%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV399^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n103.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 13:34:09 EDT 2014

% Result   : Timeout 300.01s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV399^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n103.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 09:06:46 CDT 2014
% % CPUTime  : 300.01 
% 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))) (L:((fofType->Prop)->(fofType->Prop))), (((and (forall (Xu:(fofType->Prop)) (Xv:(fofType->Prop)), ((forall (Xx:fofType), ((Xu Xx)->(Xv Xx)))->(forall (Xx:fofType), (((K Xv) Xx)->((K Xu) Xx)))))) (forall (Xu:(fofType->Prop)) (Xv:(fofType->Prop)), ((forall (Xx:fofType), ((Xu Xx)->(Xv Xx)))->(forall (Xx:fofType), (((L Xv) Xx)->((L Xu) Xx))))))->((ex (fofType->Prop)) (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw))))) of role conjecture named cTHM597_pme
% Conjecture to prove = (forall (K:((fofType->Prop)->(fofType->Prop))) (L:((fofType->Prop)->(fofType->Prop))), (((and (forall (Xu:(fofType->Prop)) (Xv:(fofType->Prop)), ((forall (Xx:fofType), ((Xu Xx)->(Xv Xx)))->(forall (Xx:fofType), (((K Xv) Xx)->((K Xu) Xx)))))) (forall (Xu:(fofType->Prop)) (Xv:(fofType->Prop)), ((forall (Xx:fofType), ((Xu Xx)->(Xv Xx)))->(forall (Xx:fofType), (((L Xv) Xx)->((L Xu) Xx))))))->((ex (fofType->Prop)) (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw))))):Prop
% Parameter fofType_DUMMY:fofType.
% We need to prove ['(forall (K:((fofType->Prop)->(fofType->Prop))) (L:((fofType->Prop)->(fofType->Prop))), (((and (forall (Xu:(fofType->Prop)) (Xv:(fofType->Prop)), ((forall (Xx:fofType), ((Xu Xx)->(Xv Xx)))->(forall (Xx:fofType), (((K Xv) Xx)->((K Xu) Xx)))))) (forall (Xu:(fofType->Prop)) (Xv:(fofType->Prop)), ((forall (Xx:fofType), ((Xu Xx)->(Xv Xx)))->(forall (Xx:fofType), (((L Xv) Xx)->((L Xu) Xx))))))->((ex (fofType->Prop)) (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw)))))']
% Parameter fofType:Type.
% Trying to prove (forall (K:((fofType->Prop)->(fofType->Prop))) (L:((fofType->Prop)->(fofType->Prop))), (((and (forall (Xu:(fofType->Prop)) (Xv:(fofType->Prop)), ((forall (Xx:fofType), ((Xu Xx)->(Xv Xx)))->(forall (Xx:fofType), (((K Xv) Xx)->((K Xu) Xx)))))) (forall (Xu:(fofType->Prop)) (Xv:(fofType->Prop)), ((forall (Xx:fofType), ((Xu Xx)->(Xv Xx)))->(forall (Xx:fofType), (((L Xv) Xx)->((L Xu) Xx))))))->((ex (fofType->Prop)) (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw))):(((eq ((fofType->Prop)->Prop)) (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw))) (fun (x:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L x))) x)))
% Found (eta_expansion_dep00 (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw))) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->Prop))=> Prop)) (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x1:(fofType->Prop))=> Prop)) (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x1:(fofType->Prop))=> Prop)) (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x1:(fofType->Prop))=> Prop)) (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw))) b)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K (L x0))) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K (L x0))) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K (L x0))) x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K (L x0))) x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->Prop)), (((eq Prop) (f x)) (((eq (fofType->Prop)) (K (L x))) x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K (L x0))) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K (L x0))) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K (L x0))) x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (K (L x0))) x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->Prop)), (((eq Prop) (f x)) (((eq (fofType->Prop)) (K (L x))) x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw))):(((eq ((fofType->Prop)->Prop)) (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw))) (fun (x:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L x))) x)))
% Found (eta_expansion_dep00 (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw))) b)
% Found ((eta_expansion_dep0 (fun (x3:(fofType->Prop))=> Prop)) (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x3:(fofType->Prop))=> Prop)) (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x3:(fofType->Prop))=> Prop)) (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x3:(fofType->Prop))=> Prop)) (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw))) b)
% Found eta_expansion000:=(eta_expansion00 (K (L x0))):(((eq (fofType->Prop)) (K (L x0))) (fun (x:fofType)=> ((K (L x0)) x)))
% Found (eta_expansion00 (K (L x0))) as proof of (((eq (fofType->Prop)) (K (L x0))) b)
% Found ((eta_expansion0 Prop) (K (L x0))) as proof of (((eq (fofType->Prop)) (K (L x0))) b)
% Found (((eta_expansion fofType) Prop) (K (L x0))) as proof of (((eq (fofType->Prop)) (K (L x0))) b)
% Found (((eta_expansion fofType) Prop) (K (L x0))) as proof of (((eq (fofType->Prop)) (K (L x0))) b)
% Found (((eta_expansion fofType) Prop) (K (L x0))) as proof of (((eq (fofType->Prop)) (K (L x0))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (((eq (fofType->Prop)) (K (L x2))) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((eq (fofType->Prop)) (K (L x2))) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((eq (fofType->Prop)) (K (L x2))) x2))
% Found (fun (x2:(fofType->Prop))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (((eq (fofType->Prop)) (K (L x2))) x2))
% Found (fun (x2:(fofType->Prop))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(fofType->Prop)), (((eq Prop) (f x)) (((eq (fofType->Prop)) (K (L x))) x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (((eq (fofType->Prop)) (K (L x2))) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((eq (fofType->Prop)) (K (L x2))) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((eq (fofType->Prop)) (K (L x2))) x2))
% Found (fun (x2:(fofType->Prop))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (((eq (fofType->Prop)) (K (L x2))) x2))
% Found (fun (x2:(fofType->Prop))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(fofType->Prop)), (((eq Prop) (f x)) (((eq (fofType->Prop)) (K (L x))) x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (K (L x2))):(((eq (fofType->Prop)) (K (L x2))) (fun (x:fofType)=> ((K (L x2)) x)))
