 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x))
% Found (fun (x:(fofType->Prop))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x))
% Found (fun (x:(fofType->Prop))=> ((eq_ref Prop) (f x))) as proof of (forall (x:(fofType->Prop)), (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x)))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x))
% Found (fun (x:(fofType->Prop))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x))
% Found (fun (x:(fofType->Prop))=> ((eq_ref Prop) (f x))) 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 x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x))
% Found (fun (x:(fofType->Prop))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x))
% Found (fun (x:(fofType->Prop))=> ((eq_ref Prop) (f x))) as proof of (forall (x:(fofType->Prop)), (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x)))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x))
% Found (fun (x:(fofType->Prop))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x))
% Found (fun (x:(fofType->Prop))=> ((eq_ref Prop) (f x))) 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 (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 eta_expansion000:=(eta_expansion00 (K x)):(((eq (fofType->Prop)) (K x)) (fun (x0:fofType)=> ((K x) x0)))
% Found (eta_expansion00 (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found ((eta_expansion0 Prop) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found (((eta_expansion fofType) Prop) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found (((eta_expansion fofType) Prop) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found (((eta_expansion fofType) Prop) (K x)) as proof of (((eq (fofType->Prop)) (K x)) 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) x)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% 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 eq_ref00:=(eq_ref0 (K x)):(((eq (fofType->Prop)) (K x)) (K x))
% Found (eq_ref0 (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found ((eq_ref (fofType->Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found ((eq_ref (fofType->Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found ((eq_ref (fofType->Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) 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) x)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x))
% Found (fun (x:(fofType->Prop))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x))
% Found (fun (x:(fofType->Prop))=> ((eq_ref Prop) (f x))) as proof of (forall (x:(fofType->Prop)), (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x)))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x))
% Found (fun (x:(fofType->Prop))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x))
% Found (fun (x:(fofType->Prop))=> ((eq_ref Prop) (f x))) as proof of (forall (x:(fofType->Prop)), (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x)))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x))
% Found (fun (x:(fofType->Prop))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x))
% Found (fun (x:(fofType->Prop))=> ((eq_ref Prop) (f x))) as proof of (forall (x:(fofType->Prop)), (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x)))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x))
% Found (fun (x:(fofType->Prop))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x))
% Found (fun (x:(fofType->Prop))=> ((eq_ref Prop) (f x))) as proof of (forall (x:(fofType->Prop)), (((eq Prop) (f x)) (((eq (fofType->Prop)) (K x)) x)))
% Found eq_ref00:=(eq_ref0 (K x)):(((eq (fofType->Prop)) (K x)) (K x))
% Found (eq_ref0 (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found ((eq_ref (fofType->Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found ((eq_ref (fofType->Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found ((eq_ref (fofType->Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) 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) x)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found eta_expansion000:=(eta_expansion00 x):(((eq (fofType->Prop)) x) (fun (x0:fofType)=> (x x0)))
% Found (eta_expansion00 x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eta_expansion0 Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion fofType) Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion fofType) Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion fofType) Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found eq_ref00:=(eq_ref0 (K x)):(((eq (fofType->Prop)) (K x)) (K x))
% Found (eq_ref0 (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found ((eq_ref (fofType->Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found ((eq_ref (fofType->Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found ((eq_ref (fofType->Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) 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) x)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found eta_expansion000:=(eta_expansion00 x):(((eq (fofType->Prop)) x) (fun (x0:fofType)=> (x x0)))
% Found (eta_expansion00 x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eta_expansion0 Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion fofType) Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion fofType) Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion fofType) Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found x01:(P x)
% Found (fun (x01:(P x))=> x01) as proof of (P x)
% Found (fun (x01:(P x))=> x01) as proof of (P0 x)
% Found x01:(P x)
% Found (fun (x01:(P x))=> x01) as proof of (P x)
% Found (fun (x01:(P x))=> x01) as proof of (P0 x)
% Found x11:(P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P0 ((K x) x0))
% Found x11:(P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P0 ((K x) x0))