% Found (eta_expansion_dep00 (K (L x2))) as proof of (((eq (fofType->Prop)) (K (L x2))) b)
% Found ((eta_expansion_dep0 (fun (x4:fofType)=> Prop)) (K (L x2))) as proof of (((eq (fofType->Prop)) (K (L x2))) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) (K (L x2))) as proof of (((eq (fofType->Prop)) (K (L x2))) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) (K (L x2))) as proof of (((eq (fofType->Prop)) (K (L x2))) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) (K (L x2))) as proof of (((eq (fofType->Prop)) (K (L x2))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) x2)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x2)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x2)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x2)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (K (L x0))):(((eq (fofType->Prop)) (K (L x0))) (fun (x:fofType)=> ((K (L x0)) x)))
% Found (eta_expansion_dep00 (K (L x0))) as proof of (((eq (fofType->Prop)) (K (L x0))) b)
% Found ((eta_expansion_dep0 (fun (x4:fofType)=> Prop)) (K (L x0))) as proof of (((eq (fofType->Prop)) (K (L x0))) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) (K (L x0))) as proof of (((eq (fofType->Prop)) (K (L x0))) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) (K (L x0))) as proof of (((eq (fofType->Prop)) (K (L x0))) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) (K (L x0))) as proof of (((eq (fofType->Prop)) (K (L x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found eta_expansion000:=(eta_expansion00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion0 Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref00:=(eq_ref0 ((K (L x0)) x1)):(((eq Prop) ((K (L x0)) x1)) ((K (L x0)) x1))
% Found (eq_ref0 ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((K (L x0)) x1)):(((eq Prop) ((K (L x0)) x1)) ((K (L x0)) x1))
% Found (eq_ref0 ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found x10:(P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P0 x0)
% Found x10:(P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P0 x0)
% Found x20:(P ((K (L x0)) x1))
% Found (fun (x20:(P ((K (L x0)) x1)))=> x20) as proof of (P ((K (L x0)) x1))
% Found (fun (x20:(P ((K (L x0)) x1)))=> x20) as proof of (P0 ((K (L x0)) x1))
% Found x20:(P ((K (L x0)) x1))
% Found (fun (x20:(P ((K (L x0)) x1)))=> x20) as proof of (P ((K (L x0)) x1))
% Found (fun (x20:(P ((K (L x0)) x1)))=> x20) as proof of (P0 ((K (L x0)) x1))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K (L x2)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x2)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x2)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x2)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 x2):(((eq (fofType->Prop)) x2) (fun (x:fofType)=> (x2 x)))
% Found (eta_expansion_dep00 x2) as proof of (((eq (fofType->Prop)) x2) b)
% Found ((eta_expansion_dep0 (fun (x4:fofType)=> Prop)) x2) as proof of (((eq (fofType->Prop)) x2) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x2) as proof of (((eq (fofType->Prop)) x2) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x2) as proof of (((eq (fofType->Prop)) x2) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x2) as proof of (((eq (fofType->Prop)) x2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion_dep0 (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref00:=(eq_ref0 ((K (L x2)) x3)):(((eq Prop) ((K (L x2)) x3)) ((K (L x2)) x3))
% Found (eq_ref0 ((K (L x2)) x3)) as proof of (((eq Prop) ((K (L x2)) x3)) b)
% Found ((eq_ref Prop) ((K (L x2)) x3)) as proof of (((eq Prop) ((K (L x2)) x3)) b)
% Found ((eq_ref Prop) ((K (L x2)) x3)) as proof of (((eq Prop) ((K (L x2)) x3)) b)
% Found ((eq_ref Prop) ((K (L x2)) x3)) as proof of (((eq Prop) ((K (L x2)) x3)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x2 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x2 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x2 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x2 x3))
% Found eq_ref00:=(eq_ref0 ((K (L x2)) x3)):(((eq Prop) ((K (L x2)) x3)) ((K (L x2)) x3))
% Found (eq_ref0 ((K (L x2)) x3)) as proof of (((eq Prop) ((K (L x2)) x3)) b)
% Found ((eq_ref Prop) ((K (L x2)) x3)) as proof of (((eq Prop) ((K (L x2)) x3)) b)
% Found ((eq_ref Prop) ((K (L x2)) x3)) as proof of (((eq Prop) ((K (L x2)) x3)) b)
% Found ((eq_ref Prop) ((K (L x2)) x3)) as proof of (((eq Prop) ((K (L x2)) x3)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x2 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x2 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x2 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x2 x3))
% Found x30:(P x2)
% Found (fun (x30:(P x2))=> x30) as proof of (P x2)
% Found (fun (x30:(P x2))=> x30) as proof of (P0 x2)
% Found x30:(P x2)
% Found (fun (x30:(P x2))=> x30) as proof of (P x2)
% Found (fun (x30:(P x2))=> x30) as proof of (P0 x2)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion_dep0 (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 ((K (L x0)) x3)):(((eq Prop) ((K (L x0)) x3)) ((K (L x0)) x3))
% Found (eq_ref0 ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found ((eq_ref Prop) ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found ((eq_ref Prop) ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found ((eq_ref Prop) ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x3))
% Found eq_ref00:=(eq_ref0 ((K (L x0)) x3)):(((eq Prop) ((K (L x0)) x3)) ((K (L x0)) x3))
% Found (eq_ref0 ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found ((eq_ref Prop) ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found ((eq_ref Prop) ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found ((eq_ref Prop) ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x3))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found x40:(P ((K (L x2)) x3))
% Found (fun (x40:(P ((K (L x2)) x3)))=> x40) as proof of (P ((K (L x2)) x3))
% Found (fun (x40:(P ((K (L x2)) x3)))=> x40) as proof of (P0 ((K (L x2)) x3))
% Found x40:(P ((K (L x2)) x3))
% Found (fun (x40:(P ((K (L x2)) x3)))=> x40) as proof of (P ((K (L x2)) x3))
% Found (fun (x40:(P ((K (L x2)) x3)))=> x40) as proof of (P0 ((K (L x2)) x3))
% Found x30:(P x0)
% Found (fun (x30:(P x0))=> x30) as proof of (P x0)
% Found (fun (x30:(P x0))=> x30) as proof of (P0 x0)
% Found x30:(P x0)
% Found (fun (x30:(P x0))=> x30) as proof of (P x0)
% Found (fun (x30:(P x0))=> x30) as proof of (P0 x0)
% Found x20:(P (x0 x1))
% Found (fun (x20:(P (x0 x1)))=> x20) as proof of (P (x0 x1))
% Found (fun (x20:(P (x0 x1)))=> x20) as proof of (P0 (x0 x1))
% Found x20:(P (x0 x1))
% Found (fun (x20:(P (x0 x1)))=> x20) as proof of (P (x0 x1))
% Found (fun (x20:(P (x0 x1)))=> x20) as proof of (P0 (x0 x1))
% Found x40:(P ((K (L x0)) x3))
% Found (fun (x40:(P ((K (L x0)) x3)))=> x40) as proof of (P ((K (L x0)) x3))
% Found (fun (x40:(P ((K (L x0)) x3)))=> x40) as proof of (P0 ((K (L x0)) x3))
% Found x40:(P ((K (L x0)) x3))
% Found (fun (x40:(P ((K (L x0)) x3)))=> x40) as proof of (P ((K (L x0)) x3))
% Found (fun (x40:(P ((K (L x0)) x3)))=> x40) as proof of (P0 ((K (L x0)) x3))
% Found eq_ref00:=(eq_ref0 ((K (L x0)) x1)):(((eq Prop) ((K (L x0)) x1)) ((K (L x0)) x1))
% Found (eq_ref0 ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((K (L x0)) x1)):(((eq Prop) ((K (L x0)) x1)) ((K (L x0)) x1))
% Found (eq_ref0 ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found x30:(P x0)
% Found (fun (x30:(P x0))=> x30) as proof of (P x0)
% Found (fun (x30:(P x0))=> x30) as proof of (P0 x0)
% Found x30:(P x0)
% Found (fun (x30:(P x0))=> x30) as proof of (P x0)
% Found (fun (x30:(P x0))=> x30) as proof of (P0 x0)
% Found x40:(P ((K (L x0)) x1))
% Found (fun (x40:(P ((K (L x0)) x1)))=> x40) as proof of (P ((K (L x0)) x1))
% Found (fun (x40:(P ((K (L x0)) x1)))=> x40) as proof of (P0 ((K (L x0)) x1))
% Found x40:(P ((K (L x0)) x1))
% Found (fun (x40:(P ((K (L x0)) x1)))=> x40) as proof of (P ((K (L x0)) x1))
% Found (fun (x40:(P ((K (L x0)) x1)))=> x40) as proof of (P0 ((K (L x0)) x1))
% Found eq_ref00:=(eq_ref0 (x2 x3)):(((eq Prop) (x2 x3)) (x2 x3))
% Found (eq_ref0 (x2 x3)) as proof of (((eq Prop) (x2 x3)) b)
% Found ((eq_ref Prop) (x2 x3)) as proof of (((eq Prop) (x2 x3)) b)
% Found ((eq_ref Prop) (x2 x3)) as proof of (((eq Prop) (x2 x3)) b)
% Found ((eq_ref Prop) (x2 x3)) as proof of (((eq Prop) (x2 x3)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K (L x2)) x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x2)) x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x2)) x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x2)) x3))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K (L x2)) x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x2)) x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x2)) x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x2)) x3))
% Found eq_ref00:=(eq_ref0 (x2 x3)):(((eq Prop) (x2 x3)) (x2 x3))
% Found (eq_ref0 (x2 x3)) as proof of (((eq Prop) (x2 x3)) b)
% Found ((eq_ref Prop) (x2 x3)) as proof of (((eq Prop) (x2 x3)) b)
% Found ((eq_ref Prop) (x2 x3)) as proof of (((eq Prop) (x2 x3)) b)
% Found ((eq_ref Prop) (x2 x3)) as proof of (((eq Prop) (x2 x3)) b)
% Found eq_ref00:=(eq_ref0 (x2 x3)):(((eq Prop) (x2 x3)) (x2 x3))
% Found (eq_ref0 (x2 x3)) as proof of (((eq Prop) (x2 x3)) b)
% Found ((eq_ref Prop) (x2 x3)) as proof of (((eq Prop) (x2 x3)) b)
% Found ((eq_ref Prop) (x2 x3)) as proof of (((eq Prop) (x2 x3)) b)
% Found ((eq_ref Prop) (x2 x3)) as proof of (((eq Prop) (x2 x3)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K (L x2)) x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x2)) x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x2)) x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x2)) x3))
% Found eq_ref00:=(eq_ref0 (x2 x3)):(((eq Prop) (x2 x3)) (x2 x3))
% Found (eq_ref0 (x2 x3)) as proof of (((eq Prop) (x2 x3)) b)
% Found ((eq_ref Prop) (x2 x3)) as proof of (((eq Prop) (x2 x3)) b)
% Found ((eq_ref Prop) (x2 x3)) as proof of (((eq Prop) (x2 x3)) b)
% Found ((eq_ref Prop) (x2 x3)) as proof of (((eq Prop) (x2 x3)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K (L x2)) x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x2)) x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x2)) x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x2)) x3))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((K (L x0)) x1)):(((eq Prop) ((K (L x0)) x1)) ((K (L x0)) x1))
% Found (eq_ref0 ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((K (L x0)) x1)):(((eq Prop) ((K (L x0)) x1)) ((K (L x0)) x1))
% Found (eq_ref0 ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K (L x0)) x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x3))
% Found eq_ref00:=(eq_ref0 (x0 x3)):(((eq Prop) (x0 x3)) (x0 x3))
% Found (eq_ref0 (x0 x3)) as proof of (((eq Prop) (x0 x3)) b)
% Found ((eq_ref Prop) (x0 x3)) as proof of (((eq Prop) (x0 x3)) b)
% Found ((eq_ref Prop) (x0 x3)) as proof of (((eq Prop) (x0 x3)) b)
% Found ((eq_ref Prop) (x0 x3)) as proof of (((eq Prop) (x0 x3)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K (L x0)) x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x3))
% Found eq_ref00:=(eq_ref0 (x0 x3)):(((eq Prop) (x0 x3)) (x0 x3))
% Found (eq_ref0 (x0 x3)) as proof of (((eq Prop) (x0 x3)) b)
% Found ((eq_ref Prop) (x0 x3)) as proof of (((eq Prop) (x0 x3)) b)
% Found ((eq_ref Prop) (x0 x3)) as proof of (((eq Prop) (x0 x3)) b)
% Found ((eq_ref Prop) (x0 x3)) as proof of (((eq Prop) (x0 x3)) b)
% Found eq_ref00:=(eq_ref0 (x0 x3)):(((eq Prop) (x0 x3)) (x0 x3))
% Found (eq_ref0 (x0 x3)) as proof of (((eq Prop) (x0 x3)) b)
% Found ((eq_ref Prop) (x0 x3)) as proof of (((eq Prop) (x0 x3)) b)
% Found ((eq_ref Prop) (x0 x3)) as proof of (((eq Prop) (x0 x3)) b)
% Found ((eq_ref Prop) (x0 x3)) as proof of (((eq Prop) (x0 x3)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K (L x0)) x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x3))
% Found eq_ref00:=(eq_ref0 (x0 x3)):(((eq Prop) (x0 x3)) (x0 x3))
% Found (eq_ref0 (x0 x3)) as proof of (((eq Prop) (x0 x3)) b)
% Found ((eq_ref Prop) (x0 x3)) as proof of (((eq Prop) (x0 x3)) b)
% Found ((eq_ref Prop) (x0 x3)) as proof of (((eq Prop) (x0 x3)) b)
% Found ((eq_ref Prop) (x0 x3)) as proof of (((eq Prop) (x0 x3)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K (L x0)) x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x3))
% Found x40:(P (x2 x3))
% Found (fun (x40:(P (x2 x3)))=> x40) as proof of (P (x2 x3))
% Found (fun (x40:(P (x2 x3)))=> x40) as proof of (P0 (x2 x3))
% Found x40:(P (x2 x3))
% Found (fun (x40:(P (x2 x3)))=> x40) as proof of (P (x2 x3))
% Found (fun (x40:(P (x2 x3)))=> x40) as proof of (P0 (x2 x3))
% Found eq_ref00:=(eq_ref0 (x0 x3)):(((eq Prop) (x0 x3)) (x0 x3))