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found x01:(P x)
% Found (fun (x01:(P x))=> x01) as proof of (P x)
% Found (fun (x01:(P x))=> x01) as proof of (P0 x)
% Found x01:(P x)
% Found (fun (x01:(P x))=> x01) as proof of (P x)
% Found (fun (x01:(P x))=> x01) as proof of (P0 x)
% Found x11:(P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P0 ((K x) x0))
% Found x11:(P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P0 ((K x) x0))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found eq_ref00:=(eq_ref0 x):(((eq (fofType->Prop)) x) x)
% Found (eq_ref0 x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found eq_ref00:=(eq_ref0 x):(((eq (fofType->Prop)) x) x)
% Found (eq_ref0 x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found x01:(P x)
% Found (fun (x01:(P x))=> x01) as proof of (P x)
% Found (fun (x01:(P x))=> x01) as proof of (P0 x)
% Found x01:(P x)
% Found (fun (x01:(P x))=> x01) as proof of (P x)
% Found (fun (x01:(P x))=> x01) as proof of (P0 x)
% Found eq_ref00:=(eq_ref0 (x x0)):(((eq Prop) (x x0)) (x x0))
% Found (eq_ref0 (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found eq_ref00:=(eq_ref0 (x x0)):(((eq Prop) (x x0)) (x x0))
% Found (eq_ref0 (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found eq_ref00:=(eq_ref0 (x x0)):(((eq Prop) (x x0)) (x x0))
% Found (eq_ref0 (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found eq_ref00:=(eq_ref0 (x x0)):(((eq Prop) (x x0)) (x x0))
% Found (eq_ref0 (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found x11:(P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P0 ((K x) x0))
% Found x11:(P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P0 ((K x) x0))
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found x01:(P x)
% Found (fun (x01:(P x))=> x01) as proof of (P x)
% Found (fun (x01:(P x))=> x01) as proof of (P0 x)
% Found x01:(P x)
% Found (fun (x01:(P x))=> x01) as proof of (P x)
% Found (fun (x01:(P x))=> x01) as proof of (P0 x)
% Found x11:(P (x x0))
% Found (fun (x11:(P (x x0)))=> x11) as proof of (P (x x0))
% Found (fun (x11:(P (x x0)))=> x11) as proof of (P0 (x x0))
% Found x11:(P (x x0))
% Found (fun (x11:(P (x x0)))=> x11) as proof of (P (x x0))
% Found (fun (x11:(P (x x0)))=> x11) as proof of (P0 (x x0))
% Found eq_ref00:=(eq_ref0 (x x0)):(((eq Prop) (x x0)) (x x0))
% Found (eq_ref0 (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found eq_ref00:=(eq_ref0 (x x0)):(((eq Prop) (x x0)) (x x0))
% Found (eq_ref0 (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found eq_ref00:=(eq_ref0 (x x0)):(((eq Prop) (x x0)) (x x0))
% Found (eq_ref0 (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found eq_ref00:=(eq_ref0 (x x0)):(((eq Prop) (x x0)) (x x0))
% Found (eq_ref0 (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found x11:(P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P0 ((K x) x0))
% Found x11:(P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P0 ((K x) x0))
% Found x11:(P (x x0))
% Found (fun (x11:(P (x x0)))=> x11) as proof of (P (x x0))
% Found (fun (x11:(P (x x0)))=> x11) as proof of (P0 (x x0))
% Found x11:(P (x x0))
% Found (fun (x11:(P (x x0)))=> x11) as proof of (P (x x0))
% Found (fun (x11:(P (x x0)))=> x11) as proof of (P0 (x x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found eq_ref00:=(eq_ref0 (x x0)):(((eq Prop) (x x0)) (x x0))
% Found (eq_ref0 (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found eq_ref00:=(eq_ref0 (x x0)):(((eq Prop) (x x0)) (x x0))
% Found (eq_ref0 (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found eq_ref00:=(eq_ref0 (x x0)):(((eq Prop) (x x0)) (x x0))
% Found (eq_ref0 (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found eq_ref00:=(eq_ref0 (x x0)):(((eq Prop) (x x0)) (x x0))
% Found (eq_ref0 (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found x11:(P (x x0))
% Found (fun (x11:(P (x x0)))=> x11) as proof of (P (x x0))
% Found (fun (x11:(P (x x0)))=> x11) as proof of (P0 (x x0))
% Found x11:(P (x x0))
% Found (fun (x11:(P (x x0)))=> x11) as proof of (P (x x0))
% Found (fun (x11:(P (x x0)))=> x11) as proof of (P0 (x x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found eq_ref00:=(eq_ref0 (x x0)):(((eq Prop) (x x0)) (x x0))
% Found (eq_ref0 (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found eq_ref00:=(eq_ref0 (x x0)):(((eq Prop) (x x0)) (x x0))
% Found (eq_ref0 (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found eq_ref00:=(eq_ref0 (x x0)):(((eq Prop) (x x0)) (x x0))
% Found (eq_ref0 (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found eq_ref00:=(eq_ref0 (x x0)):(((eq Prop) (x x0)) (x x0))
% Found (eq_ref0 (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found x11:(P (x x0))
% Found (fun (x11:(P (x x0)))=> x11) as proof of (P (x x0))
% Found (fun (x11:(P (x x0)))=> x11) as proof of (P0 (x x0))
% Found x11:(P (x x0))
% Found (fun (x11:(P (x x0)))=> x11) as proof of (P (x x0))
% Found (fun (x11:(P (x x0)))=> x11) as proof of (P0 (x 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 (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 eq_ref00:=(eq_ref0 (K x)):(((eq (fofType->Prop)) (K x)) (K x))
% Found (eq_ref0 (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found ((eq_ref (fofType->Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found ((eq_ref (fofType->Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found ((eq_ref (fofType->Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) 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) x)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((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 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 eta_expansion_dep000:=(eta_expansion_dep00 (K x)):(((eq (fofType->Prop)) (K x)) (fun (x0:fofType)=> ((K x) x0)))