% Found (eq_ref0 (x0 x3)) as proof of (((eq Prop) (x0 x3)) b)
% Found ((eq_ref Prop) (x0 x3)) as proof of (((eq Prop) (x0 x3)) b)
% Found ((eq_ref Prop) (x0 x3)) as proof of (((eq Prop) (x0 x3)) b)
% Found ((eq_ref Prop) (x0 x3)) as proof of (((eq Prop) (x0 x3)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K (L x0)) x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x3))
% Found eq_ref00:=(eq_ref0 (x0 x3)):(((eq Prop) (x0 x3)) (x0 x3))
% Found (eq_ref0 (x0 x3)) as proof of (((eq Prop) (x0 x3)) b)
% Found ((eq_ref Prop) (x0 x3)) as proof of (((eq Prop) (x0 x3)) b)
% Found ((eq_ref Prop) (x0 x3)) as proof of (((eq Prop) (x0 x3)) b)
% Found ((eq_ref Prop) (x0 x3)) as proof of (((eq Prop) (x0 x3)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K (L x0)) x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x3))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found x40:(P (x0 x3))
% Found (fun (x40:(P (x0 x3)))=> x40) as proof of (P (x0 x3))
% Found (fun (x40:(P (x0 x3)))=> x40) as proof of (P0 (x0 x3))
% Found x40:(P (x0 x3))
% Found (fun (x40:(P (x0 x3)))=> x40) as proof of (P (x0 x3))
% Found (fun (x40:(P (x0 x3)))=> x40) as proof of (P0 (x0 x3))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found x40:(P (x0 x3))
% Found (fun (x40:(P (x0 x3)))=> x40) as proof of (P (x0 x3))
% Found (fun (x40:(P (x0 x3)))=> x40) as proof of (P0 (x0 x3))
% Found x40:(P (x0 x3))
% Found (fun (x40:(P (x0 x3)))=> x40) as proof of (P (x0 x3))
% Found (fun (x40:(P (x0 x3)))=> x40) as proof of (P0 (x0 x3))
% Found x40:(P (x0 x1))
% Found (fun (x40:(P (x0 x1)))=> x40) as proof of (P (x0 x1))
% Found (fun (x40:(P (x0 x1)))=> x40) as proof of (P0 (x0 x1))
% Found x40:(P (x0 x1))
% Found (fun (x40:(P (x0 x1)))=> x40) as proof of (P (x0 x1))
% Found (fun (x40:(P (x0 x1)))=> x40) as proof of (P0 (x0 x1))
% Found eq_ref00:=(eq_ref0 ((K (L x2)) x3)):(((eq Prop) ((K (L x2)) x3)) ((K (L x2)) x3))
% Found (eq_ref0 ((K (L x2)) x3)) as proof of (((eq Prop) ((K (L x2)) x3)) b)
% Found ((eq_ref Prop) ((K (L x2)) x3)) as proof of (((eq Prop) ((K (L x2)) x3)) b)
% Found ((eq_ref Prop) ((K (L x2)) x3)) as proof of (((eq Prop) ((K (L x2)) x3)) b)
% Found ((eq_ref Prop) ((K (L x2)) x3)) as proof of (((eq Prop) ((K (L x2)) x3)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x2 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x2 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x2 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x2 x3))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x2 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x2 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x2 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x2 x3))
% Found eq_ref00:=(eq_ref0 ((K (L x2)) x3)):(((eq Prop) ((K (L x2)) x3)) ((K (L x2)) x3))
% Found (eq_ref0 ((K (L x2)) x3)) as proof of (((eq Prop) ((K (L x2)) x3)) b)
% Found ((eq_ref Prop) ((K (L x2)) x3)) as proof of (((eq Prop) ((K (L x2)) x3)) b)
% Found ((eq_ref Prop) ((K (L x2)) x3)) as proof of (((eq Prop) ((K (L x2)) x3)) b)
% Found ((eq_ref Prop) ((K (L x2)) x3)) as proof of (((eq Prop) ((K (L x2)) x3)) b)
% Found eq_ref00:=(eq_ref0 (K (L x0))):(((eq (fofType->Prop)) (K (L x0))) (K (L x0)))
% Found (eq_ref0 (K (L x0))) as proof of (((eq (fofType->Prop)) (K (L x0))) b)
% Found ((eq_ref (fofType->Prop)) (K (L x0))) as proof of (((eq (fofType->Prop)) (K (L x0))) b)
% Found ((eq_ref (fofType->Prop)) (K (L x0))) as proof of (((eq (fofType->Prop)) (K (L x0))) b)
% Found ((eq_ref (fofType->Prop)) (K (L x0))) as proof of (((eq (fofType->Prop)) (K (L x0))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found 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 (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw)))
% Found ((eta_expansion0 Prop) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw)))
% Found (((eta_expansion (fofType->Prop)) Prop) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw)))
% Found (((eta_expansion (fofType->Prop)) Prop) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw)))
% Found (((eta_expansion (fofType->Prop)) Prop) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw)))
% 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 (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw)))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw)))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw)))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x3))
% Found eq_ref00:=(eq_ref0 ((K (L x0)) x3)):(((eq Prop) ((K (L x0)) x3)) ((K (L x0)) x3))
% Found (eq_ref0 ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found ((eq_ref Prop) ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found ((eq_ref Prop) ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found ((eq_ref Prop) ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x3))
% Found eq_ref00:=(eq_ref0 ((K (L x0)) x3)):(((eq Prop) ((K (L x0)) x3)) ((K (L x0)) x3))
% Found (eq_ref0 ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found ((eq_ref Prop) ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found ((eq_ref Prop) ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found ((eq_ref Prop) ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found x1:(P (K (L x0)))
% Instantiate: b:=(K (L x0)):(fofType->Prop)
% Found x1 as proof of (P0 b)
% Found eta_expansion000:=(eta_expansion00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion0 Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x3))
% Found eq_ref00:=(eq_ref0 ((K (L x0)) x3)):(((eq Prop) ((K (L x0)) x3)) ((K (L x0)) x3))
% Found (eq_ref0 ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found ((eq_ref Prop) ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found ((eq_ref Prop) ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found ((eq_ref Prop) ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x3))
% Found eq_ref00:=(eq_ref0 ((K (L x0)) x3)):(((eq Prop) ((K (L x0)) x3)) ((K (L x0)) x3))
% Found (eq_ref0 ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found ((eq_ref Prop) ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found ((eq_ref Prop) ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found ((eq_ref Prop) ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found x1:(P (K (L x0)))
% Instantiate: f:=(K (L x0)):(fofType->Prop)
% Found x1 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (x0 x)))
% Found x1:(P (K (L x0)))
% Instantiate: f:=(K (L x0)):(fofType->Prop)
% Found x1 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (x0 x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (K (L x0))):(((eq (fofType->Prop)) (K (L x0))) (fun (x:fofType)=> ((K (L x0)) x)))
% Found (eta_expansion_dep00 (K (L x0))) as proof of (((eq (fofType->Prop)) (K (L x0))) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) (K (L x0))) as proof of (((eq (fofType->Prop)) (K (L x0))) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (K (L x0))) as proof of (((eq (fofType->Prop)) (K (L x0))) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (K (L x0))) as proof of (((eq (fofType->Prop)) (K (L x0))) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (K (L x0))) as proof of (((eq (fofType->Prop)) (K (L x0))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found eq_ref00:=(eq_ref0 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 (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw)))
% Found ((eta_expansion_dep0 (fun (x3:(fofType->Prop))=> Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw)))
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x3:(fofType->Prop))=> Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw)))
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x3:(fofType->Prop))=> Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw)))
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x3:(fofType->Prop))=> Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (Xw:(fofType->Prop))=> (((eq (fofType->Prop)) (K (L Xw))) Xw)))
% Found eta_expansion000:=(eta_expansion00 (K (L x2))):(((eq (fofType->Prop)) (K (L x2))) (fun (x:fofType)=> ((K (L x2)) x)))
% Found (eta_expansion00 (K (L x2))) as proof of (((eq (fofType->Prop)) (K (L x2))) b)
% Found ((eta_expansion0 Prop) (K (L x2))) as proof of (((eq (fofType->Prop)) (K (L x2))) b)
% Found (((eta_expansion fofType) Prop) (K (L x2))) as proof of (((eq (fofType->Prop)) (K (L x2))) b)
% Found (((eta_expansion fofType) Prop) (K (L x2))) as proof of (((eq (fofType->Prop)) (K (L x2))) b)
% Found (((eta_expansion fofType) Prop) (K (L x2))) as proof of (((eq (fofType->Prop)) (K (L x2))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) x2)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x2)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x2)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x2)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((K (L x0)) x1)):(((eq Prop) ((K (L x0)) x1)) ((K (L x0)) x1))
% Found (eq_ref0 ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found eq_ref00:=(eq_ref0 ((K (L x0)) x1)):(((eq Prop) ((K (L x0)) x1)) ((K (L x0)) x1))
% Found (eq_ref0 ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eta_expansion000:=(eta_expansion00 (K (L x0))):(((eq (fofType->Prop)) (K (L x0))) (fun (x:fofType)=> ((K (L x0)) x)))
% Found (eta_expansion00 (K (L x0))) as proof of (((eq (fofType->Prop)) (K (L x0))) b)
% Found ((eta_expansion0 Prop) (K (L x0))) as proof of (((eq (fofType->Prop)) (K (L x0))) b)
% Found (((eta_expansion fofType) Prop) (K (L x0))) as proof of (((eq (fofType->Prop)) (K (L x0))) b)
% Found (((eta_expansion fofType) Prop) (K (L x0))) as proof of (((eq (fofType->Prop)) (K (L x0))) b)
% Found (((eta_expansion fofType) Prop) (K (L x0))) as proof of (((eq (fofType->Prop)) (K (L x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found eta_expansion000:=(eta_expansion00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion0 Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found eta_expansion000:=(eta_expansion00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion0 Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref00:=(eq_ref0 ((K (L x0)) x1)):(((eq Prop) ((K (L x0)) x1)) ((K (L x0)) x1))
% Found (eq_ref0 ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((K (L x0)) x1)):(((eq Prop) ((K (L x0)) x1)) ((K (L x0)) x1))
% Found (eq_ref0 ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found x1:(P x0)
% Instantiate: x0:=(K (L b)):(fofType->Prop)
% Found x1 as proof of (P0 b)
% Found eta_expansion000:=(eta_expansion00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion0 Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found x3:(P (K (L x2)))
% Instantiate: b:=(K (L x2)):(fofType->Prop)
% Found x3 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 x2):(((eq (fofType->Prop)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (fofType->Prop)) x2) b)
% Found ((eq_ref (fofType->Prop)) x2) as proof of (((eq (fofType->Prop)) x2) b)
% Found ((eq_ref (fofType->Prop)) x2) as proof of (((eq (fofType->Prop)) x2) b)
% Found ((eq_ref (fofType->Prop)) x2) as proof of (((eq (fofType->Prop)) x2) b)
% Found eq_ref00:=(eq_ref0 x0):(((eq (fofType->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found x1:(P x0)
% Instantiate: b:=x0:(fofType->Prop)
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (K (L x0))):(((eq (fofType->Prop)) (K (L x0))) (K (L x0)))
% Found (eq_ref0 (K (L x0))) as proof of (((eq (fofType->Prop)) (K (L x0))) b)
% Found ((eq_ref (fofType->Prop)) (K (L x0))) as proof of (((eq (fofType->Prop)) (K (L x0))) b)
% Found ((eq_ref (fofType->Prop)) (K (L x0))) as proof of (((eq (fofType->Prop)) (K (L x0))) b)
% Found ((eq_ref (fofType->Prop)) (K (L x0))) as proof of (((eq (fofType->Prop)) (K (L x0))) b)
% Found eq_ref00:=(eq_ref0 ((K (L x0)) x1)):(((eq Prop) ((K (L x0)) x1)) ((K (L x0)) x1))
% Found (eq_ref0 ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((K (L x0)) x1)):(((eq Prop) ((K (L x0)) x1)) ((K (L x0)) x1))
% Found (eq_ref0 ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((K (L x0)) x1)):(((eq Prop) ((K (L x0)) x1)) ((K (L x0)) x1))
% Found (eq_ref0 ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((K (L x0)) x1)):(((eq Prop) ((K (L x0)) x1)) ((K (L x0)) x1))
% Found (eq_ref0 ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found x3:(P (K (L x2)))
% Instantiate: f:=(K (L x2)):(fofType->Prop)
% Found x3 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) (x2 x4))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (x2 x4))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (x2 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) (x2 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (x2 x)))
% Found x3:(P (K (L x2)))
% Instantiate: f:=(K (L x2)):(fofType->Prop)