% Found (eta_expansion_dep00 (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) 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) x)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% 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 x0:(P (K x))
% Instantiate: b:=(K x):(fofType->Prop)
% Found x0 as proof of (P0 b)
% Found eta_expansion000:=(eta_expansion00 x):(((eq (fofType->Prop)) x) (fun (x0:fofType)=> (x x0)))
% Found (eta_expansion00 x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eta_expansion0 Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion fofType) Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion fofType) Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion fofType) Prop) x) as proof of (((eq (fofType->Prop)) x) 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 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 Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found x0:(P (K x))
% Instantiate: f:=(K x):(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x00)):(((eq Prop) (f x00)) (f x00))
% Found (eq_ref0 (f x00)) as proof of (((eq Prop) (f x00)) (x x00))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) (x x00))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) (x x00))
% Found (fun (x00:fofType)=> ((eq_ref Prop) (f x00))) as proof of (((eq Prop) (f x00)) (x x00))
% Found (fun (x00:fofType)=> ((eq_ref Prop) (f x00))) as proof of (forall (x0:fofType), (((eq Prop) (f x0)) (x x0)))
% Found x0:(P (K x))
% Instantiate: f:=(K x):(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x00)):(((eq Prop) (f x00)) (f x00))
% Found (eq_ref0 (f x00)) as proof of (((eq Prop) (f x00)) (x x00))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) (x x00))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) (x x00))
% Found (fun (x00:fofType)=> ((eq_ref Prop) (f x00))) as proof of (((eq Prop) (f x00)) (x x00))
% Found (fun (x00:fofType)=> ((eq_ref Prop) (f x00))) as proof of (forall (x0:fofType), (((eq Prop) (f x0)) (x x0)))
% Found x0:(P (K x))
% Instantiate: b:=(K x):(fofType->Prop)
% Found x0 as proof of (P0 b)
% Found eta_expansion000:=(eta_expansion00 x):(((eq (fofType->Prop)) x) (fun (x0:fofType)=> (x x0)))
% Found (eta_expansion00 x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eta_expansion0 Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion fofType) Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion fofType) Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion fofType) Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found eq_ref00:=(eq_ref0 (K x)):(((eq (fofType->Prop)) (K x)) (K x))
% Found (eq_ref0 (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found ((eq_ref (fofType->Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found ((eq_ref (fofType->Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found ((eq_ref (fofType->Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) 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) x)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% Found x0:(P (K x))
% Instantiate: f:=(K x):(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x00)):(((eq Prop) (f x00)) (f x00))
% Found (eq_ref0 (f x00)) as proof of (((eq Prop) (f x00)) (x x00))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) (x x00))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) (x x00))
% Found (fun (x00:fofType)=> ((eq_ref Prop) (f x00))) as proof of (((eq Prop) (f x00)) (x x00))
% Found (fun (x00:fofType)=> ((eq_ref Prop) (f x00))) as proof of (forall (x0:fofType), (((eq Prop) (f x0)) (x x0)))
% Found x0:(P (K x))
% Instantiate: f:=(K x):(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x00)):(((eq Prop) (f x00)) (f x00))
% Found (eq_ref0 (f x00)) as proof of (((eq Prop) (f x00)) (x x00))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) (x x00))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) (x x00))
% Found (fun (x00:fofType)=> ((eq_ref Prop) (f x00))) as proof of (((eq Prop) (f x00)) (x x00))
% Found (fun (x00:fofType)=> ((eq_ref Prop) (f x00))) as proof of (forall (x0:fofType), (((eq Prop) (f x0)) (x x0)))
% 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_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 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 x)):(((eq (fofType->Prop)) (K x)) (fun (x0:fofType)=> ((K x) x0)))
% Found (eta_expansion00 (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found ((eta_expansion0 Prop) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found (((eta_expansion fofType) Prop) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found (((eta_expansion fofType) Prop) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found (((eta_expansion fofType) Prop) (K x)) as proof of (((eq (fofType->Prop)) (K x)) 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) x)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% Found eta_expansion000:=(eta_expansion00 (K x)):(((eq (fofType->Prop)) (K x)) (fun (x0:fofType)=> ((K x) x0)))
% Found (eta_expansion00 (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found ((eta_expansion0 Prop) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found (((eta_expansion fofType) Prop) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found (((eta_expansion fofType) Prop) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found (((eta_expansion fofType) Prop) (K x)) as proof of (((eq (fofType->Prop)) (K x)) 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) x)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found eta_expansion000:=(eta_expansion00 x):(((eq (fofType->Prop)) x) (fun (x0:fofType)=> (x x0)))