% Found x3 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) (x2 x4))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (x2 x4))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (x2 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) (x2 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (x2 x)))
% Found x3:(P (K (L x0)))
% Instantiate: b:=(K (L x0)):(fofType->Prop)
% Found x3 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 x0):(((eq (fofType->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found x10:(P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P0 x0)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) 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) x0)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) x0)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) x0)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) x0)
% Found x10:(P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P0 x0)
% Found x10:(P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P0 x0)
% Found x10:(P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P0 x0)
% Found eta_expansion000:=(eta_expansion00 (K (L x2))):(((eq (fofType->Prop)) (K (L x2))) (fun (x:fofType)=> ((K (L x2)) x)))
% Found (eta_expansion00 (K (L x2))) as proof of (((eq (fofType->Prop)) (K (L x2))) b)
% Found ((eta_expansion0 Prop) (K (L x2))) as proof of (((eq (fofType->Prop)) (K (L x2))) b)
% Found (((eta_expansion fofType) Prop) (K (L x2))) as proof of (((eq (fofType->Prop)) (K (L x2))) b)
% Found (((eta_expansion fofType) Prop) (K (L x2))) as proof of (((eq (fofType->Prop)) (K (L x2))) b)
% Found (((eta_expansion fofType) Prop) (K (L x2))) as proof of (((eq (fofType->Prop)) (K (L x2))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) x2)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x2)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x2)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x2)
% Found x20:(P ((K (L x0)) x1))
% Found (fun (x20:(P ((K (L x0)) x1)))=> x20) as proof of (P ((K (L x0)) x1))
% Found (fun (x20:(P ((K (L x0)) x1)))=> x20) as proof of (P0 ((K (L x0)) x1))
% Found x20:(P ((K (L x0)) x1))
% Found (fun (x20:(P ((K (L x0)) x1)))=> x20) as proof of (P ((K (L x0)) x1))
% Found (fun (x20:(P ((K (L x0)) x1)))=> x20) as proof of (P0 ((K (L x0)) x1))
% Found x20:(P ((K (L x0)) x1))
% Found (fun (x20:(P ((K (L x0)) x1)))=> x20) as proof of (P ((K (L x0)) x1))
% Found (fun (x20:(P ((K (L x0)) x1)))=> x20) as proof of (P0 ((K (L x0)) x1))
% Found x20:(P ((K (L x0)) x1))
% Found (fun (x20:(P ((K (L x0)) x1)))=> x20) as proof of (P ((K (L x0)) x1))
% Found (fun (x20:(P ((K (L x0)) x1)))=> x20) as proof of (P0 ((K (L x0)) x1))
% Found x3:(P (K (L x0)))
% Instantiate: f:=(K (L x0)):(fofType->Prop)
% Found x3 as proof of (P0 f)
% Found x3:(P (K (L x0)))
% Instantiate: f:=(K (L x0)):(fofType->Prop)
% Found x3 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) (x0 x4))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (x0 x4))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (x0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) (x0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (x0 x)))
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) (x0 x4))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (x0 x4))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (x0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) (x0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (x0 x)))
% Found x1:(P x0)
% Instantiate: f:=x0:(fofType->Prop)
% Found x1 as proof of (P0 f)
% Found x1:(P x0)
% Instantiate: f:=x0:(fofType->Prop)
% Found x1 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((K (L x0)) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((K (L x0)) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((K (L x0)) x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((K (L x0)) x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((K (L x0)) x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((K (L x0)) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((K (L x0)) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((K (L x0)) x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((K (L x0)) x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((K (L x0)) x)))
% Found x2:(P ((K (L x0)) x1))
% Instantiate: b:=((K (L x0)) x1):Prop
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found x2:(P ((K (L x0)) x1))
% Instantiate: b:=((K (L x0)) x1):Prop
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eta_expansion000:=(eta_expansion00 (K (L x0))):(((eq (fofType->Prop)) (K (L x0))) (fun (x:fofType)=> ((K (L x0)) x)))
% Found (eta_expansion00 (K (L x0))) as proof of (((eq (fofType->Prop)) (K (L x0))) b)
% Found ((eta_expansion0 Prop) (K (L x0))) as proof of (((eq (fofType->Prop)) (K (L x0))) b)
% Found (((eta_expansion fofType) Prop) (K (L x0))) as proof of (((eq (fofType->Prop)) (K (L x0))) b)
% Found (((eta_expansion fofType) Prop) (K (L x0))) as proof of (((eq (fofType->Prop)) (K (L x0))) b)
% Found (((eta_expansion fofType) Prop) (K (L x0))) as proof of (((eq (fofType->Prop)) (K (L x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x3:(P x2)
% Instantiate: x2:=(K (L b)):(fofType->Prop)
% Found x3 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 x2):(((eq (fofType->Prop)) x2) (fun (x:fofType)=> (x2 x)))
% Found (eta_expansion_dep00 x2) as proof of (((eq (fofType->Prop)) x2) b)
% Found ((eta_expansion_dep0 (fun (x5:fofType)=> Prop)) x2) as proof of (((eq (fofType->Prop)) x2) b)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) x2) as proof of (((eq (fofType->Prop)) x2) b)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) x2) as proof of (((eq (fofType->Prop)) x2) b)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) x2) as proof of (((eq (fofType->Prop)) x2) b)
% Found x1:(P x0)
% Instantiate: x0:=(K a):(fofType->Prop)
% Found x1 as proof of (P0 a)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) (L x0))
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) (L x0))
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) (L x0))
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) (L x0))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K (L x2)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x2)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x2)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x2)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 x2):(((eq (fofType->Prop)) x2) (fun (x:fofType)=> (x2 x)))