% Found (eta_expansion00 x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eta_expansion0 Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion fofType) Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion fofType) Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion fofType) Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found eta_expansion000:=(eta_expansion00 x):(((eq (fofType->Prop)) x) (fun (x0:fofType)=> (x x0)))
% Found (eta_expansion00 x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eta_expansion0 Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion fofType) Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion fofType) Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion fofType) Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found x0:(P x)
% Instantiate: x:=(K b):(fofType->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 x):(((eq (fofType->Prop)) x) x)
% Found (eq_ref0 x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% 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 x):(((eq (fofType->Prop)) x) (fun (x0:fofType)=> (x x0)))
% Found (eta_expansion_dep00 x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) x) as proof of (((eq (fofType->Prop)) x) 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_expansion000:=(eta_expansion00 b):(((eq ((fofType->Prop)->Prop)) b) (fun (x:(fofType->Prop))=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found ((eta_expansion0 Prop) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (K U)) U)))
% Found (((eta_expansion (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 (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 (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 (K x)):(((eq (fofType->Prop)) (K x)) (K x))
% Found (eq_ref0 (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found ((eq_ref (fofType->Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found ((eq_ref (fofType->Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found ((eq_ref (fofType->Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) 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) x)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x)
% Found x0:(P (K x))
% Instantiate: b:=(K x):(fofType->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 x):(((eq (fofType->Prop)) x) x)
% Found (eq_ref0 x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found x0:(P x)
% Instantiate: b:=x:(fofType->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (K x)):(((eq (fofType->Prop)) (K x)) (K x))
% Found (eq_ref0 (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found ((eq_ref (fofType->Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found ((eq_ref (fofType->Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found ((eq_ref (fofType->Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found eta_expansion000:=(eta_expansion00 x):(((eq (fofType->Prop)) x) (fun (x0:fofType)=> (x x0)))
% Found (eta_expansion00 x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eta_expansion0 Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion fofType) Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion fofType) Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion fofType) Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found eta_expansion000:=(eta_expansion00 x):(((eq (fofType->Prop)) x) (fun (x0:fofType)=> (x x0)))
% Found (eta_expansion00 x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eta_expansion0 Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion fofType) Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion fofType) Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion fofType) Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found x0:(P x)
% Instantiate: x:=(K b):(fofType->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 x):(((eq (fofType->Prop)) x) x)
% Found (eq_ref0 x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found x0:(P (K x))
% Instantiate: f:=(K x):(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x00)):(((eq Prop) (f x00)) (f x00))
% Found (eq_ref0 (f x00)) as proof of (((eq Prop) (f x00)) (x x00))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) (x x00))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) (x x00))
% Found (fun (x00:fofType)=> ((eq_ref Prop) (f x00))) as proof of (((eq Prop) (f x00)) (x x00))
% Found (fun (x00:fofType)=> ((eq_ref Prop) (f x00))) as proof of (forall (x0:fofType), (((eq Prop) (f x0)) (x x0)))
% Found x0:(P (K x))
% Instantiate: f:=(K x):(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x00)):(((eq Prop) (f x00)) (f x00))
% Found (eq_ref0 (f x00)) as proof of (((eq Prop) (f x00)) (x x00))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) (x x00))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) (x x00))
% Found (fun (x00:fofType)=> ((eq_ref Prop) (f x00))) as proof of (((eq Prop) (f x00)) (x x00))
% Found (fun (x00:fofType)=> ((eq_ref Prop) (f x00))) as proof of (forall (x0:fofType), (((eq Prop) (f x0)) (x x0)))
% Found eta_expansion000:=(eta_expansion00 x):(((eq (fofType->Prop)) x) (fun (x0:fofType)=> (x x0)))
% Found (eta_expansion00 x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eta_expansion0 Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion fofType) Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion fofType) Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion fofType) Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (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) x)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) x)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) x)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) x)
% Found x01:(P x)
% Found (fun (x01:(P x))=> x01) as proof of (P x)
% Found (fun (x01:(P x))=> x01) as proof of (P0 x)
% Found x01:(P x)
% Found (fun (x01:(P x))=> x01) as proof of (P x)
% Found (fun (x01:(P x))=> x01) as proof of (P0 x)
% Found x01:(P x)
% Found (fun (x01:(P x))=> x01) as proof of (P x)