% Found (eta_expansion_dep00 x2) as proof of (((eq (fofType->Prop)) x2) b)
% Found ((eta_expansion_dep0 (fun (x4:fofType)=> Prop)) x2) as proof of (((eq (fofType->Prop)) x2) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x2) as proof of (((eq (fofType->Prop)) x2) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x2) as proof of (((eq (fofType->Prop)) x2) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x2) as proof of (((eq (fofType->Prop)) x2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K (L x2)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x2)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x2)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x2)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 x2):(((eq (fofType->Prop)) x2) (fun (x:fofType)=> (x2 x)))
% Found (eta_expansion_dep00 x2) as proof of (((eq (fofType->Prop)) x2) b)
% Found ((eta_expansion_dep0 (fun (x4:fofType)=> Prop)) x2) as proof of (((eq (fofType->Prop)) x2) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x2) as proof of (((eq (fofType->Prop)) x2) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x2) as proof of (((eq (fofType->Prop)) x2) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x2) as proof of (((eq (fofType->Prop)) x2) b)
% Found x1:(P x0)
% Instantiate: x0:=(K a):(fofType->Prop)
% Found x1 as proof of (P0 (K a))
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) (L x0))
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) (L x0))
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) (L x0))
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) (L x0))
% Found eq_ref00:=(eq_ref0 x2):(((eq (fofType->Prop)) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (fofType->Prop)) x2) b)
% Found ((eq_ref (fofType->Prop)) x2) as proof of (((eq (fofType->Prop)) x2) b)
% Found ((eq_ref (fofType->Prop)) x2) as proof of (((eq (fofType->Prop)) x2) b)
% Found ((eq_ref (fofType->Prop)) x2) as proof of (((eq (fofType->Prop)) x2) b)
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((K (L x0)) x1)):(((eq Prop) ((K (L x0)) x1)) ((K (L x0)) x1))
% Found (eq_ref0 ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((K (L x0)) x1)):(((eq Prop) ((K (L x0)) x1)) ((K (L x0)) x1))
% Found (eq_ref0 ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((K (L x0)) x1)):(((eq Prop) ((K (L x0)) x1)) ((K (L x0)) x1))
% Found (eq_ref0 ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((K (L x0)) x1)):(((eq Prop) ((K (L x0)) x1)) ((K (L x0)) x1))
% Found (eq_ref0 ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found x3:(P x2)
% Instantiate: b:=x2:(fofType->Prop)
% Found x3 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (K (L x2))):(((eq (fofType->Prop)) (K (L x2))) (fun (x:fofType)=> ((K (L x2)) x)))
% Found (eta_expansion_dep00 (K (L x2))) as proof of (((eq (fofType->Prop)) (K (L x2))) b)
% Found ((eta_expansion_dep0 (fun (x5:fofType)=> Prop)) (K (L x2))) as proof of (((eq (fofType->Prop)) (K (L x2))) b)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) (K (L x2))) as proof of (((eq (fofType->Prop)) (K (L x2))) b)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) (K (L x2))) as proof of (((eq (fofType->Prop)) (K (L x2))) b)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) (K (L x2))) as proof of (((eq (fofType->Prop)) (K (L x2))) b)
% Found x3:(P x0)
% Instantiate: x0:=(K (L b)):(fofType->Prop)
% Found x3 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion_dep0 (fun (x5:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion_dep0 (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion_dep0 (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found x10:(P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P0 x0)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found x10:(P1 x0)
% Found (fun (x10:(P1 x0))=> x10) as proof of (P1 x0)
% Found (fun (x10:(P1 x0))=> x10) as proof of (P2 x0)
% Found x10:(P1 x0)
% Found (fun (x10:(P1 x0))=> x10) as proof of (P1 x0)
% Found (fun (x10:(P1 x0))=> x10) as proof of (P2 x0)
% Found eq_ref00:=(eq_ref0 ((K (L x2)) x3)):(((eq Prop) ((K (L x2)) x3)) ((K (L x2)) x3))
% Found (eq_ref0 ((K (L x2)) x3)) as proof of (((eq Prop) ((K (L x2)) x3)) b)
% Found ((eq_ref Prop) ((K (L x2)) x3)) as proof of (((eq Prop) ((K (L x2)) x3)) b)
% Found ((eq_ref Prop) ((K (L x2)) x3)) as proof of (((eq Prop) ((K (L x2)) x3)) b)
% Found ((eq_ref Prop) ((K (L x2)) x3)) as proof of (((eq Prop) ((K (L x2)) x3)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x2 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x2 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x2 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x2 x3))
% Found eq_ref00:=(eq_ref0 x0):(((eq (fofType->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref00:=(eq_ref0 ((K (L x2)) x3)):(((eq Prop) ((K (L x2)) x3)) ((K (L x2)) x3))
% Found (eq_ref0 ((K (L x2)) x3)) as proof of (((eq Prop) ((K (L x2)) x3)) b)
% Found ((eq_ref Prop) ((K (L x2)) x3)) as proof of (((eq Prop) ((K (L x2)) x3)) b)
% Found ((eq_ref Prop) ((K (L x2)) x3)) as proof of (((eq Prop) ((K (L x2)) x3)) b)
% Found ((eq_ref Prop) ((K (L x2)) x3)) as proof of (((eq Prop) ((K (L x2)) x3)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x2 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x2 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x2 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x2 x3))
% Found eq_ref00:=(eq_ref0 ((K (L x2)) x3)):(((eq Prop) ((K (L x2)) x3)) ((K (L x2)) x3))
% Found (eq_ref0 ((K (L x2)) x3)) as proof of (((eq Prop) ((K (L x2)) x3)) b)
% Found ((eq_ref Prop) ((K (L x2)) x3)) as proof of (((eq Prop) ((K (L x2)) x3)) b)
% Found ((eq_ref Prop) ((K (L x2)) x3)) as proof of (((eq Prop) ((K (L x2)) x3)) b)
% Found ((eq_ref Prop) ((K (L x2)) x3)) as proof of (((eq Prop) ((K (L x2)) x3)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x2 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x2 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x2 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x2 x3))
% Found eq_ref00:=(eq_ref0 ((K (L x2)) x3)):(((eq Prop) ((K (L x2)) x3)) ((K (L x2)) x3))
% Found (eq_ref0 ((K (L x2)) x3)) as proof of (((eq Prop) ((K (L x2)) x3)) b)
% Found ((eq_ref Prop) ((K (L x2)) x3)) as proof of (((eq Prop) ((K (L x2)) x3)) b)