% Found (fun (x01:(P x))=> x01) as proof of (P0 x)
% Found x01:(P x)
% Found (fun (x01:(P x))=> x01) as proof of (P x)
% Found (fun (x01:(P x))=> x01) as proof of (P0 x)
% Found x0:(P (K x))
% Instantiate: b:=(K x):(fofType->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 x):(((eq (fofType->Prop)) x) x)
% Found (eq_ref0 x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found x0:(P x)
% Instantiate: b:=x:(fofType->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (K x)):(((eq (fofType->Prop)) (K x)) (K x))
% Found (eq_ref0 (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found ((eq_ref (fofType->Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found ((eq_ref (fofType->Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found ((eq_ref (fofType->Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (K x)):(((eq (fofType->Prop)) (K x)) (fun (x0:fofType)=> ((K x) x0)))
% Found (eta_expansion_dep00 (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) x)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x)
% Found x11:(P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P0 ((K x) x0))
% Found x11:(P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P0 ((K x) x0))
% Found x11:(P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P0 ((K x) x0))
% Found x11:(P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P0 ((K x) x0))
% Found x0:(P x)
% Instantiate: f:=x:(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P x)
% Instantiate: f:=x:(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x00)):(((eq Prop) (f x00)) (f x00))
% Found (eq_ref0 (f x00)) as proof of (((eq Prop) (f x00)) ((K x) x00))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((K x) x00))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((K x) x00))
% Found (fun (x00:fofType)=> ((eq_ref Prop) (f x00))) as proof of (((eq Prop) (f x00)) ((K x) x00))
% Found (fun (x00:fofType)=> ((eq_ref Prop) (f x00))) as proof of (forall (x0:fofType), (((eq Prop) (f x0)) ((K x) x0)))
% Found eq_ref00:=(eq_ref0 (f x00)):(((eq Prop) (f x00)) (f x00))
% Found (eq_ref0 (f x00)) as proof of (((eq Prop) (f x00)) ((K x) x00))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((K x) x00))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((K x) x00))
% Found (fun (x00:fofType)=> ((eq_ref Prop) (f x00))) as proof of (((eq Prop) (f x00)) ((K x) x00))
% Found (fun (x00:fofType)=> ((eq_ref Prop) (f x00))) as proof of (forall (x0:fofType), (((eq Prop) (f x0)) ((K x) x0)))
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found x1:(P ((K x) x0))
% Instantiate: b:=((K x) x0):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (x x0)):(((eq Prop) (x x0)) (x x0))
% Found (eq_ref0 (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found x1:(P ((K x) x0))
% Instantiate: b:=((K x) x0):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (x x0)):(((eq Prop) (x x0)) (x x0))
% Found (eq_ref0 (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found x0:(P (K x))
% Instantiate: f:=(K x):(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P (K x))
% Instantiate: f:=(K x):(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x00)):(((eq Prop) (f x00)) (f x00))
% Found (eq_ref0 (f x00)) as proof of (((eq Prop) (f x00)) (x x00))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) (x x00))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) (x x00))
% Found (fun (x00:fofType)=> ((eq_ref Prop) (f x00))) as proof of (((eq Prop) (f x00)) (x x00))
% Found (fun (x00:fofType)=> ((eq_ref Prop) (f x00))) as proof of (forall (x0:fofType), (((eq Prop) (f x0)) (x x0)))
% Found eq_ref00:=(eq_ref0 (f x00)):(((eq Prop) (f x00)) (f x00))
% Found (eq_ref0 (f x00)) as proof of (((eq Prop) (f x00)) (x x00))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) (x x00))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) (x x00))
% Found (fun (x00:fofType)=> ((eq_ref Prop) (f x00))) as proof of (((eq Prop) (f x00)) (x x00))
% Found (fun (x00:fofType)=> ((eq_ref Prop) (f x00))) as proof of (forall (x0:fofType), (((eq Prop) (f x0)) (x x0)))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found eta_expansion000:=(eta_expansion00 x):(((eq (fofType->Prop)) x) (fun (x0:fofType)=> (x x0)))
% Found (eta_expansion00 x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eta_expansion0 Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion fofType) Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion fofType) Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion fofType) Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found eq_ref00:=(eq_ref0 x):(((eq (fofType->Prop)) x) x)
% Found (eq_ref0 x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) 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 x01:(P x)
% Found (fun (x01:(P x))=> x01) as proof of (P x)
% Found (fun (x01:(P x))=> x01) as proof of (P0 x)
% Found x01:(P x)
% Found (fun (x01:(P x))=> x01) as proof of (P x)
% Found (fun (x01:(P x))=> x01) as proof of (P0 x)
% Found x01:(P x)
% Found (fun (x01:(P x))=> x01) as proof of (P x)
% Found (fun (x01:(P x))=> x01) as proof of (P0 x)
% Found x01:(P x)
% Found (fun (x01:(P x))=> x01) as proof of (P x)
% Found (fun (x01:(P x))=> x01) as proof of (P0 x)
% Found x01:(P b)
% Found (fun (x01:(P b))=> x01) as proof of (P b)
% Found (fun (x01:(P b))=> x01) as proof of (P0 b)
% Found x0:(P x)
% Instantiate: x:=(K a):(fofType->Prop)
% Found x0 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) x)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) x)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) x)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) x)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) x)
% Found x0:(P x)
% Instantiate: x:=(K a):(fofType->Prop)
% Found x0 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) x)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) x)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) x)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) x)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) x)
% Found eq_ref00:=(eq_ref0 (K x)):(((eq (fofType->Prop)) (K x)) (K x))
% Found (eq_ref0 (K x)) as proof of (((eq (fofType->Prop)) (K x)) b0)
% Found ((eq_ref (fofType->Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b0)
% Found ((eq_ref (fofType->Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b0)
% Found ((eq_ref (fofType->Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (fofType->Prop)) b0) (fun (x:fofType)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (fofType->Prop)) b0) x)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (fofType->Prop)) b0) x)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) x)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) x)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) x)
% Found eta_expansion000:=(eta_expansion00 (K x)):(((eq (fofType->Prop)) (K x)) (fun (x0:fofType)=> ((K x) x0)))
% Found (eta_expansion00 (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found ((eta_expansion0 Prop) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found (((eta_expansion fofType) Prop) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found (((eta_expansion fofType) Prop) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found (((eta_expansion fofType) Prop) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) x)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) 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 x11:(P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P0 ((K x) x0))
% Found x11:(P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P0 ((K x) 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 x0:(P x)
% Instantiate: f:=x:(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found x11:(P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P0 ((K x) x0))
% Found x0:(P x)
% Instantiate: f:=x:(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found x11:(P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P0 ((K x) x0))
% Found eq_ref00:=(eq_ref0 (f x00)):(((eq Prop) (f x00)) (f x00))
% Found (eq_ref0 (f x00)) as proof of (((eq Prop) (f x00)) ((K x) x00))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((K x) x00))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((K x) x00))
% Found (fun (x00:fofType)=> ((eq_ref Prop) (f x00))) as proof of (((eq Prop) (f x00)) ((K x) x00))
% Found (fun (x00:fofType)=> ((eq_ref Prop) (f x00))) as proof of (forall (x0:fofType), (((eq Prop) (f x0)) ((K x) x0)))
% Found eq_ref00:=(eq_ref0 (f x00)):(((eq Prop) (f x00)) (f x00))
% Found (eq_ref0 (f x00)) as proof of (((eq Prop) (f x00)) ((K x) x00))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((K x) x00))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((K x) x00))
% Found (fun (x00:fofType)=> ((eq_ref Prop) (f x00))) as proof of (((eq Prop) (f x00)) ((K x) x00))
% Found (fun (x00:fofType)=> ((eq_ref Prop) (f x00))) as proof of (forall (x0:fofType), (((eq Prop) (f x0)) ((K x) x0)))
% Found x0:(P x)
% Instantiate: x:=(K b):(fofType->Prop)
% Found x0 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 x):(((eq (fofType->Prop)) x) (fun (x0:fofType)=> (x x0)))
% Found (eta_expansion_dep00 x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found eq_ref00:=(eq_ref0 x):(((eq (fofType->Prop)) x) x)
% Found (eq_ref0 x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found eta_expansion000:=(eta_expansion00 x):(((eq (fofType->Prop)) x) (fun (x0:fofType)=> (x x0)))
% Found (eta_expansion00 x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eta_expansion0 Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion fofType) Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion fofType) Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion fofType) Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found x1:(P ((K x) x0))
% Instantiate: b:=((K x) x0):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (x x0)):(((eq Prop) (x x0)) (x x0))
% Found (eq_ref0 (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found x1:(P ((K x) x0))
% Instantiate: b:=((K x) x0):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (x x0)):(((eq Prop) (x x0)) (x x0))
% Found (eq_ref0 (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 x):(((eq (fofType->Prop)) x) (fun (x0:fofType)=> (x x0)))
% Found (eta_expansion_dep00 x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found x01:(P b)
% Found (fun (x01:(P b))=> x01) as proof of (P b)
% Found (fun (x01:(P b))=> x01) as proof of (P0 b)
% Found x01:(P b)
% Found (fun (x01:(P b))=> x01) as proof of (P b)
% Found (fun (x01:(P b))=> x01) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found eq_ref00:=(eq_ref0 x):(((eq (fofType->Prop)) x) x)
% Found (eq_ref0 x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found eq_ref00:=(eq_ref0 x):(((eq (fofType->Prop)) x) x)
% Found (eq_ref0 x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) 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 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 x01:(P b)
% Found (fun (x01:(P b))=> x01) as proof of (P b)
% Found (fun (x01:(P b))=> x01) as proof of (P0 b)
% Found x0:(P x)
% Instantiate: b:=x:(fofType->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (K x)):(((eq (fofType->Prop)) (K x)) (K x))
% Found (eq_ref0 (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found ((eq_ref (fofType->Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found ((eq_ref (fofType->Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found ((eq_ref (fofType->Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found x02:(P x)