% Found ((eq_ref Prop) ((K (L x2)) x3)) as proof of (((eq Prop) ((K (L x2)) x3)) b)
% Found ((eq_ref Prop) ((K (L x2)) x3)) as proof of (((eq Prop) ((K (L x2)) x3)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x2 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x2 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x2 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x2 x3))
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eta_expansion_dep0 (fun (x4:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) 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) x2)
% Found ((eta_expansion_dep0 (fun (x4:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) x2)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) x2)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) x2)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) x2)
% Found x20:(P1 ((K (L x0)) x1))
% Found (fun (x20:(P1 ((K (L x0)) x1)))=> x20) as proof of (P1 ((K (L x0)) x1))
% Found (fun (x20:(P1 ((K (L x0)) x1)))=> x20) as proof of (P2 ((K (L x0)) x1))
% Found x20:(P1 ((K (L x0)) x1))
% Found (fun (x20:(P1 ((K (L x0)) x1)))=> x20) as proof of (P1 ((K (L x0)) x1))
% Found (fun (x20:(P1 ((K (L x0)) x1)))=> x20) as proof of (P2 ((K (L x0)) x1))
% Found x3:(P x0)
% Instantiate: b:=x0:(fofType->Prop)
% Found x3 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (K (L x0))):(((eq (fofType->Prop)) (K (L x0))) (fun (x:fofType)=> ((K (L x0)) x)))
% Found (eta_expansion_dep00 (K (L x0))) as proof of (((eq (fofType->Prop)) (K (L x0))) b)
% Found ((eta_expansion_dep0 (fun (x5:fofType)=> Prop)) (K (L x0))) as proof of (((eq (fofType->Prop)) (K (L x0))) b)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) (K (L x0))) as proof of (((eq (fofType->Prop)) (K (L x0))) b)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) (K (L x0))) as proof of (((eq (fofType->Prop)) (K (L x0))) b)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) (K (L x0))) as proof of (((eq (fofType->Prop)) (K (L x0))) b)
% 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 x0):(((eq (fofType->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (fofType->Prop)) x0) b0)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b0)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b0)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b0)
% Found x30:(P x2)
% Found (fun (x30:(P x2))=> x30) as proof of (P x2)
% Found (fun (x30:(P x2))=> x30) as proof of (P0 x2)
% Found x30:(P x2)
% Found (fun (x30:(P x2))=> x30) as proof of (P x2)
% Found (fun (x30:(P x2))=> x30) as proof of (P0 x2)
% Found x30:(P x2)
% Found (fun (x30:(P x2))=> x30) as proof of (P x2)
% Found (fun (x30:(P x2))=> x30) as proof of (P0 x2)
% Found x30:(P x2)
% Found (fun (x30:(P x2))=> x30) as proof of (P x2)
% Found (fun (x30:(P x2))=> x30) as proof of (P0 x2)
% Found x3:(P x0)
% Instantiate: x0:=(K (L b)):(fofType->Prop)
% Found x3 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion_dep0 (fun (x5:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion_dep0 (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found x20:(P ((K (L x0)) x1))
% Found (fun (x20:(P ((K (L x0)) x1)))=> x20) as proof of (P ((K (L x0)) x1))
% Found (fun (x20:(P ((K (L x0)) x1)))=> x20) as proof of (P0 ((K (L x0)) x1))
% Found eq_ref00:=(eq_ref0 ((K (L x0)) x1)):(((eq Prop) ((K (L x0)) x1)) ((K (L x0)) x1))
% Found (eq_ref0 ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found x20:(P ((K (L x0)) x1))
% Found (fun (x20:(P ((K (L x0)) x1)))=> x20) as proof of (P ((K (L x0)) x1))
% Found (fun (x20:(P ((K (L x0)) x1)))=> x20) as proof of (P0 ((K (L x0)) x1))
% Found eq_ref00:=(eq_ref0 ((K (L x0)) x1)):(((eq Prop) ((K (L x0)) x1)) ((K (L x0)) x1))
% Found (eq_ref0 ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found ((eq_ref Prop) ((K (L x0)) x1)) as proof of (((eq Prop) ((K (L x0)) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found x2:(P (x0 x1))
% Instantiate: x0:=(K (L b)):(fofType->Prop)
% Found x2 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion_dep0 (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found eq_ref00:=(eq_ref0 x0):(((eq (fofType->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found x2:(P (x0 x1))
% Instantiate: x0:=(K (L b)):(fofType->Prop)
% Found x2 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion_dep0 (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (K (L x0)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion_dep0 (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref00:=(eq_ref0 ((K (L x0)) x3)):(((eq Prop) ((K (L x0)) x3)) ((K (L x0)) x3))
% Found (eq_ref0 ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found ((eq_ref Prop) ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found ((eq_ref Prop) ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found ((eq_ref Prop) ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x3))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 ((K (L x0)) x3)):(((eq Prop) ((K (L x0)) x3)) ((K (L x0)) x3))
% Found (eq_ref0 ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found ((eq_ref Prop) ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found ((eq_ref Prop) ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found ((eq_ref Prop) ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x3))
% Found eq_ref00:=(eq_ref0 ((K (L x0)) x3)):(((eq Prop) ((K (L x0)) x3)) ((K (L x0)) x3))
% Found (eq_ref0 ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found ((eq_ref Prop) ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found ((eq_ref Prop) ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found ((eq_ref Prop) ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x3))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found eq_ref00:=(eq_ref0 ((K (L x0)) x3)):(((eq Prop) ((K (L x0)) x3)) ((K (L x0)) x3))
% Found (eq_ref0 ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found ((eq_ref Prop) ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found ((eq_ref Prop) ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found ((eq_ref Prop) ((K (L x0)) x3)) as proof of (((eq Prop) ((K (L x0)) x3)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x3))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((K (L x0)) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) 
% EOF
%------------------------------------------------------------------------------