% Found (fun (x02:(P x))=> x02) as proof of (P x)
% Found (fun (x02:(P x))=> x02) as proof of (P0 x)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found eq_ref00:=(eq_ref0 x):(((eq (fofType->Prop)) x) x)
% Found (eq_ref0 x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found x0:(P x)
% Instantiate: x:=(K a):(fofType->Prop)
% Found x0 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) x)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) x)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) x)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) x)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) x)
% Found x0:(P x)
% Instantiate: x:=(K a):(fofType->Prop)
% Found x0 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) x)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) x)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) x)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) x)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) x)
% Found x01:(P1 x)
% Found (fun (x01:(P1 x))=> x01) as proof of (P1 x)
% Found (fun (x01:(P1 x))=> x01) as proof of (P2 x)
% Found x01:(P1 x)
% Found (fun (x01:(P1 x))=> x01) as proof of (P1 x)
% Found (fun (x01:(P1 x))=> x01) as proof of (P2 x)
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found x0:(P x)
% Instantiate: x:=(K b):(fofType->Prop)
% Found x0 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 x):(((eq (fofType->Prop)) x) (fun (x0:fofType)=> (x x0)))
% Found (eta_expansion_dep00 x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found eq_ref00:=(eq_ref0 x):(((eq (fofType->Prop)) x) x)
% Found (eq_ref0 x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found eq_ref00:=(eq_ref0 x):(((eq (fofType->Prop)) x) x)
% Found (eq_ref0 x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found eta_expansion000:=(eta_expansion00 x):(((eq (fofType->Prop)) x) (fun (x0:fofType)=> (x x0)))
% Found (eta_expansion00 x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eta_expansion0 Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion fofType) Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion fofType) Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion fofType) Prop) x) as proof of (((eq (fofType->Prop)) x) b)
% Found x11:(P1 ((K x) x0))
% Found (fun (x11:(P1 ((K x) x0)))=> x11) as proof of (P1 ((K x) x0))
% Found (fun (x11:(P1 ((K x) x0)))=> x11) as proof of (P2 ((K x) x0))
% Found x11:(P1 ((K x) x0))
% Found (fun (x11:(P1 ((K x) x0)))=> x11) as proof of (P1 ((K x) x0))
% Found (fun (x11:(P1 ((K x) x0)))=> x11) as proof of (P2 ((K x) x0))
% Found x01:(P b)
% Found (fun (x01:(P b))=> x01) as proof of (P b)
% Found (fun (x01:(P b))=> x01) as proof of (P0 b)
% Found x01:(P b)
% Found (fun (x01:(P b))=> x01) as proof of (P b)
% Found (fun (x01:(P b))=> x01) as proof of (P0 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) (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 eq_ref00:=(eq_ref0 b0):(((eq (fofType->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found eq_ref00:=(eq_ref0 x):(((eq (fofType->Prop)) x) x)
% Found (eq_ref0 x) as proof of (((eq (fofType->Prop)) x) b0)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b0)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b0)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b0)
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found x01:(P x)
% Found (fun (x01:(P x))=> x01) as proof of (P x)
% Found (fun (x01:(P x))=> x01) as proof of (P0 x)
% Found x01:(P x)
% Found (fun (x01:(P x))=> x01) as proof of (P x)
% Found (fun (x01:(P x))=> x01) as proof of (P0 x)
% Found x01:(P x)
% Found (fun (x01:(P x))=> x01) as proof of (P x)
% Found (fun (x01:(P x))=> x01) as proof of (P0 x)
% Found x01:(P x)
% Found (fun (x01:(P x))=> x01) as proof of (P x)
% Found (fun (x01:(P x))=> x01) as proof of (P0 x)
% Found eq_ref00:=(eq_ref0 (K x)):(((eq (fofType->Prop)) (K x)) (K x))
% Found (eq_ref0 (K x)) as proof of (((eq (fofType->Prop)) (K x)) b0)
% Found ((eq_ref (fofType->Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b0)
% Found ((eq_ref (fofType->Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b0)
% Found ((eq_ref (fofType->Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (fofType->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (fofType->Prop)) b0) x)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) x)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) x)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) x)
% Found x12:(P ((K x) x0))
% Found (fun (x12:(P ((K x) x0)))=> x12) as proof of (P ((K x) x0))
% Found (fun (x12:(P ((K x) x0)))=> x12) as proof of (P0 ((K x) x0))
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found x12:(P ((K x) x0))
% Found (fun (x12:(P ((K x) x0)))=> x12) as proof of (P ((K x) x0))
% Found (fun (x12:(P ((K x) x0)))=> x12) as proof of (P0 ((K x) x0))
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found x0:(P x)
% Instantiate: b:=x:(fofType->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (K x)):(((eq (fofType->Prop)) (K x)) (K x))
% Found (eq_ref0 (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found ((eq_ref (fofType->Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found ((eq_ref (fofType->Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found ((eq_ref (fofType->Prop)) (K x)) as proof of (((eq (fofType->Prop)) (K x)) b)
% Found x02:(P x)
% Found (fun (x02:(P x))=> x02) as proof of (P x)
% Found (fun (x02:(P x))=> x02) as proof of (P0 x)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K x))
% Found eq_ref00:=(eq_ref0 x):(((eq (fofType->Prop)) x) x)
% Found (eq_ref0 x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found x1:(P (x x0))
% Instantiate: x:=(K b):(fofType->Prop)
% Found x1 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 x):(((eq (fofType->Prop)) x) (fun (x0:fofType)=> (x x0)))
% Found (eta_expansion_dep00 x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found x1:(P (x x0))
% Instantiate: x:=(K b):(fofType->Prop)
% Found x1 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 x):(((eq (fofType->Prop)) x) (fun (x0:fofType)=> (x x0)))
% Found (eta_expansion_dep00 x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found eq_ref00:=(eq_ref0 (x x0)):(((eq Prop) (x x0)) (x x0))
% Found (eq_ref0 (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found eq_ref00:=(eq_ref0 (x x0)):(((eq Prop) (x x0)) (x x0))
% Found (eq_ref0 (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found eq_ref00:=(eq_ref0 (x x0)):(((eq Prop) (x x0)) (x x0))
% Found (eq_ref0 (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found eq_ref00:=(eq_ref0 (x x0)):(((eq Prop) (x x0)) (x x0))
% Found (eq_ref0 (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found eq_ref00:=(eq_ref0 (x x0)):(((eq Prop) (x x0)) (x x0))
% Found (eq_ref0 (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found x1:(P0 b)
% Instantiate: b:=((K x) x0):Prop
% Found (fun (x1:(P0 b))=> x1) as proof of (P0 ((K x) x0))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of ((P0 b)->(P0 ((K x) x0)))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (x x0)):(((eq Prop) (x x0)) (x x0))
% Found (eq_ref0 (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found x1:(P0 b)
% Instantiate: b:=((K x) x0):Prop
% Found (fun (x1:(P0 b))=> x1) as proof of (P0 ((K x) x0))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of ((P0 b)->(P0 ((K x) x0)))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (x x0)):(((eq Prop) (x x0)) (x x0))
% Found (eq_ref0 (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found eq_ref00:=(eq_ref0 (x x0)):(((eq Prop) (x x0)) (x x0))
% Found (eq_ref0 (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found eq_ref00:=(eq_ref0 (x x0)):(((eq Prop) (x x0)) (x x0))
% Found (eq_ref0 (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found eq_ref00:=(eq_ref0 (x x0)):(((eq Prop) (x x0)) (x x0))
% Found (eq_ref0 (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found eq_ref00:=(eq_ref0 (x x0)):(((eq Prop) (x x0)) (x x0))
% Found (eq_ref0 (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found eq_ref00:=(eq_ref0 (x x0)):(((eq Prop) (x x0)) (x x0))
% Found (eq_ref0 (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found eq_ref00:=(eq_ref0 (x x0)):(((eq Prop) (x x0)) (x x0))
% Found (eq_ref0 (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found eq_ref00:=(eq_ref0 (x x0)):(((eq Prop) (x x0)) (x x0))
% Found (eq_ref0 (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found ((eq_ref Prop) (x x0)) as proof of (((eq Prop) (x x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K x) x0))
% Found eq_ref00:=(eq_ref0 x):(((eq (fofType->Prop)) x) x)
% Found (eq_ref0 x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 x):(((eq (fofType->Prop)) x) (fun (x0:fofType)=> (x x0)))
% Found (eta_expansion_dep00 x) as proof of (((eq (fofType->Prop)) x) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x) as proof of (((eq (fofType->Prop)) x) b)
% Found x01:(P1 x)
% Found (fun (x01:(P1 x))=> x01) as proof of (P1 x)
% Found (fun (x01:(P1 x))=> x01) as proof of (P2 x)
% Found x01:(P1 x)
% Found (fun (x01:(P1 x))=> x01) as proof of (P1 x)
% Found (fun (x01:(P1 x))=> x01) as proof of (P2 x)
% Found x0:(P x)
% Instantiate: f:=x:(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P x)
% Instantiate: f:=x:(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 x):(((eq (fofType->Prop)) x) x)
% Found (eq_ref0 x) as proof of (((eq (fofType->Prop)) x) b0)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b0)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b0)
% Found ((eq_ref (fofType->Prop)) x) as proof of (((eq (fofType->Prop)) x) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (fofType->Prop)) b0) (fun (x:fofType)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found x11:(P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P0 ((K x) x0))
% Found x11:(P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P0 ((K x) x0))
% Found x11:(P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P0 ((K x) x0))
% Found x11:(P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P ((K x) x0))
% Found (fun (x11:(P ((K x) x0)))=> x11) as proof of (P0 ((K x) x0))
% Found eq_ref00:=(eq_ref0 (f x00)):(((eq Prop) (f x00)) (f x00))
% Found (eq_ref0 (f x00)) as proof of (((eq Prop) (f x00)) ((K x) x00))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((K x) x00))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((K x) x00))
% Found (fun (x00:fofType)=> ((eq_ref Prop) (f x00))) as proof of (((eq Prop) (f x00)) ((K x) x00))
% Found (fun (x00:fofType)=> ((eq_ref Prop) (f x00))) as proof of (forall (x0:fofType), (((eq Prop) (f x0)) ((K x) x0)))
% Found eq_ref00:=(eq_ref0 (f x00)):(((eq Prop) (f x00)) (f x00))
% Found (eq_ref0 (f x00)) as proof of (((eq Prop) (f x00)) ((K x) x00))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((K x) x00))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((K x) x00))
% Found (fun (x00:fofType)=> ((eq_ref Prop) (f x00))) as proof of (((eq Prop) (f x00)) ((K x) x00))
% Found (fun (x00:fofType)=> ((eq_ref Prop) (f x00))) as proof of (forall (x0:fofType), (((eq Prop) (f x0)) ((K x) x0)))
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found eq_ref00:=(eq_ref0 ((K x) x0)):(((eq Prop) ((K x) x0)) ((K x) x0))
% Found (eq_ref0 ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found ((eq_ref Prop) ((K x) x0)) as proof of (((eq Prop) ((K x) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x x0))
% Found eq_ref00:=(e
% EOF
%------------------------------------------------------------------------------